spectral-based color separation algorithm development - citeseer
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SPECTRAL-BASED COLOR SEPARATION ALGORITHMDEVELOPMENT FOR MULTIPLE-INK COLOR
REPRODUCTION
by
Di-Yuan Tzeng
B.S. Chinese Culture University, Taipei, Taiwan (1988)
M.A. Central Connecticut State University (1994)
A dissertation submitted in partial fulfillmentof the requirements for the degree of Ph.D. in theChester F. Carlson Center for Imaging Science
of the College of ScienceRochester Institute of Technology
September 1999
Signature of the Author ____________________________________________________
Accepted by ____________________________________________________Coordinator, Ph.D. Degree Program Date
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CHESTER F. CARLSONCENTER FOR IMAGING SCIENCE
COLLEGE OF SCIENCEROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER, NY
CERTIFICATE OF APPROVAL
Ph.D. DEGREE DISSERTATION
The Ph.D. Degree Dissertation of Di-Yuan Tzenghas been examined and approved by thedissertation committee as satisfactory for
the dissertation requirement for thePh.D. degree in Imaging Science
Dr. Roy S. Berns, Dissertation Advisor
Dr. Mark D. Fairchild
Dr. Jonathan S. Arney
Mr. Hubert D. Wood
Date
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DISSERTATION RELEASE PERMISSIONROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER, NY YORK
SPECTRAL-BASED COLOR SEPARATION ALGORITHM DEVELOPMENT FORMULTIPLE-INK COLOR REPRODUCTION
I, Di-Yuan Tzeng, hereby grant permission to Wallace Memorial Library of R.I.T. toreproduce my dissertation in whole or in part. Any reproduction will not be forcommercial use or profit.
Di-Yuan Tzeng
Date
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SPECTRAL-BASED COLOR SEPARATION ALGORITHMDEVELOPMENT FOR MULTIPLE-INK COLOR
REPRODUCTION
by
Di-Yuan Tzeng
Submitted to the Center for Imaging Sciencein partial fulfill of the requirements for the Ph. D.
degree as Rochester Institute of Technology
September 1999
ABSTRACT
Conventional four-color printing systems are limited by an insufficient number of degreesof freedom for tuning the visible region of the spectrum; as a consequence, they are oftenlimited to metameric color reproductions. That is, color matches defined for a singleobserver and illuminant (usually CIE illuminant D50 and the 1931 standard observer) areoften unstable when viewed under other illuminants or by other observers. For criticalcolor-matching applications, such as catalog sales and artwork reproductions, the resultsare usually disappointing due to typical uncontrolled lighting and viewing. Furthermore,the existing multiple-ink printing systems, which all focus on expanding color gamut, donot alleviate metamerism since their separation algorithms are trichromatic in nature. Theadvantage of an increased number of degrees of freedom is not exploited.A research and development program has been initiated at the Rochester Institute ofTechnology’s Munsell Color Science Laboratory to develop a spectral-based colorreproduction system. Research has included multi-spectral acquisition systems andspectral-based printing. The current research is concerned with bridging these analysis andsynthesis stages of color reproduction. The goal of the doctoral research was to minimize
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metamerism between originals and their corresponding reproductions, thus bestapproaching spectral matches. Accomplishing this goal required first estimating thespectral properties of the colorants used to create the original object or set of objects.After the possible colorants were statistically uncovered, they were correlated to anexisting ink database for determining an optimal ink set. An algorithm was developed forpredicting ink overprints, known as the Neugebauer secondary, tertiary, quaternaryprimaries, etc., which is the required information for color synthesis using a halftoneprinting process. Once all the required Neugebauer primaries were determined, a spectral-based printing model minimizing metamerism was derived to calculate the correspondingcolor separations for each selected ink. The various research components were testedcomputationally and experimentally. Finally, a DuPont Waterproof® system was used as arepresentation of halftone printing process to output each color separation in order to testseveral computational subsystems. This completes the chain of a spectral-based outputsystem. Five modules of this chain were developed and discussed in this dissertation.
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DEDICATION
This dissertation is dedicated to my parents, Chien-feng and Ai-Tsu, my beloved wife,Wei-Chien, my dearest son, Andrew Tsu-Jin, my second coming son, Yo-Jin, my youngerbrother, Chih-Yuan, and my youngest brother, Li-Yuan.
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ACKNOWLEDGEMENTS
I would like to thank Dr. Roy S. Berns, my advisor and a good friend, who saw the best ofme and helped to make this document and its associated research the single mostsignificant accomplishment of my life. His tireless guidance and inspiring advise has taughtme so much more than he realizes. I could not come to this point without his wisdom.
I thank my parents for giving me the strength and foundation materially and spiritually thathave accompanied me to come to this point. I could not have done this without theirunselfish support and so many sleepless nights. I greatly thank them for never giving up onme when I was a wild teenager.
I would like to thank my wife, Wei-Chien, for her entire devotion to my family taking fullcare of our family such that I can fully concentrate on pursuing this academic achievement.
I would like to thank my two sons, Andrew Tsu-Jin and Yo-Jin, who all came on thecrucial time of my academic life and brought me luck. Wei-chien was pregnant with Tsu-Jin when I had the Ph.D. comprehensive exam and passed it later. Now she is in her fifth-month of pregnancy with Yo-Jin. The advent of Yo-Jin brought me through my defensetoward my Ph.D. degree.
I gratefully acknowledge the financial support of the Munsell Color Science Laboratory,the Center for Imaging Science, and E. I. Du Pont de Nemours and Company.
I thank Dr. Mark D. Fairchild, Dr. Jonathan S. Arney, and Mr. Hubert D. Wood for theirgenerous efforts monitoring the quality of this doctoral research and their informativeadvise.
I would like to express my appreciation to Dr. Tony Liang and his colleagues at DuPontfor their tireless sample preparations to help substantiate this research.
Finally, I would like to thank all my friends and colleagues in the Munsell Color ScienceLaboratory for their innumerable comments, inspirations, and encouragement to thisresearch, especially, Dr. Noboru Ohta; Dr. Ethan Montag; Dr. Francisco Imai; Dr. PeterBurns; Mr. Gus Braun, who is going to be a Ph.D. in a short time; Mr. Dave Wyble, whohelped with the proofreading of this dissertation; and Mrs. Colleen Desimone.
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Table of Contents
I. INTRODUCTION 1A. OVERALL PROBLEM 3B. OVERALL SOLUTION 7C. SCOPE OF DISSERTATION 9
1. Colorant Estimation 102. Optimal Ink Selection 123. Ink Overprint Prediction 134. Spectral-Based Six-Color Separation Minimizing Metamerism 175. Multiple-Ink Direct Printing 18
II. BACKGROUND 20A. MULTIPLE-INK COLOR SYSTEMS 20B. KUBELKA-MUNK TURBID MEDIA THEORY 21C. COMPUTER COLORANT MATCHING 26D. VECTOR REPRESENTATION FOR COLORANTS 28
Vector Subspace 32E. LINEAR MODELING TECHNIQUES 34
Principal Component Analysis (PCA) 34 Principal Component Analysis for Color Science Applications 43 Interpretations for Coloration Processing Applications by PCA 46 Multivariate Normality for Effective Data Reduction 51
F. SPECTRAL PRINTING MODELS 57Basic Assumptions 58Murray-Davies Theory 59Dot Gain 61Neugebauer Theory 62
III. LINEAR COLORANT MIXING SPACES 66A. REFLECTION AND ABSORPTION SPACES 67B. TRANSFORMATION BETWEEN REFLECTANCE AND ΦΦΦΦ SPECTRA 70C. DIMENSIONALITY REDUCTION: NORMAL VERSUS NON-NORMAL POPULATIONS 73D. VERIFICATIONS 77E. CONCLUSIONS 85
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IV. COLORANT ESTIMATION OF ORIGINAL OBJECTS 86A. APPROXIMATELY LINEAR COLORANT MIXING SPACE 87B. PRINCIPAL COMPONENT ANALYSIS 88C. COLORANT ESTIMATION 90D. JUSTIFICATION OF EIGENVECTOR RECONSTRUCTION WITHOUT SAMPLE MEAN 92E. VERIFICATIONS 93
Testing the Constrained-Rotation Engine by a Virtual Sample Population 93 Colorant Estimation for the Kodak Q60C Target 95 Colorant Estimation for the Still Life Painting 97 Colorant Estimation for 105 Mixtures Using Ψ Space 102
F. CONCLUSIONS 107
V. OPTIMAL INK-SELECTION 109A. FIRST ORDER INK-SELECTION BY VECTOR CORRELATION 110B. CONTINUOUS TONE APPROXIMATION 113C. VERIFICATIONS AND RESULTS 116
Deriving a Linear Color Mixing Space for Continuous Tone Approximation 116 Vector Correlation Analysis in Ψ Space 120 Colorimetric and Spectral Performance by Continuous Tone Approximation 123
D. CONCLUSIONS 127
VI. SPECTRAL REFLECTANCE PREDICTION OF INK OVERPRINT USING KUBELKA-MUNK TURBIDMEDIA THEORY 129
A. PRIVIOUS RESEARCH 131B. TECHNICAL APPROACH 132C. EXPERIMENTAL 137D. RESULTS 139E. DISCUSSIONS 144F. CONCLUSIONS 145
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VII. SPECTRAL-BASED SIX-COLOR SEPARATION MINIMIZING METAMERISM 146
A. SIX-COLOR YULE-NIELSEN MODIFIED SPECTRAL NUEGEBAUER EQUATION 147B. AN ALTERNATIVE APPROACH USING SIX-COLOR HALFTONE PRINTING PROCESS MINIMIZING METAMERISM 151
Subdivision of Six-Color Modeling 152 Forward Four-Color Halftone Spectral Printing Models 153
1. The first order forward model 1532. Second Order improvement (modeling for ink- and optical-trapping) 1573. Alternative second order improvement 1624. Modeling by matrix transformation 168
Proper Four-Color Sub-Model Selection 169 Backward Printing Models for Six-Color Separation Minimizing Metamerism 171
C. EXPERIMENTAL AND VERIFICATION 174Sample Preparation 175
1. Preparation for ramps 1762. Preparation of the verification target (5x5x5x5 combinatorial design for mixtures) 1763. Sample measurements 1774. Accuracy metric 178
Determining the Yule-Nielsen n-Factor 178Accuracy for the First Order Six-Color Forward Printing Model 180Second Order Modification (By Iino and Berns’ Suggestion) 184Alternative Second Order Modification (Proposed Algorithm) 189Modeling by Matrix Transformation 199Six-Color Separation Minimizing Metamerism 202
D. SPECTRAL PERFORMANCE COMPARISONS FOR THREE-, FOUR-, AND SIX-COLOR PRINTING PROCESSES 206E. CONCLUSIONS 209
VIII. MULTIPLE-INK DIRECT PRINTING 212A. VERIFICATIONS 212B. CONCLUSIONS 215
IX. CONCLUSIONS, DISCUSSIONS, AND SUGGESTIONS FOR FUTURE RESEARCH 217
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X. REFERENCES 228
XI. APPENDICES 236
APPENDIX A : MATLAB PROGRAMS FOR THE“MUNSELL” LIBRARY 237
APPENDIX B : MATLAB PROGRAMS FOR THE COLORANTESTIMATION SUBSYSTEM 248
APPENDIX C : MATLAB PROGRAMS FOR THE INK SELECTIONSUBSYSTEM 251
APPENDIX D : MATLAB PROGRAMS FOR THE INK OVERPRINTPREDICTION SUBSYSTEM 254
APPENDIX E : MATLAB PROGRAMS FOR THE PROPOSEDSIX-COLOR FORWARD PRINTING MODEL 271
APPENDIX F : MATLAB PROGRAMS FOR THE SPECTRAL-BASED SIX-COLOR SEPARATIONMINIMIZNG METAMERISM 285
APPENDIX G : THE SIX ESTIMATED COLORANTS FOR THE105 MIXTURES OF THE POSTER COLORS 294
APPENDIX H : THE ESTIMATED SPECTRAL ABSORPTION ANDSCATTERING COEFFICIENTS FOR THEWATERPROOF® CMYRGB PRIMARIES 295
APPENDIX I : THE REFLECTANCE SPECTRA OF THE ORIGINALGRETAG MACBETH COLOR CHECKER 297
APPENDIX J : THE REFLECTANCE SPECTRA OF THEPREDICTED GRETAG MACBETH COLORCHECKER BY THE PROPOSED SIX-COLORSEPARATION ALGORITHM 301
APPENDIX K : THE REFLECTANCE SPECTRA OF THEREPRODUCED GRETAG MACBETH COLORCHECKER USING DUPONT WATERPROOF ® SYSTEM 305
APPENDIX L : THE REFLECTANCE SPECTRA OF THE PREDICTEDGRETAG MACBETH COLOR CHECKER USINGFUJIX PICTROGRAPH 3000 309
APPENDIX M : THE REFLECTANCE SPECTRA OF THEPREDICTED GRETAG MACBETH COLORCHECKER USING KODAK PROFESSIONAL8670 PS THERMAL PRINTER 313
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APPENDIX N : THE REFLECTANCE SPECTRA OF THEPREDICTED GRETAG MACBETH COLORCHECKER USING DUPONT WATERPROOF ®
WITH CMYK PRIMARIES 317APPENDIX O : THE ACCURACY OF THE GRETAG SPECTROLINO
SPECTROPHOTOMETER 321
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LIST OF TABELS
Table 1-1: The number of overprints of a k-primary halftone printing process. 16
Table 2-1: A sample set (shown up to five samples) of absorptionmeasurements from 400 nm to 700 nm at 20 nm intervals. 41
Table 2-2: The principal components as the new coordinates in the eigenvector coordinate system of the samples in Table 2-1. 42
Table 3-1: The colorimetric and spectral accuracy of the multivariate normaland non-normal sample sets reconstructed with different numberof dimensions, where Stdev stands for the standard deviation andRMS representing the totalroot mean square error of thereconstructed reflectance spectra. 77
Table 3-2: The percent variance by eigenvector analysis in both reflectanceand absorption space of IT8.7/2 reflection target. 78
Table 3-3: The colorimetric and spectral performance of three eigenvectorreconstruction in both spaces for an IT8.7/2 reflection target. 80
Table 3-4: The statistical performance by six-eigenvector reconstruction for141 acrylic colors in R, Ψ and Φ spaces. 81
Table 3-5: The statistical performance by six-eigenvector reconstruction for105 poster colors in R, Ψ ,and Φ spaces. 82
Table 4-1: The spectral and colorimetric accuracy of the three-eigenvectorreconstruction for the Kodak Q60C. 95
Table 4-2: The spectral and colorimetric accuracy of the six-eigenvectorreconstruction for the still life painting. 98
Table 4-3: The spectral and colorimetric accuracy of the six estimatedcolorants for the still life painting. 100
Table 4-4: The colorimetric and spectral accuracy of the six eigenvectorreconstruction for the 105 mixtures. 105
Table 5-1: The colorimetric and spectral accuracy for the four SWOPprimaries synthesizing the 928 samples of IT8/7.3 target in theproposed linear colorant mixing space. 118
Table 5-2: The correation coefficeints and the chroma of the 18 inks withthe five chromatic statistical primaries. 122
Table 5-3: The spectral and colorimetric accuracy of the three optimal ink sets. 123
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Table 5-4: The Pantone color names of the three optimal ink sets. 124Table 5-5: The spectral and colorimetric accuracy of the three worst performing
ink sets. 125Table 5-6: The ink combinations of the three worst performing ink sets. 125
Table 6-1: The colorimetric accuracy, spectral accuracy, and the statisticalthickness for the six primaries. 140
Table 6-2: The colorimetric and spectral accuracy of the 25 overprints. 142
Table 7-1: The effective dot areas and the correction scalar, q, of cyan fixedat 50% theoretical dot area by overlapping the secondary magenta ink at various theoretical dot areas (Iino and Berns, 1998). 161
Table 7-2: The determined q scalars by proposed modification and thedot-gain of the primary (magenta) given that the secondary (cyan) is present. 164
Table 7-3: The colorimetric and spectral accuracy of n = 2.2 in predicting all72 samples of the six primary ramps where Stdev stands for thestandard deviation and RMS represents the root-mean-square error in unit of reflectance factor. 179
Table 7-4: The colorimetric and spectral accuracy of the first order forwardmodel in predicting the 6,250 samples of the verification target. 181
Table 7-5: The colorimetric and spectral accuracy in predicting the verificationof 6,250 sample by the algorithms suggested by Iino and Berns. 186
Table 7-6: The theoretical dot areas and their correction scalar by Eq. (7-14)without logical correction. 190
Table 7-7: The theoretical dot areas and their correction scalar by Eq. (7-14)with logical correction. 192
Table 7-8: The colorimeric and spectral accuracy of the proposed algorithmsin predicting the verification target of 6,250 sample mixtures. 193
Table 7-9: The colorimeric and spectral accuracy of the proposed algorithmsmodified by Eq. (7-20) in predicting the verification target of 6,250sample mixtures. 196
Table 7-10: The colorimetric and spectral performance of the verificationtarget predicted by the matrix method. 201
Table 7-11: The predicted theoretical dot areas, colorimetric and spectralerrors of the 24 colors in Macbeth color checker where M.I.represents the metamerism index. 203
Table 7-12: The statistical results in predicting the 24 colors using theproposed six-color printing model. 204
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Table 7- 13: The colorimetric and spectral performance of the 22 in gamutcolors. 204
Table 7-14: The color difference of the predicted Macbeth Color Checkerunder illuminant A and F2 by four different printing processes,where the color names corresponding to the bold faced entriesare the out of colorimetric gamut colors of each device. 208
Table 8-1: The colorimetric and spectral accuracy of original vs. reproductionand prediction vs. reproduction for the Gretag Macbeth ColorChecker. 213
Table 8-2: The statistical colorimetric and spectral accuracy correspondingto the predicted and reproduced Gretag Macbeth Color Checker. 214
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LIST OF FIGURES
Fig. 1-1: An application of paint catalog sales by the conventional four-colorprinting. Notice that the paint chip approximately matches the printedpaint chip under daylight illuminant and mismatches the printed paintchip under the Incandescent illuminant. 5
Fig. 1-2: The color gamuts projected on the xy chromaticity plane for Pantone Hexachrome, conventional four-color printing process, and CRT. 6Fig. 1-3: The outline of a multi-spectral color reproduction system. 8Fig. 1-4: The structure chart of the research development for spectral printing. 10Fig. 1-5: The structure diagram of colorant estimation. 12Fig. 1-6: The optimal ink selection scheme for low-error spectral reproduction. 13Fig. 1-7: The microscopic structure of halftone color formulation of a CMY
printing device, where Rλ, color is the spectral reflectance factor of acolor appearing in the figure. 14
Fig. 1-8: The structure of spectral-based six-color printing process whereYNSN stands for Yule-Nielsen modified spectral Neugebauer equation. 19
Fig. 2-1: The three dimensional vector, P3, in ℜℜℜℜ3. 29Fig. 2-2: Spectral reflectance curve of the example. 31Fig. 2-3: Example for a set of three dimensional vectors localized in various
vector subspaces. 33Fig. 2-4: Scatter plot of a two dimensional sample set. 35Fig. 2-5: The representation of sample set of Figure 4 in the eigenvector
coordinate system. 38Fig. 2-6: The mean (thick line), the first eigenvector (thin line), and the
second eigenvector (dashed line) of the 622 daylight measurements(Judd, MacAdam, and Wyszecki, 1964). 45
Fig. 2-7: Spectral distribution of typical daylight at correlated colortemperatures 4800 K, 5500 K, 6500 K, 7500 K, and 10,000 K(Judd, MacAdam, and Wyszecki, 1964). 46
Fig. 2-8: Three sets of absorption spectra of cyan, magenta, and yellow dyes(left to right) at eleven different concentrations. 47
Fig. 2-9: The first six eigenvectors obtained for the virtual sample set. (Asmany as thirty-one eigenvectors can be shown.) 48
Fig. 2-10: The six eigenvectors obtained from an IT8.7/2 reflection target. 50Fig. 2-11: Scatter plot of the 500 random samples and their eigenvectors from
a uniform distribution. 52
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Fig. 2-12: Normal probability plot of absorption coefficients at 500 nm fromthe virtual normal population obtained by linear combinations of threelinearly independent dyes. 55
Fig. 2-13: Normal probability plot of absorption coefficients at 500 nm of theexponential population obtained by linear combinations of threelinearly independent dyes. 55
Fig. 2-14: The two-dimensional scatter plot of absorption coefficients at550 nm vs. 500 nm from the virtual normal population obtained bylinear combinations of three independent dyes. 57
Fig. 2-15: The outline of a 2x2 halftone cell. 58Fig. 2-16: The 3x3 halftone cells covered by three different dot area coverage. 60Fig. 2-17: Non-ideal halftone dot shapes vary due to the mechanical dot gain effect. 61Fig. 2-18: The cause of optical dot gain. 62
Fig. 3-1: The possible field of view of a spectrophotometer. 71Fig. 3-2: The normal plot of simulated reflectance factors at each sample
wavelength obtained by linear combinations of six approximatelynormal distributions. 75
Fig. 3-3:The normal plot of simulated reflectance factors at each samplewavelength generated by linear combinations of six non-normaldistributions. 76
Fig. 3-4: The normal plot of marginal distributions of IT8.7/2 reflection targetin reflectance space. 79
Fig. 3-5: The Kubelka-Munk inverse transformation, Eq. (3-3), for opaquecolor. 83
Fig. 3-6: An example of enhanced spectral error after the transformation byEq. (3-3). 84
Fig. 3-7: The transformation from Ψ to R space by Eq. (3-7). 84
Fig. 4-1: The still life painting of a floral arrangement creating with six opaquecolorants. 87
Fig. 4-2: The six eigenvectors obtained from the still life painting. 89Fig. 4-3: The six acrylic paints used for the computer generated sample
population. 93Fig. 4-4: The all-positive eigenvectors as the estimated dye spectra (thick line)
and the local eigenvectors (dotted lines). 96Fig. 4-5: The six all-positive eigenvectors as the estimated colorants for the still
life painting. 99Fig. 4-6: The estimated colorants (solid lines) and the original colorants
(astroidal lines) used for the still life painting. 100
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Fig. 4-7: The 105 mixtures created by hand mixing six opaque poster paints. 103Fig. 4-8: The spectral reflectance factors of the six poster colors used for
creating the 105 opaque mixtures. 103Fig. 4-9: The offset vector,
v
a , for transforming the 105 mixtures to Ψ space. 104Fig. 4-10: The six eigenvectors of the 105 mixtures in Ψ space. 105Fig. 4-11: The reflectance factors of the six original colorants (solid lines) and
statistical primaries (dashed lines) derived by a constrained rotationfrom the six eigenvectors. 106
Fig. 5-1: The SWOP specified process CMYK primaries and paper substrate. 117Fig. 5-2: The colorimetric and spectral error vs. w values for the empirical
transformation, Eq. (5-2), where M.I. represents metamerism index. 117Fig. 5-3: The four spectra reconstructed with the highest colorimetric and
spectral errors based on the linear colorant mixing space where solidline is the measured spectrum and the dashed line is the reconstructedspectrum . 119
Fig. 5-4: The six statistical primaries derived from the 105 mixtures by thecolorant estimation module. 120
Fig. 5-5: The Pantone 14 basic colors and the process CMYK as the inkdatabase (Color name order corresponding to each spectrum is fromleft to right and top to bottom). 121
Fig. 5-6: The three spectral reconstructions of a sample corresponding to themaximum prediction error by the three optimal ink sets. 125
Fig. 5-7: The three spectral predictions of the sample, shown in Fig. 5-6, bythe three worst performed ink sets. 126
Fig. 5-8: The two reconstructed spectra by set 23 and set 26 for the sampleused as the example in Fig. 5-6. 127
Fig. 6-1: The microscopic structure of color formation by a halftone printingprocess where Rλ,color represents the spectral reflectance factor ofa color appearing in the square area. 129
Fig. 6-2: An ink film applied on black and white contrast paper. 133Fig. 6-3: The diagram of a three-ink-layer overprint. 136Fig. 6-4: The six primaries printed on contrast paper. 138Fig. 6-5: Twenty-five overprints printed on coated paper. 138Fig. 6-6: The spectral absorption (solid line) and scattering (dashed line
multiplied by ten times) curves of the six primaries. 139Fig. 6-7: The difference spectra between measured and predicted primaries. 140Fig. 6-8: Histogram of the metamerism indices for prediction of the 25
overprints. 142
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Fig. 6-9: The difference spectra of four overprints best predicted with highaccuracy. 143
Fig. 6-10: The difference spectra of four overprints predicted with relativelylow accuracy. 143
Fig. 6-11: The estimated two optical constants for cyan ink and correspondingdifference spectrum in units of reflectance factor. 145
Fig. 7-1: The structure chart for the development of six-color separationminimizing metamerism. 152
Fig. 7-2: The structure of a general forward halftone printing model where f( )is a mathematical function or LUT describing the dot-gain effect andNλ,n( ) is the function of Yule-Nielsen modified spectral Neugebauerequation. 154
Fig. 7-3: The algorithm structure of determining the Yule-Nielsen n-factorwhere INV(YNMD) stands for the inverse function, Eq. (7-3), ofn-factor corrected spectral Murray-Davies equation. 156
Fig. 7-4: A set of theoretical to effective dot area transfer functions determinedfrom a CMYK halftone printing process. 157
Fig. 7-5: The dot-gain functions of the CMYK ramps given in Fig. 7-4. 158Fig. 7-6: The family of dot-gain curves of a cyan ramp when magenta ink
presents at 0%, 25%, 50%, and 75% fractional dot areas. 159Fig. 7-7: The dot-gain loci of a cyan ramp when a magenta ink is present at
different theoretical dot areas where the locus goes through a3, b3,and c3 are the dot gains esimated by the first-order model, Iino andBerns' algoritms, and the proposed algorithms, respectively. 166
Fig. 7- 8: The functions of dot gain corrrection scalar by Iino and Berns (left)and the proposed (right) algorithms. 167
Fig. 7-9: The structure of the six-color backward spectral printing model usingcyan, magenta, yellow, green, orange and black ink as printingprimaries. 173
Fig. 7-10: The reflectance spectra of the printed six primaries and substrate. 175Fig. 7-11: The mean prediction accuracy of all 72 samples of the six primary
ramps as function of n-factor. 178Fig. 7-12: The measured and the predicted reflectance spectra by Eqs. (7-3)
and (2-33) using n = 2.2 where the solid lines are measured spectraand the dashed lines are the predicted spectra. 179
Fig. 7-13: The theoretical to effective transfer functions of the six primaries. 180Fig. 7-14: Histogram of the colorimetric performance using the first-order
forward printing model in predicting the 6,250 samples of theverification target. 181
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Fig. 7-15: The vector plot of the 100 predicted samples with the highestcolorimeric errors by the first-order forward model. The vector tailrepresents the measured coordinate and the vector head representsthe prediction. 183
Fig. 7-16: Four example spectra showing under prediction by the first ordermodel. 183
Fig. 7-17: The three example functions of dot-gain correction scalar determinedfor CMYK sub-model based on Iino and Berns' algorithms. 185
Fig. 7-18: Histogram of the colorimetric error in units of ∆E*94 for the 6,250
samples predicted by Iino and Berns' algorithms. 186Fig. 7-19: The vector plot of L* vs. a* for the 100 samples used as examples
in Fig. 7-15 predicted by Iino and Berns' algorithms. 187Fig. 7-20: The spectral predictions of the four example samples used as
examples in Fig. 7-16 by the Iino and Berns' algorithms. 188Fig. 7-21: The three example functions of dot-gain correction scalar
determined for CMYK sub-model based on proposed algorithms. 190Fig. 7-22: Histogram of the colorimeric accuracy of the proposed algorithms
in predicting the verification target of 6,250 sample mixtures. 193Fig. 7-23: The vector plot of L* vs. a* for the 100 samples used as examples
in Fig. 7-15 predicted by the proposed algorithms. 194Fig. 7-24: The spectral prediction of the four samples used as examples in
Fig. 7-16 by the proposed algorithms. 195Fig. 7-25: Histogram of the colorimeric accuracy of the proposed algorithms
modified by Eq. (7-20) in predicting the verification target of 6,250sample mixtures. 197
Fig. 7-26: The vector plot of L* vs. a*, L* vs. b*, and b* vs a* for the 300samples whose colorimetric error is predicted higher than 2.41 unitsof ∆E*94. 198
Fig. 7-27: The spectral prediction by the proposed algorithms modified byEq. (7-20) for the four samples used as examples in Fig. 7-16. 199
Fig. 7-28: Histogram of the colorimeric accuracy of the matrix method inpredicting the verification target of 6,250 sample mixtures. 201
Fig. 7-29: The four best predicted spectra in terms of metamerism index ofthe 24 colors. 205
Fig. 7-30: The four worst predicted spectra in terms of metamerism index ofthe 24 colors. 206
Fig. 8-1: The original, predicted, and reproduced spectral reflectance factorsof the six Gretag Macbeth Colors. 215
1
I. INTRODUCTION
Conventional graphic reproduction at the analysis stage, in general, uses broad
band RGB filters to determine color densities (Dr, Dg, and Db) of an original. It then
converts the measured color densities to effective halftone fractional dot areas
corresponding to process cyan, magenta, yellow and black (CMYK) inks. This is done by
using empirically determined ink tables constructed by exhaustive sampling and measuring
prints of a particular printing process (Pobboravsky and Pearson, 1972; Viggiano, 1985).
High-end color scanners for the printing industry were initially invented by Hardy
and Wurzburg in 1948, known as the Hardy and Wurzburg scanner, and by Murray and
Morse in 1941 known as the P.D.I. scanner. The Hardy and Wurzburg scanner was
designed with spectral sensitivities equal to the CIE color matching functions,
x y and z( ), ( ), ( ),λ λ λ or linear transformations of them, in order to record tristimulus
values of originals and evaluate the amount of CMY inks needed to be delivered onto
paper substrate through electronic computing circuits (Hardy and Wurzburg, 1948; Hardy
and Dench, 1948). Whereas, the P.D.I. scanner was devised based on the masking theory
instead of the Neugebauer theory (Murray and Morse, 1941; Hunt, 1995).
The later modified versions such as the Crossfield Diascan, Hell Chromagraph, and
Linotype-Paul Linoscan, are generally designed with three narrow-band spectral
sensitivities centered around short (450 nm), medium (550 nm), and long (650 nm)
2
wavelength regions throughout the visible spectrum to evaluate the color densities (Dr, Dg,
and Db) or CMYK concentrations of an original. These scanners function more closely to
densitometers instead of colorimeters since their spectral sensitivities are not linearly
related to CIE color matching functions. High-end scanners used in the printing industry,
which evaluate color densities (Dr, Dg, and Db) or CMYK concentrations of originals, do
not unambiguously record the color information of originals. As a consequence, only a
metameric reproduction can be achieved by a printing process if colorants used in
reproduction are different from that of original (Berns and Shyu, 1995). Thus, the
outcome of the pre-press color acquisition together with the four process colors by the
conventional printing is intrinsically metameric.
Although metameric reproduction by CMYK color reproduction of conventional
printing technology accomplishes pleasing results, the color mismatch under illuminants
other than that standardized by the printing industry is always problematic for critical
color-matching applications such as catalog sales, art-work reproduction, and computer-
aided design.
Nowadays, the demanded quality of color reproduction is skyrocketing.
Consumers are more willing to invest extra capital in pursuing high fidelity color
reproduction for its appealing results with respect to printing technology. Generally, the
terms Hi-Fi, multispectral, and multiple-ink are synonyms. The so called “high fidelity”
color reproduction utilizes more than four primary inks to expand device gamuts (Carli
3
and Davis, 1991). It is believed that a larger device gamut has a better chance to match
colors from the real world because, theoretically, a system can exactly reproduce the
colorimetric values that fall within its color gamut. Previous research by Ostromoukohv
suggested a way to enhance chromaticity gamut via heptatone multicolor printing process
(Ostromoukhov, 1993). He employed the Neugebauer equation together with seven
printing primaries to match the colorimetric information (tristimulus values) of an
original. This approach is a prevalent scheme for state-of-the-art Hi-Fi color systems.
However, the colorimetric matching reproduction still suffers from both illuminant and
observer metamerism (Grum and Bartleson, 1980). Kohler and Berns pointed out that
illuminant metamerism could be reduced by using five or more colored inks based on
spectral color reproduction (Kohler and Berns, 1993). The primary goal of this doctoral
research is to devise color separation algorithms for multiple-ink printing systems that are
capable of reconstructing the spectral information from original objects such that the
metamerism is minimized between original objects and their color reproductions.
A. OVERALL PROBLEM
One manifestation of metamerism occurs when two surfaces, each produced with
different colorants as judged by their distinct spectral reflectance factors integrate to the
same color perception under a given lighting condition and to a different color perception
under other illuminants with respect to the human visual system. Metamerism can be used
4
as an advantage for color devices such as CRTs and printers to reproduce trichromatic
matches between originals and reproductions under a predefined viewing condition. That
is, metamerism enables CRTs and printers to accomplish colorimetric synthesis in a
relatively inexpensive fashion since the color match is defined as the identical colorimetric
values between originals and their reproductions. However, all sorts of materials are not
necessarily made from phosphors and inks. The color synthesis by phosphors and inks do
not render color unambiguously when only a metameric color match can be achieved.
Ambiguity means that the color match is only defined for a standard viewing environment.
Color mismatch, due to uncontrollable lighting conditions, affects the accuracy of color
communication (Berns, 1998).
Current four-color printing systems are limited to metameric reproduction due to
an insufficient number of degrees of freedom. This means that the possible spectral
variations from originals can not be synthesized by only four inks given that the originals
are likely constituted by more than four distinct colorants. Figure 1-1 exemplifies a catalog
sales for a paint retail store. Paint samples were printed on the catalog to convey the
chromatic information and try to attract the preference of consumers. A paint chip is
placed underneath the catalog to represent the paint sold in a paint store. Standard
daylight is assumed as the illumination for paint stores and incandescent illuminant is
assumed to be the typical indoor illumination used by general consumers. Once a
consumer matches his or her color preference under home illumination from the catalog
5
and purchases the paint based on a visual match by the catalog under the illumination in a
paint store, it is possible that the consumer will return the paint after he or she realizes
there are mismatches for his or her applications indicating by that the paint matches the
printed sample under daylight illuminant and mismatches under the incandescent
illuminant. This is a typical problem of metamerism. The paint industry can not just sell
paint by catalog because of metameric catalog reproduction.
Daylight illuminant Incandescent illuminant
Fig. 1-1: An application of paint catalog sales by the conventional four-color printing.Notice that the paint chip approximately matches the printed paint chip under daylight
illuminant and mismatches the printed paint chip under the Incandescent illuminant.
Since the insufficient number of degrees of freedom lead to metameric
reproduction, the intuitive action for increasing degrees of freedom by adopting two or
more inks should alleviate metamerism. Although the current multiple-ink printing
6
systems, such as Pantone Hexachrome, Küppers 7-Color, and DuPont Hyper Color,
employ six or more inks for high fidelity color reproduction, their color separation
algorithms are still based on trichromatic color reproduction. The increased degrees of
freedom are used for expanding device color gamut (Herbert, 1993; Boll, 1994; Stollnitz,
Ostromoukhov, and Salesin, 1998; Viggiano, 1998). The problem of metamerism is not
alleviated by these systems. The advantage of more number of degrees of freedom is not
fully exploited. Figure 1-2 demonstrates the color gamuts, plotted in terms of xy
chromaticities, of Hexachrome, RGB monitior, and four-color printing process,
respectively.
Hexachrome
RGB Monitor
Process 4C
x
y
Fig. 1-2: The color gamuts projected on the xy chromaticity plane for PantoneHexachrome, conventional four-color printing process, and CRT.
7
Although the comparison should be made by evaluating the three gamut volumes, the
projection of each device gamut onto the xy chromaticity plane is shown for that the
Hexachrome system has a larger gamut projection relative to RGB monitor and
conventional four-color printing.
B. OVERVIEW OF SOLUTION
In order to reduce the degree of metamerism, a spectral approach, as opposed to a
trichromatic approach, is the answer for critical color matching applications. The goal of
spectral reproduction is to reproduce the best spectral match whose color-match remains
constant under most illuminants within a visual tolerance. This ensures that the colors of
merchandise reproduced in a catalog, or the colors of an original painting reproduced in an
art book, will unambiguously be rendered under various illuminating and viewing
conditions.
A research and development program has been initiated at the Munsell Color
Science Laboratory in Rochester Institute of Technology for developing a spectral-based
color reproduction system (Berns, Imai, Burns, and Tzeng, 1998). Research has included
multi-spectral acquisition systems (Burns and Berns, 1996; Burns, 1997; Burns and Berns,
1997a; Burns and Berns, 1997b) and spectral-based printing algorithms (Iino and Berns,
1997; Iino and Berns, 1998a; Iino and Berns, 1998b). The current research effort is
devoted for the development of the spectral output system which is comprised of an ink-
8
selection module, and the implementation of spectral-based printing algorithms for a
multiple-ink printing device, shown as Fig. 1-3.
Multi-SpectralAcquisition
Spectal-BasedPrinting
Fig. 1-3: The outline of a multi-spectral color reproduction system.
Since the limitation of four-color reproduction for spectral reconstruction is due to
insufficient number of degrees of freedom for adjusting the reproduced spectra, printers
need extra primary inks to shape the region of the spectrum that is beyond the limitations
of four-color processes. Intuitively, the extra inks extend the capability and accuracy for
spectral reproduction, when chosen properly. A properly selected ink set implies that the
adoption of a fixed ink set by previously mentioned multiple-ink systems are not
sufficiently equipped for spectral reproduction since the spectral variations may be beyond
the synthesis capability of those ink sets. In another words, original objects are unlikely
9
created by those ink sets. Thus, the overall solution for multiple-ink printing devices focus
on minimizing metamerism by developing spectral-based color separation algorithms
which employ a greater number of degrees of freedom as well as the dynamic ink selection
to overcome the shortcoming of current multiple-ink systems. It is expected that the
greater number of degrees of freedom together with the flexibility of dynamic ink selection
can well describe the spectral variations from an arbitrary original object.
C. SCOPE OF DISERTATION
The scope of this dissertation was to develop a spectral-based color separation
algorithm for a multiple-ink printing process and to derive a spectral reconstruction
scheme based on the spectral Neugebauer equation at the synthesis stage. The current
research aims to identify a set of six inks for spectral halftone printing, corresponding to
the fact that six-color capabilities are widely available around the world. According to
NAFTA statistics, six-ink printing capabilities are available at over seven thousand sites in
North America alone. The choice of the number six is based on this production
convenience.
The research development of spectral six-color printing is divided into modules of
colorant estimation (Tzeng and Berns, 1998), optimal ink selection (Tzeng and Berns,
1999a), prediction of ink overprint (Tzeng and Berns, 1999b), spectral-based color
separation minimizing metamerism, and direct digital or conventional printing. Structure
10
of the proposed research development is depicted in Fig. 1-4 by assuming that the input of
this spectral printing system is a spectral image acquired by a multi-spectral acquisition
device.
ColorantEstimation
Spectral-BasedColor SeparationMinimizingMetamerism
Ink SelectionAlgorithm
Multiple-InkDirect Digital orConventionalPrinting
Ink OverprintPrediction
SpectralImage
Fig. 1-4: The structure chart of the research development for spectral printing.
Function of each module is designated as follows:
1. Colorant Estimation
To minimize metamerism between spectral inputs and their corresponding
reproductions, it is desirable to understand the spectral properties of the colorants possibly
utilized for creating the original objects. Then the use of a new set of colorants with
similar or identical spectral properties for synthesis should achieve the minimum or zero
spectral error. Since, in practice, it is not likely that the colorant information for original
11
objects is available from their creators, the estimation has to resort to statistical
approaches. Any set of spectral samples can be captured by a multi-spectral acquisition
device or measured via a spectrophotometer or spectroradiometer with finite bandwidth
across the visible spectrum. Then the set of spectral samples is distributed in a multi-
dimensional vector space. Principal component analysis (PCA) can provide a gauge to
statistically decompose these spectral samples into fewer statistical dimensions. These
dimensions dominate the major sample variations of these spectral measurements. In
another words, majority of samples vary along the directions of these statistical
dimensions. It can be viewed as if there existed a set of colorants whose spectral
properties coincide with the directions of the major sample variation, therefore, the whole
sample variations are the exact combinations of the set of imaginary colorants, which are
the eigenvectors derived by PCA. Since the eigenvectors are the only link to the physical
colorants which can be used for synthesis, a transformation to a set of all-positive vector
representations as the estimated primary colorants is necessary to account for the all-
positive spectral properties of real materials.
Thus, at the analysis stage, this research will perform statistical analysis via PCA
on original images, which are the spectral inputs prior to color separation, followed by a
constrained transformation to statistically estimate the possible primaries that can be used
for low-error spectral synthesis. That is, the colorant estimation procedure statistically
estimates a set of possible primary colorants, whose linear mixtures are the closest
12
approximation for each pixel of spectral inputs. Ideally, if the estimated primary colorants
exist in a current ink database, then the use of the exact ink set for halftone printing will
yield ideal spectral reproduction. A detailed chart of this module is shown in Fig. 1-5.
The six rotated normalized all-positive eigenvectors
0
0.2
0.4
0.6
0.8
1
1.2
400
430
460
490
520
550
580
610
640
670
700
Wavelength
(Abs
orpt
ion
/ Sca
tterin
g)
Sampledwavelength
Spectral image
Kubelka-Munkor empiricaltransformationinto a linearcolor mixingrepresentation
Six eigenvectors Statistical primaries
Constrainedtransformation
400 500 600 700-1
-0.5
0
0.5
1The 1st eigenvector.
K/S
400 500 600 700-1
-0.5
0
0.5
1The 2nd eigenvector.
400 500 600 700-1
-0.5
0
0.5
1The 3rd eigenvector.
400 500 600 700-1
-0.5
0
0.5
1The 4th eigenvector.
Wavelength
K/S
400 500 600 700-1
-0.5
0
0.5
1The 5th eigenvector.
Wavelength400 500 600 700
-1
-0.5
0
0.5
1The 6th eigenvector.
Wavelength
400 450 500 550 600 650 7000
0.5
1
1.5
2
2.5
Wavelength
K/S
Principal componentanalysis
Fig. 1-5: The structure diagram of colorant estimation.
2. Optimal Ink Selection
It is not likely to have an ink set with identical spectral properties to the estimated
primaries exist in the ink data base. The optimal ink set which minimizes the spectral error
between original objects and reproductions could be manufactured by colorant chemists to
obtain the desired spectral characteristic. However, the optimal ink set is image
13
dependent. Frequently, it is not time and cost efficient to manufacture an optimal ink set
for industrial applications whenever spectral color reproduction is encountered. It is more
practical to select an optimal ink set from a large ink database, currently in manufacture.
Hence, devising an optimal ink selection method in order to choose the corresponding inks
for the best spectral reconstruction is necessary. An optimal ink selection scheme, shown
as Fig. 1-6, will be devised based on vector correlation analysis by searching through the
existing ink database such as Pantone formula colors to identify the most similar ink set.
Th e six rotate d normali zed all - posi tive ei ge nvectors
0
0.2
0.4
0.6
0.8
1
1.2
400
430
460
490
520
550
580
610
640
670
700
W ave length
(Abs
orpt
ion
/ Sca
tterin
g)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
400
430
460
490
520
550
580
610
640
670
700
Wavelength
Ref
lect
or fa
ctor
Rhodamine red
Purple
Reflex blue
process black
process yellow
process cyan
Ink selection by vectorcorrelation analysis
Optimal ink-set
Statistical primaries
Pantone colors
Fig. 1-6: The optimal ink selection scheme for low-error spectral reproduction.
3. Ink Overprint Prediction
At the synthesis stage of this research, the chosen optimal ink set will be used as
the printing primaries. The spectral expansion of the Neugebauer equation together with
14
the Yule-Nielsen n-factor correction will be designated as the halftone printer model. This
model is capable of reconstructing the spectra of an original with high accuracy
(Pobboravsky and Pearson, 1972; Rolleston and Balasubramanian, 1993). Consider the
color formulation depicted as Fig. 1-7 by a CMY halftone printing device, the overall
color stimulus over a printed spot is contributed by the colors appearing in that spot.
Fig. 1-7: The microscopic structure of halftone color formulation of a CMY printingdevice, where Rλ, color is the spectral reflectance factor of a color appearing in the figure.
Assuming the spot is circumscribed by the thick line square in Fig. 1-7, then the
overall color stimulus over the thick line square in terms of spectral reflectance factor is
the linear sum of the spectral reflectance factor of each color inside the square.
Apparently, the corresponding modulation is proportional to the percentage of the
occurrence of each color inside the square. This percentage of occurrence is quantified by
Rλ,Y
Rλ,M
Rλ,C
Rλ,White
Rλ,R
Rλ,G
Rλ,B
Rλ,CMY
15
the fractional dot area of each color. These colors inside the square are not just printing
primary colors (cyan, magenta, and yellow) but also overprints (red, green, blue, and
three-color black). The linear sum is exactly the three-color spectral Neugebauer equation,
shown as Eq. (1-1) and its Yule-Nielsen n-factor (Yule and Nielsen, 1951) corrected
spectral Neugebauer equation, shown as Eq. (1-2).
Rλ = aCRλ,C + aMRλ,M + aYRλ,Y
+ aRRλ,R + aGRλ,G + aBRλ,B + aCMYRλ,CMY , (1-1)
+ (1- aC - aM - aY - aR - aG - aB - aCMY)Rλ,white,
Rλ = [aCRλ,C1/n + aMRλ,M
1/n + aYRλ,Y1/n
+ aRRλ,R1/n + aGRλ,G
1/n + aBRλ,B1/n + aCMYRλ,CMY
1/n , (1-2)
+ (1- aC - aM - aY - aR - aG - aB - aCMY)Rλ,white1/n]n,
where λ is wavelength, R represents reflectance factor, a is fractional dot area, and the
capitalized subscripts represent the color names of primaries, secondary, and tertiary
primaries. These primaries are also named the Neugebauer primaries.
To utilize the original or n-factor modified spectral Neugebauer equation, it is
necessary to posses the knowledge for the spectra of the Neugebauer primaries, i.e., the
spectra of secondary, tertiary, quaternary primaries (for four-color halftone printing), and
so forth. The spectral reflectance factors of overprints are usually obtained by printing and
measuring. Recognizing that with a change of materials (i.e., inks and paper), the spectral
reflectance factor on top of the printed samples includes a change in overprints. In order
16
to characterize a halftone printing process, the ramps for primaries and overprints need to
be printed and measured when new materials are applied. There are 2k-k-1 overprints for a
k-color printing process. The effort required for printing and measuring of ramps may not
be significant when a four-color printing process is encountered since the number of over
printings is only eleven. Table 1-1 lists the number of overprints corresponding to three to
eight-color printing processes. It shows that as the number of primaries increases linearly,
the number of corresponding overprints increases exponentially. The use of analytical
prediction of overprints can avoid the necessity of printing and measuring upon changing
the ink set selected by the ink-selection procedure. The computation for the entire
integrated multiple-ink printing system can be automated without interruption due to the
change of ink set and paper.
A reasonable prediction process for overprints is to use the Kubelka-Munk turbid
media theory for translucent materials to predict the spectra of ink overprints. Once all the
spectral information of Neugebauer primaries is compiled, the spectral estimation is
enabled by the use of the Yule-Nielsen modified spectral Neugebauer equation.
Table 1-1: The number of overprints of a k-primary halftone printing process.
Number of primaries Number of overprints3 44 115 266 577 1208 247
17
4. Spectral-Based Six-Color Separation Minimizing Metamerism
When printing solid inks on top of each other, the ability to print the later wet ink
on top of formerly printed wet ink is call ink trapping. Ink trapping failure is caused by the
wetness and total amount of ink piling on a printed spot. The SWOP standard constrains
the ink trapping limit in terms of total fraction dot area to be 300%. Several printing
applications extending this limitation to 340% to 400% depending on the physical and
chemical design of the ink materials. Based on this physical limitation of ink trapping, it is
impossible to overprint six solid primaries in one location. Hence, the ink limiting is the
primary concern of the design of the six color printing process. In addition, consider an
input color with an arbitrary spectrum, if the subtractive synthesis requires more than four
independent colors then this particular color is of low spectral reflectance factor. Hence, it
is able to be approximated by a black and three chromatic inks for its spectral synthesis.
Therefore, the ink limiting predefining the six-color printing model is based on ten four-
color printing sub-models since choosing one black ink and three chromatic ink out of
remaining five from the optimal ink set is ten (C(5, 3)).
Often, given a color with a known spectrum, the real world problem is to
determine the amount of ink, or the percentage of dot areas, which needs to be delivered
onto a paper support. The Yule-Nielsen modified spectral Neugebauer equation is
required to be inverted to solve for the effective dot areas for a known color. Since the
analytical inversion of Neugebauer equation is impossible to accomplish, it must be
18
approximated by a numerical approach (Mahy and Delabastita, 1996). For this research
development, MATLAB and its optimization toolbox are utilized for inverting the Yule-
Nielsen modified spectral Neugebauer equation (MATLAB, 1996).
With the derived effective fractional dot areas, spectral reconstruction of an
original will be performed using the forward Yule-Nielsen modified Neugebauer equation
at the synthesis stage. The performance of this process will be evaluated in terms of
colorimetric and spectral accuracy. If the minimized spectral error still reveals low
colorimetric accuracy then the post-process of slightly trading in spectral accuracy to
exchange for the high colorimetric accuracy under the standard illuminant D50 will be
adopted. This will result in the closest possible match across multiple light sources.
Hence, the degree of metamerism is reduced.
5. Multiple-Ink Direct Printing
The six-color separations are output by a multiple-color proofing system with
stochastic screening at 175 LPI (or equivalent) screen frequency to avoid the Moiré
pattern. Figure 1-8 shows the processes of color separation minimizing metamerism and
multiple-ink direct printing. Accomplishment of the processes described above completes
the research of the spectral-based color separation algorithms development for multiple-
ink color reproduction. Detailed technical contents are described in the later chapters.
19
Color separationby YNSN
Direct printingor proofing
Original painting
Spectral reproduction
Fig. 1-8: The structure of spectral-based six-color printing process where YNSN standsfor Yule-Nielsen modified spectral Neugebauer equation.
20
II. BACKGROUND
Current research developments involve colorant estimation, ink selection, ink
overprint prediction, and spectral-based printing minimizing metamerism. The theoretical
underpinnings include multiple-ink printing systems, Kubelka-Munk turbid media theory,
computer colorant matching, multivariate statistics including principal component analysis,
and the Neugebauer theory for modeling the halftone printing process. This chapter
provides a review for these topics.
A. MULTIPLE-INK COLOR SYSTEMS
The concept of utilizing multiple inks for color reproduction is not new (Leekley,
Cox, and Gordon, 1953). It originated during the 19th century for color enhancement
(Friedman, 1978). State-of-the-art multiple-ink color reproduction systems such as Hi-Fi
color systems utilize more colors than that of conventional CMYK color reproduction
processes. The most well known Hi-Fi color systems are DuPont Hyper Color, Küppers 7-
Color (Küppers, 1986), K&E (BASF) 7-Color, Fogra 7-Color, and Pantone Hexachrome
(Di Bernardo and Matarazzo, 1995; Herbert and Di Bernardo, 1998). DuPont Hyper
Color applies the process CMY inks with two different density levels and one process
black for color reproduction. Pantone Hexachrome was designed with fluorescent
mixtures to form cyan, magenta, yellow, black, orange, and green as the printing
21
primaries. Designing with the usage of fluorescent inks is to drive the reproduction gamut
as large as possible. The remaining systems all use CMYK plus RGB for color
reproduction.
Since present multiple-ink systems are all originally contrived to expand the
reproduction color gamut, color matches are still limited to standard ANSI viewing
conditions. Nevertheless, multiple-ink systems are quite capable of reproducing highly
saturated colors well beyond that of a conventional CMYK printing process (Takaghi,
Ozeki, Ogata, and Minato, 1994; Granger, 1996).
B. KUBELKA-MUNK TURBID MEDIA THEORY
Kubelka and Munk examined the reflectance of a material which had a thin layer of
colorant in optical contact with its diffuse opaque substrate (Kubelka and Munk, 1931;
Kubelka, 1948). They assumed that the layer of colorant could be further divided into a
large number of sublayers parallel to the surface of the entire colorant layer. Then,
sublayers were homogenous with identical optical properties to each other. Assuming the
thickness of the entire layer is X, then the thickness of the sublayers is differentially
defined as dx. Kubelka and Munk further assumed two diffuse light fluxes i, a downward
flux, and j, an upward flux. The magnitude of the downward flux, i is decreased by the
absorption and scattering of the colorant sublayer. The effect of the scattering process is
to reverse the portion of downward flux, i, to the upward direction toward the surface of
22
the colorant layer. The upward flux, j, is further absorbed and scattered back to the
downward direction by the sublayer. Hence, the portion of the upward flux, j, scattered
back to the downward direction, needs to be added to the remainder of the downward
flux. Continuing in this fashion, the differential equations to account for the downward
and upward flux was initially set up as
di (S K)idx Sjdx− = − + + (2-1)
and
dj (S K)jdx Sidx = − + + , (2-2)
where K is the absorption coefficient and S is the scattering coefficient. The negative sign
in Eqs. (2-1) and (2-2) to account for the downward direction had been defined as the
negative direction (Wyszecki and Stiles, 1982).
To solve the above differential equation in terms of reflectance factor, R, of the
material, let ρ be the ratio of j to i then, from the quotient rule of differentiation (Allen,
1980),
d
dx
d j i
dx
i(dj dx j di dx
i
ρ= =
−( / ) / ) ( / )2 . (2-3)
Substituting Eqs. (2-1) and (2-2) to Eq. (2-3), Eq. (4) is obtained as
d
dxS K S S
ρρ ρ= − + +2 2( ) . (2-4)
Equation (2-4) is a separable first-order differential equation. The boundary conditions
are that ρ = Rg when x = 0, and ρ = R when x = X, where Rg is the reflectance factor of a
23
substrate, R is the surface reflectance factor of a material, and X is the thickness of the
colorant layer of the corresponding material. Rearranging Eq. (2-4) together with the
boundary conditions, it turns out as Eq. (2-5),
dxd
S K S S
X
R
R
g0 22∫ ∫=− + +
ρρ ρ( )
. (2-5)
Solving for Eq. (2-5) in terms of R, the famous Kubelka-Munk equation results in
RRg a b bSX
a Rg b bSX=
− −− +
1 ( coth( ))
coth( ), (2-6)
where a is equal to 1+K/S and b is equal to (a2 - 1)1/2.
For an opaque colorant layer, known as the complete hiding case, light flux
traveling in the colorant layer keeps being scattered and never reaches the substrate. It is
equivalent to treating the thickness X of the entire colorant layer as infinitely large.
Another assumption by Kubelka and Munk is that there is no fluorescence in the colorant
layer. Based on these assumptions, Eq. (2-6) can be further simplified to Eq. (2-7),
R K S K S K S∞ = + − +1 22( / ) ( / ) ( / ) , (2-7)
and its inverse, Eq. (2-8),
K S R R/ ( ) /= − ∞ ∞1 22 , (2-8)
for the opaque material where R∞ denotes as the surface reflectance factor of an opaque
material with thickness.
24
In the application of a transparent layer in optical contact with a highly scattering
support, such as the photographic paper, the scattering of the colorant layer is ideally
zero. Therefore, a portion of downward flux passing through the colorant layer is
absorbed and not scattered. The remaining flux reaches the substrate and gets reflected
upward. The second absorption proceeds as the upward flux travels again through the
colorant layer. Hence, Eq. (2-6) approaches Eq. (2-9) as S approaches zero. That is,
limS
gKXR R e
→
−=0
2 . (2-9)
For most applications predicting color mixing for a transparent colorant layer in optical
contact with an opaque support, the thickness X of the colorant layer is frequently
assumed to be unity. Thus, the inverse of Eq. (2-9) in terms of absorption coefficient K is
obtain as:
KR
Rg
= −0 5. ln( ). (2-10)
It has been shown that the absorption and scattering coefficients are linearly
related to concentration (Allen, 1980; McDonald, 1987; Shan and Gandhi, 1990). Hence,
the colorant mixing, mathematically described by Eq. (2-11), for the transparent material
in optical contact with an opaque support is simply the mixture of the absorption
coefficients normalized to unit concentration, k, of primary colorants modulated by their
concentrations.
K c k c k c k c kmix i i= + + + +1 1 2 2 3 3 L , (2-11)
25
where ci is the concentration of the ith colorant. The reflectance factor of such material
can be calculated by Eq. (2-9). For the opaque material, the reflectance factor of color
mixture is a function of the mixtures of the absorption coefficient, shown as Eq. (2-12),
and the scattering coefficient, shown as Eq. (2-13).
K k c k c k c kmix t i i= + + + +1 1 2 2 L , (2-12)
where t denotes the substrate that suspends the primary colorant inside the colorant layer
and kt is the absorption coefficient of the substrate.
S s c s c s c smix t i i= + + + +1 1 2 2 L . (2-13)
Coefficient s represents the scattering normalized to its unit concentration. Hence the ratio
of the absorption to scattering coefficients in the opaque colorant layer is
K
S
k c k c k c k
s c s c s c smix
t i i
t i i
=
+ + + ++ + + +
1 1 2 2
1 1 2 2
L
L. (2-14)
Approaches leading to Eq. (2-14) are known as Kubelka-Munk two constant theory.
Consider that when the individual scattering influence contributed by each primary
colorant within the colorant layer is far less than the scattering power of the substrate.
Equation (2-14) can be simplified to Eq. (2-15), known as Kubleka-Munk single constant
theory,
26
K
S
k c k c k c k
s
k
sc
k
sc
k
sc
k
s
mix
t i i
t
ti
i
=
+ + + +
=
+
+
+ +
1 1 2 2
11
22
L
L
. (2-15)
The reflectance factor of a mixture of opaque colorants can be calculated by Eq. (2-7).
C. COMPUTER COLORANT MATCHING
Computers have long been used for dye recipe formulation in the textile and paint
industries. The early commercial applications were first proposed in 1961 (Anderson,
Atherton, and Derbyshire, 1961). Several alternative approaches were later refined in 1963
when computers were practically available. Up-to-date, computer methods for colorant
formulation have been vastly developed and published. Allen disclosed a common
algorithm for tristimulus matching in the applications of computer colorant formulation
(Allen, 1966; 1974). This algorithm is based on Kubelka-Munk turbid theory for a dyed
highly scattering opaque substrate. The matrix equation of the single constant theory for
the first prediction of concentration vector c corresponding to a matched standard color is
defined by Eq. (2-16).
c = (TEDΦΦΦΦ)-1 TED(f(a) - f(t)) (2-16)
where T is a matrix of one of the CIE color matching functions, E is a matrix of the
spectral power distribution of a CIE standard illuminant, f(a) is a matrix of the linearized
function of the reflectance of an opaque sample fabricated by colorant mixtures in optical
27
contact with an opaque substrate, f(t) is a matrix of the linearized function of reflectance of
an opaque substrate, D is a diagonal matrix of a derivative weighting function, and ΦΦΦΦ is a
matrix of the ratios of spectral absorptivities and unit scattering coefficients of three basis
colorants. In addition, their matrix forms are shown by the following:
T E f (a)
f (t)
= = =
=
x x x
y y y
z z z
E
E
E
f R a
f R a
f R a
f R t
f R t
f R t
400 410 700
400 410 700
400 410 700
400 0 0
0 410 0
0 0 700
400
410
700
400
410
700
L
L
L
L
L
M M O M
L
M
M
, ,
( )( )
( )( )
( )( )
,
( )( )
( )( )
( )( )
= =, ,
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
D
d
d
d
400 0 0
0 410 0
0 0 700
4001
4002
4003
4101
4102
4103
7001
7002
7003
L
L
M M O M
L
M M MΦΦΦΦ
φ φ φ
φ φ φ
φ φ φ
.
The linearized function for opaque samples is f(Rλ) = (K/S)λ = (1-Rλ)2/(2Rλ) and
the Jacobian suggested by Allen is dλ = (-2Rλ2)/(1- Rλ
2). Equation (2-16) is used to
estimate concentrations of the matched standard. This is only an approximation. The real
solution requires an iterative process by defining the tristimulus and concentration
difference vectors, ∆∆∆∆
∆∆∆∆
t c=
=
X
Y
Z
c
c
c
,
1
2
3
, respectively, where ∆t represents the
tristimulus difference and ∆c is the concentration difference vector between the standard
28
and the prediction at the current stage of iteration. Both values are bipolar and are
interrelated in the iterative correction process as Eq. (2-17),
∆∆∆∆c = (TEDφφφφ)-1 ∆∆∆∆t . (2-17)
The value of ∆t is used as the conditional specification for the iterative process. If ∆t is
within a goodness criteria then the iteration stops. If not, the ∆c at current iteration is
added to the old concentration to get the new concentration used for the next iteration.
This process continues until a goodness criteria is met. (The computer colorant matching
technique for the Kubelka-Munk two-constant theory is not shown in this review.
Interested reader can refer to Allen’s 1974 publication.)
D. VECTOR REPRESENTATION FOR COLORANTS
Generally speaking, a vector is a description for measurements at several features
of an event. For instance, an event can represent a point, P3 = (x, y, z), in a three
dimensional Cartesian coordinate system, denoted as ℜℜℜℜ3, where ℜℜℜℜ is the set of real
numbers. Since the standard basis vectors (Marsden and Tromba), which are orthonormal,
of ℜℜℜℜ3 are i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1), x, y, and z are the three features
describing the magnitudes of P3 along the directions of i, j , and k, respectively.
(“Orthonormal vectors” are a set of vectors with unit length which are perpendicular to
each other. “Basis vectors” are a set of vectors which are not only linearly independent to
each other but also span the entire vector space (Anton, 1991).) Although P3 is a
29
coordinate in ℜℜℜℜ3, it also can be viewed as a vector emerging from the origin since P3 can
be expressed as a linear combination of the i, j , and k, i.e.,
P3 = x⋅ i + y⋅ j + z⋅ k . (2-18)
Figure 2-1 denotes the visualization of a three dimensional vector P3 in ℜℜℜℜ3 where x, y, and
z are the scalar magnitudes (or projection length) along the i, j , and k, respectively.
i
k
j
P3 = (x, y, z)
x
z
y
(0, 0, 0)
Fig. 2-1: The three dimensional vector, P3, in ℜℜℜℜ3.
By extension, a point Pn = (x1, x2, x3,…, xn) in an n dimensional Cartesian coordinate
system, ℜℜℜℜn , has n feature measurements, x1, x2, x3,…, xn, describing the magnitudes of Pn
along the directions of n standard basis vectors in ℜℜℜℜn† , respectively. Hence, for any
† The standard basis vectors in ℜℜℜℜn are i1 = (1, 0, 0,…,0), i2 = (0, 1, 0,…,0), i3 = (0, 0, 1,…,0), …, in = (0,0, 0,…,1).
30
coordinate, Pn, treated as a vector in ℜℜℜℜn can be expressed as the linear combination of the
standard basis vector in ℜℜℜℜn, i.e.,
P in = ⋅=
∑ x jj
n
j1
. (2-19)
The basic mathematical operations applied to vectors are vector addition and scalar
multiplication. For two vectors, Pn,1 and Pn,2, in ℜℜℜℜn, the vector addition is defined as
P P i i in n, , , , , ,( )1 2 11
21
1 21
+ = ⋅ + ⋅ = + ⋅= = =
∑ ∑ ∑x x x xjj
n
j jj
n
j j jj
n
j . (2-20)
Equation (2-20) indicates that the vector addition operates by adding the two vectors
componentwise. The scalar multiplication is defined as
a a x a xjj
n
j jj
n
j⋅ = ⋅ ⋅ = ⋅ ⋅= =
∑ ∑P i in1 1
( ) , (2-21)
where a is a real number. Equation (2-21) shows that scalar multiplication modulates
every component of the vector, Pn, with the same scalar. It is useful to examine whether
measured spectral information can be represented as a vector; then all the merits of vector
algebra can be used for color science applications.
Given a sample with spectral reflectance factor, Rλ, measured within the visible
spectrum between 400 nm and 700 nm at 10 nm intervals where λ is a sampled
wavelength, Rλ is merely an array of thirty-one numbers; for example,
Rλ,yellow = (0.0153, 0.0154, 0.0152, 0.0148, 0.0153, 0.0161, 0.0165, 0.0166, 0.0191,
0.0301, 0.0905, 0.2706, 0.5216, 0.685, 0.7541, 0.7825, 0.8014, 0.8128,
31
0.8202, 0.8284, 0.832, 0.8341, 0.836, 0.8386, 0.8403, 0.8461, 0.8451,
0.8429, 0.8459, 0.849, 0.8491).
Rλ,yellow is usually plotted versus measured wavelength, shown in Fig. 2-2.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
400 450 500 550 600 650 700
Wavelength
Ref
lect
ance
fact
or
Fig. 2-2: Spectral reflectance curve of the example.
Although Rλ,yellow is sampled discretely at a 10 nm bandpass, it is viewed as a spectral
curve for the reason of smooth transition across 10 nm spectral neighborhood for typical
materials. Color scientists are used to examining and analyzing the thirty-one component
array on such a two dimensional layout of spectral measurements. In a coloration process,
various amounts of colorants are used for manufacturing a desired color. The modulated
spectral curves might be treated by multiplying the spectral curve of the colorant at 100%
of its maximum concentration for any percentage of modulation. Every component of a
32
measured array is multiplied by the same percentage such that, as a consequence, the
modulated spectral curves are parallel.
Similarly, in color science applications, the previous mentioned event can be a
spectral curve such as Rλ,yellow which has thirty-one measured reflectance factors at each
sampled wavelength. Measurements at each sampled wavelength specifies a dimension of
reflectance variations. Thus, Rλ,yellow can be treated as a vector in ℜℜℜℜ31 with the set of
standard basis vectors, i1, i2, i3,…, i31. Modulation by the amount of a colorant is
equivalent to scalar multiplication of the basic vector operation. The measured spectral
reflectance factors can be further transformed to different representations such as spectral
absorption where the vector addition and scalar multiplication are well defined. The
determination of the optimal transformation is discussed in a later section. Therefore,
adopted vector representations of the spectral information for color objects can further
employ vector algebra and linear modeling techniques to analyze originals and synthesize
reproductions.
Vector Subspace
For any set of n dimensional vectors, it is distributed in the n dimensional vector
space or sometimes localized in a lower dimensional vector subspace. Figure 2-3
demonstrates the three possible distributions for three sets of three dimensional vectors.
33
0
0.5
1
0
0.5
1
0
0.2
0.4
0.6
0.8
1
0
0.5
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.5
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Fig. 2-3: Example for a set of three dimensional vectors localized in various vectorsubspaces.
Each set of vectors are normalized and translated into the cube of [0, 1]3 which represents
the universe of ℜℜℜℜ3. The left-most figure shows that the uniform distribution of the three
dimensional vectors spanning the entire three dimensional vector space. The middle- and
right-most figures indicate that two sets of three dimensional vectors localize in a two and
one dimensional vector subspaces, respectively. A set of spectral measurements is
probably localized in a lower dimensional vector subspace since the spectral variations are
highly correlated across neighborhood spectral regions. The dimensionality of the set of
spectral measurements can be analyzed and approximated by principal component analysis.
34
E. LINEAR MODELING TECHNIQUES
Linear modeling techniques based on principle component analysis (PCA) have
been vastly applied for estimating the spectral reflectance factors of objects (Jaaskelainen,
Parkkien, and Toyooka, 1990; Maloney, 1986; Vrhel, Gershon ,and Iwan, 1994; Vrhel
and Trussel, 1992; García-Beltrán, Nieves, Hernández-Andrés, and Romero, 1998;
Tajima, 1998). Previous research was aimed at determining a smaller numbers of basis
vectors (or eigenvectors) dominating the data variation from a particular set of
measurements. These basis vectors are usually an acceptable representation of the original
objects. Seemingly, the greater the number of basis vectors, the higher the accuracy of the
spectral reconstruction. Applications based on PCA have been generated to characterize
multispectral CCD cameras for capturing spectral reflectance factors of a scene (Burns,
1997; Burns and Berns, 1997a; Burns and Berns, 1997b; Burns and Berns, 1996;
Haneishi, Hasegawa, Tsumura, and Miyake, 1997).
Principal Component Analysis (PCA)
Principal component analysis explores the variance-covariance or correlation
structure of a sample set in vector form. It primarily serves the purpose of data (or
dimensionality) reduction and interpretation (Johnson and Wichern, 1992). Data reduction
is accomplished by neglecting the unimportant directions along where samples’ variances
are insignificantly small. Since major sample variations are along several significant
directions, the number of these directions approximates the dimensionality of the sample
35
set. Figure 2-4 demonstrates a set of two dimensional samples migrating along a
significant direction in which i = (1, 0) and j = (0, 1) are two standard basis vectors of the
Cartesian coordinate system and e1 and e2 are two orthonormal vectors specifying the
directions of a rotated coordinate system.
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
x axis with standard basis vector i=(1,0)
y axis
with
stan
dard
bas
is ve
ctor j
=(0,
1)
Fig. 2-4: Scatter plot of a two dimensional sample set.
e2
Sample mean
e1
36
Figure 2-4 indicates strongly that samples vary significantly along direction of e1,
whereas, direction of e2 accounts for significantly less variation. The variation along
directions of e1 and e2 reveals the covariance structure between samples. Let i and j be the
directions for feature measurements in terms of, for example, reflectance factors separately
measured at two sampled wavelengths. The corresponding coefficients, x and y, of i and j
can be treated as two random variables of a random measurement at the two features,
denoted as feature x and feature y, which describe an arbitrary sample. The example set of
two dimensional samples in Fig. 2-4 specifies one trend that as feature x of the samples
increases, feature y increases, and vise versa. Hence, features x and y are positively
correlated. The other trend is that as the feature x increases, the feature y decreases, and
vise versa, indicating that features x and y are negatively correlated. The direction of e1 is
the weighted average (or weighted sum) direction of x and y, and the direction of e2 is the
contrast (or weighted difference) direction of x and y. Observation for algebraic
interpretation can be made by finding a line equation of ax + by = c which is parallel to the
direction of e1 where a and b are positive real numbers and c is a real number. Thus, a
sample along the direction of e1 is exactly described by the weighted sum of features x and
y as the equation ax + by = c shows. Similarly, a line equation of dx - ey = f can be found
to be parallel to the direction of e2 where d and e are positive real number and f is a real
number. Hence, a sample along e2 is exactly described by the difference of weighted
features x and y. These justifications are useful for data interpretation depending on the
37
physical meanings of the features x and y (recall that x and y are representing the
reflectance factors measured at any two different sampled wavelengths for this example).
Since there is less variation along e2, it might be considered as an insignificant direction (or
dimension). The entire two dimensional sample set can be approximated by modulating the
vector, e1. Thus, data (dimensionality) reduction is achieved by eliminating the dimension
with negligible variance.
The e1 and e2 are known as the two eigenvectors of the two dimensional sample
set. Having samples in Fig. 2-4 represented in the new (or eigenvector) coordinate system
whose basis vectors are e1 and e2, then e1 and e2, as opposed to the standard basis vectors
in ℜℜℜℜ2, become two new metrics of feature measurements describing each sample. Each
sample has been relabeled by the coefficients of the linear combination of e1 and e2. That
is, if a sample P = xi + yj = le1 + me2 where x, y, l, and m are real numbers, then P is
labeled as (x, y) in the Cartesian coordinate system and labeled as (l, m) in the eigenvector
coordinate system. Figure 2-5 depicts the appearance of sample set represented in the
eigenvector coordinate system. The sample set in the eigenvector coordinate system is
decorrelated, i.e., the l and m features of a sample are uncorrelated. Hence, the modulation
of e1 is not affecting the modulation of e2.
38
-1 -0.5 0 0.5 1 1.5 2 2.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
l axis with eigenvector vector e1
m a
xis
with
eig
enve
ctor
vec
tor
e2
Fig. 2-5: The representation of sample set of Figure 4 in the eigenvector coordinatesystem.
Given a sample set, V, with q number of samples and n features, the sample
variance-covariance matrix of size nxn is calculated as
S V V V V=−
− −=∑1
1 1q ii
q
iT( )( ) , (2-22)
where V is the mean vector of size nx1 of the sample set. Notice that n features specify
that the sample set is in an n dimensional vector space. Diagonal entries of S are variances
measured along the standard basis vectors, i1, i2, i3,…, in, in the n dimensional Cartesian
39
coordinate system. S can be further transformed to a diagonal matrix, ΛΛΛΛ, for uncovering
the variances along the significant directions. Equation (2-23) specifies the matrix
diagonalization transformation.
ΛΛΛΛ = − −( )E SET 1 1 , (2-23)
where E is an nxn matrix whose n column vectors are the eigenvectors of S. The
transformation by Eq. (2-23) decorrelates the sample variance-covariance matrix, S, of a
sample set according to the eigenvectors such that the new sample representations are
uncorrelated and the corresponding variance are maximized along the basis vectors, e1, e2,
e3,…, en, in the eigenvector coordinate system. The off-diagonal terms of ΛΛΛΛ are zeros
which further symbolize the sample features represented in the eigenvector coordinate
system are uncorrelated. The diagonal terms, L1, L2, L3,…, Ln, of ΛΛΛΛ, the so called
eigenvalues, are the variances measured with respect to the eigenvectors. Eigenvalues of
the sample set can be attained from the sample variance-covariance matrix by the
characteristic equation,
det( )L ⋅ − =I S 0, (2-24)
where det is the determinant of a square matrix, L is an arbitrary eigenvalue, and I is the
nxn identity matrix. There are n eigenvalues solved by Eq. (2-24). With these eigenvalues,
their corresponding eigenvectors can be derived by Eq. (2-23).
Eigenvalues can be zeros when a set of sample, V, is actually distributed in an m-
dimensional subspace, where m ≤ n. Eigenvectors whose corresponding eigenvalues are
40
zero explain no variance. Thus, an arbitrary sample, Vsample, in the sample set, V, can be
exactly described by a linear combination of the significant m eigenvectors derived from
the variance-covariance matrix of V, i.e.,
V esample ii
m
ib==
∑1
, (2-25)
where bi is the coefficient for reconstructing Vsample. In this case, m eigenvectors explain
hundred percent of the variance. The percent variance explained by m eigenvectors is
calculated by
Percent Variance
L
L
ii
m
ii
n= =
=
∑
∑1
1
% . (2-26)
Percent variance can be used as a gauge to estimate the dimensionality of a sample set.
Given an n dimensional sample set, if 99.00% (depends on the type of applications and the
required accuracy for reconstruction) of total variance is explained by m eigenvectors
where m ≤ n then the sample set can be approximately reconstructed by the significant m
eigenvectors together with the sample mean vector, V sample mean, i.e.,
V e Vsample ii
m
i sample meanb≅ +=∑
1
. (2-27)
Therefore, data reduction is accomplished by an m-dimensional approximation.
Principal components are often mistakenly referred to as eigenvectors (Vrhel and
Trussel, 1992; Tajima, 1998; Haneishi, Hasegawa, Tsumura, and Miyake, 1997; García-
41
Beltrán, Nieves, Hernández-Andrés , and Romero, 1998). The following is given to clarify
the difference between principal components and eigenvectors. Having eigenvectors, e1,
e2, e3,…, en, and their corresponding eigenvalues, L1, L2, L3,…, Ln such that L1 ≥ L2 ≥ L3
≥ … ≥ Ln, principal components are defined as the new coordinates of samples onto
eigenvector coordinate system. Hence, there are n principal components for an n
dimensional sample set. The ith principal component, Pci, is the collection by projecting
samples, V, of size nxp, onto the ith eigenvector, i.e.,
Pc e Vi iT= , (2-28)
where i = 1…n. Hence, the whole sample set is decomposed into n uncorrlated
components, Pc1, Pc2, … Pcn, by Eq. (2-28) whose variances are exactly the n
eigenvalues, L1, L2, … ,Ln. Assuming the data in Table 2-1 is the measured spectral
absorption coefficients, shown up to five samples, sampled between 400 nm and 700 nm at
each 20 nm interval. The principal components calculated by Eq. (2-28) are shown in
Table 2-2.
Table 2-1: A sample set (shown up to five samples) of absorption measurements from 400nm to 700 nm at 20 nm intervals.
Wavelength Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Variance400 0.569 0.334 0.360 0.663 0.921 0.058420 0.757 0.556 0.650 1.127 1.552 0.169440 0.670 0.551 0.693 1.230 1.732 0.248460 0.642 0.595 0.739 1.267 1.743 0.246480 0.812 0.814 0.906 1.353 1.628 0.136500 1.127 1.161 1.186 1.548 1.509 0.042520 1.439 1.481 1.451 1.763 1.467 0.019540 1.618 1.619 1.557 1.840 1.432 0.022
42
560 1.563 1.463 1.380 1.621 1.246 0.022580 1.326 1.042 0.944 1.102 0.866 0.031600 1.222 0.719 0.581 0.679 0.550 0.074620 1.267 0.577 0.387 0.461 0.384 0.139640 1.341 0.520 0.281 0.346 0.297 0.201660 1.377 0.499 0.232 0.294 0.258 0.234680 1.322 0.469 0.198 0.256 0.228 0.225700 1.178 0.417 0.166 0.219 0.197 0.182
Table 2-2: The principal components as the new coordinates in the eigenvector coordinatesystem of the samples in Table 2-1.
Principal component Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 VariancePc1 4.582 3.063 2.725 3.661 3.701 0.504Pc2 0.066 -1.066 -1.573 -2.348 -2.774 1.245Pc3 -1.226 -1.538 -1.381 -1.211 -0.163 0.294Pc4 0.069 0.052 -0.016 -0.043 -0.072 0.004Pc5 -0.001 -0.050 -0.079 -0.051 -0.031 0.001Pc6 -0.012 -0.018 -0.020 -0.003 0.007 0.000Pc7 -0.003 0.000 0.012 0.005 -0.001 0.000Pc8 0.021 0.017 0.024 0.022 0.022 0.000Pc9 -0.004 -0.012 -0.001 0.005 0.001 0.000Pc10 0.001 0.005 0.003 0.004 0.002 0.000Pc11 0.003 0.003 0.005 0.003 0.003 0.000Pc12 -0.001 -0.002 0.003 0.001 0.002 0.000Pc13 -0.001 -0.001 -0.001 -0.001 0.000 0.000Pc14 0.000 0.001 -0.001 0.000 0.000 0.000Pc15 0.000 -0.002 -0.001 -0.001 -0.001 0.000Pc16 0.002 0.002 0.001 0.002 0.001 0.000
Variances along standard basis (or absorption measured at each sampled wavelength) and
along eigenvectors are listed as the last of columns of Tables 2-1 and 2-2, respectively.
Variances along each sampled wavelength are not informative. Attention is specially paid
in the eigenvector coordinate system. Large percentage of variances explained by the first
three principal components retain as much information as that of Table 2-1. Thus, the
sample information is approximately preserved by retaining only three eigenvectors and
43
three principal components, as a consequence, data reduction is achieved. That is, the five
samples can be approximated by the linear combination of the first three eigenvectors
together with the first three principal components as the coefficients of reconstruction.
The further justification to explore the concept of the last sentence. Let E equal
the nxn matrix of eigenvectors as the column vectors and Pc equal the nxp matrix of
principal components of a sample set, V. The matrix E is known as an orthogonal matrix
with properties such that
EET = ETE = I , (2-29)
where I is the nxn identity matrix. Equation (2-28) is rewritten as
Pc = ETV. (2-30)
Multiplying E on sides of Eq. (2-30) yielded
EPc = EETV = IV = V . (2-31)
Hence, an arbitrary sample set, V, can be reconstructed by linear combinations of
eigenvectors modulated by their corresponding principal components as claimed. The
principal components are not eigenvectors but exactly the coefficients, bi, used in Eqs. (2-
25) and (2-27).
Principal Component Analysis for Color Science Applications
Color science research employing PCA was published as early as 1954 (Morris and
Morrissey, 1954). This early research devised an objective method for determining
equivalent neutral densities of color film images. The basic concept was to determine the
44
minimum number of spectral density curves as the basis vectors necessary to account for
spectral variability of typical processed film with different exposures. They calculated the
three most significant eigenvectors and corresponding eigenvalues based on forty-two
patches of Ektachrome film. Then the three eigenvectors were used to spectrally
reconstruct the spectral densities of single-layer-coated films (e.g., cyan, or magenta, or
yellow dyes). The eigenvector-fitted dye density spectra were considered as the “assumed
dyes” which were the closest representations of spectral densities of three dyes accounting
for the minimum residual variance from normal manufacturing and processing.
Later, Simonds, in a landmark article, exemplified the applications of PCA for
photographic and optical response data (Simonds, 1963). He showed a visualization of
how a measured response curve was decomposed by six curves, that is, the response curve
can be synthesized by a linear combination of the six curves provided by his
demonstration. Mathematical basics and numerical examples were also discussed.
The spectral distribution of typical daylight as a function of correlated color
temperature was determined using PCA (Judd, MacAdam, and Wyszecki, 1964). The
compiled six hundred and twenty-two samples of daylight (249 daylight measurements
from Rochester, United States, 274 daylight measurements from Enfield, England, and 99
daylight measurements from Ottawa, Canada) were used to derive the mean daylight
vector and two eigenvectors of daylight, shown in Fig. 2-6. By Eq. (2-27), the mean and
first two eigenvectors were used to reconstruct the daylight distributions, plotted in Fig. 2-
45
7, at correlated color temperatures 4800 K, 5500 K, 6500 K, 7500 K, and 10000 K by
specially chosen scalar multiples in Eq. (2-27), bi, for reproducing exactly the same
chromaticities of the specified correlated color temperatures.
-200
0
200
400
600
800
1000
1200
1400
300
330
360
390
420
450
480
510
540
570
600
630
660
690
720
750
780
810
Wavelength
Rel
ativ
e irr
adia
nce
Fig. 2-6: The mean (thick line), the first eigenvector (thin line), and the second eigenvector(dashed line) of the 622 daylight measurements (Judd, MacAdam, and Wyszecki, 1964).
Figure 2-6 reveals three aspects of important information. First, the average
daylight is blue-greenish. Second, the first eigenvector indicates that, among six hundred
and twenty-two measurements, daylight spectral distributions vary mainly along the
yellow-blue direction which corresponds to the sky colors changing from yellowish to
bluish during a day. Finally, the second eigenvector explains the greenish to purplish
variation of daylight. This green-purple variation, according to the interpretation of Judd
et al., may be caused by the water vapor in the sky.
46
0
200
400
600
800
1000
1200
1400
1600
1800
300
330
360
390
420
450
480
510
540
570
600
630
660
690
720
750
780
810
Wavelength
Rel
ativ
e irr
adia
nce
Fig. 2-7: Spectral distribution of typical daylight at correlated color temperatures 4800 K,5500 K, 6500 K, 7500 K, and 10,000 K (Judd, MacAdam, and Wyszecki, 1964).
Interpretations for Coloration Processing Applications by PCA
Assuming objects are measured within the visible spectrum between 400 nm and
700 nm at 10 nm intervals, every measured sample is a vector of thirty-one components.
Thus, a measured sample set is thirty-one dimensional. Vector representations of colored
objects will adopt a type of spectral information whose vector addition and scalar
multiplication are defined. Hence, temporarily, the matrix-vector notation, ΦΦΦΦ, is employed
as the additive and scalar multiplicative spectral information. Consider three sets of
spectral absorption spectra of cyan, magenta, and yellow dyes modulated at different
concentrations, depicted in Fig. 2-8.
4800 K5500 K
6500 K
7500 K
10000 K
47
4 00 5 00 6 00 7 000
0 .5
1
1 .5
2
2 .5
3
Ab
so
rpta
nc
e
4 00 5 00 6 00 7 000
0 .5
1
1 .5
2
2 .5
W ave leng th4 00 5 00 6 00 7 000
0 .5
1
1 .5
2
2 .5
Fig. 2-8: Three sets of absorption spectra of cyan, magenta, and yellow dyes (left to right)at eleven different concentrations.
A “virtual” sample set, Φλ,mixture can be created by each combination of spectra at six
different percentages (0%, 20%, 40% 60%, 80% and 100%) of the three. Thus, the a
priori knowledge about this virtual sample set of two hundred and sixteen samples is that
it is theoretically distributed in a three dimensional subspace of ℜℜℜℜ31 since it is formed from
a combination of three dyes. Hence, visualizing in a thirty-one dimensional space, the
underlying variations of three-dye mixtures are along the cyan, magenta, and yellow
dimensions as shown in Fig. 2-8. Spectra of cyan, magenta, and yellow dyes at one
hundred percent concentration, Φλ,cyan, Φλ,magenta, and Φλ,yellow, respectively, are the basis
vectors of the three-dye coordinate system. Spectra of mixtures can not be easily
48
decomposed with respect to Φλ,cyan, Φλ,magenta, and Φλ,yellow if they are initially unknown.
Alternatively, if the Φλ,mixture could be scatter plotted in ℜℜℜℜ31 then the appearance of the
sample set would have samples varying along one weighted average direction and several
contrast directions, as mentioned previously. Since the visualization is impossible for a
thirty-one dimensional scatter plot, the feasible alternative is to observe the spectral curves
of the eigenvectors, shown in Fig. 2-9, derived from Φλ,mixture (spectral absorption
coefficients in this case).
400 500 600 700-1
-0.5
0
0.5
1The 1st eigenvector
Ab
sorp
tanc
e
400 500 600 700-1
-0.5
0
0.5
1The 2nd eigenvector
400 500 600 700-1
-0.5
0
0.5
1The 3rd eigenvector
400 500 600 700-1
-0.5
0
0.5
1The 4th eigenvector
Wavelength
Ab
sorp
tanc
e
400 500 600 700-1
-0.5
0
0.5
1The 5th eigenvector
Wavelength400 500 600 700
-1
-0.5
0
0.5
1The 6th eigenvector
Wavelength
Fig. 2-9: The first six eigenvectors obtained for the virtual sample set. (As many as thirty-one eigenvectors can be shown.)
The first eigenvector points out the weighted average direction along which the
samples are distributed. This implies that there are neutral colors in the Φλ,mixture judged by
49
the appearance of a flat spectrum, i. e., majority of samples are distributed along the
direction of weighted-average absoprtion. The second eigenvector reveals that samples
vary from short wavelength regions to long wavelength regions indicating the existence of
bluish, blue-greenish, yellowish, and reddish samples in Φλ,mixture. Similarly, the third
eigenvector describes that samples varying along the middle wavelength regions and the
corresponding complementary spectral regions indicates the existence of greenish and
purplish color samples in Φλ,mixture. These three eigenvectors explain one hundred percent
of total variance and the corresponding directions reveal the color information of samples
according to the previous interpretation. The rest of the eigenvectors (the fourth to sixth
are plotted to demonstrate their content) are associated with zero variance, that is, no
sample varies along such directions. Therefore, the interpretation of them is meaningless.
In practice, a measured sample set does not provide the statistical results of exactly the
same dimensionality as the known number of colorants used to construct the sample set.
For example, Fig. 2-10 depicts the six eigenvectors determined from the IT8.7/2 reflection
target, Kodak Q60C for Ektacolor paper, a test target of two hundred and sixty-four color
patches sampling the photographic paper’s color gamut (McDowell, 1993).
50
400 500 600 700-1
-0.5
0
0.5
1The 1st eigenvector
Ab
so
rpta
nce
400 500 600 700-1
-0.5
0
0.5
1The 2nd eigenvector
400 500 600 700-1
-0.5
0
0.5
1The 3rd eigenvector
400 500 600 700-1
-0.5
0
0.5
1The 4th eigenvector
Wavelength
Ab
so
rpta
nce
400 500 600 700-1
-0.5
0
0.5
1The 5th eigenvector
Wavelength400 500 600 700
-1
-0.5
0
0.5
1The 6th eigenvector
Wavelength
Fig. 2-10: The six eigenvectors obtained from an IT8.7/2 reflection target.
The first three eigenvectors explain 99.96% of total variance. The interpretation of the
first three eigenvectors was described previously. Interest is focused on the extra
dimensions of the statistical estimation. As the variance explained by the ith eigenvector
decreases, the oscillating appearance of the corresponding eigenvector increases. This can
be realized by inspecting the fourth to the ith eigenvectors with nonzero variance.
Explanations of the high degree of oscillation can be attributed to noise caused by
manufacturing, photographic processing, and spectrophotometric measurements. Since the
rapidly oscillating eigenvectors indicate the directions of samples with spectral properties
varying rapidly across the neighborhood spectral regions, no such colorants exist in the
51
physical world. Such eigenvectors must describe the noise behavior of the sample set.
Fortunately, the noisy behavior only contributes to insignificant statistical results. Thus,
the reconstruction of a sample set achieved by the significant eigenvectors fulfills the goal
of data (dimensionality) reduction.
Multivariate Normality for Effective Data Reduction
It is worth speculating on the multivariate normality of a given set of multivariate
measurements although it is not a requirement for deriving eigenvector-eigenvalue pairs.
Multivariate normality ensures that eigenvectors and the associated eigenvalues derived
from measured samples is close to the eigenvector-eigenvalue pairs of the entire
population since manufacturing and sampling procedures may not be optimally performed
(Johnson and Wichern, 1992a; Anderson, 1984; Anderson, 1963; Grishick, 1939). If a
population is multivariate normally distributed then the 99% of the population should
distribute inside an ellipse, ellipsoid, and hyper-ellipsoid for two, three, and higher
dimensional populations, respectively. Half lengths of the axes of an ellipsoid in an
eigenvector coordinate system are directly proportional to the corresponding eigenvalues.
If a population is not of ellipsoidal shape then data reduction may not be optimally applied
for reconstructing the samples from this population since the non-ellipsoidal shape of the
population may imply a bizarre structure among the random variables representing the
sample features (in this case, the random variable can be, for instance, the Φλ measured at
52
each sampled wavelength). To give a clearer insight, a set of 500 random samples was
generated from a bivariate uniform distribution, plotted in Fig. 2-11.
0 1 2 3 4 5 60
1
2
3
4
5
6
7
Random variable X
Ran
dom
var
iabl
e Y
Fig. 2-11: Scatter plot of the 500 random samples and their eigenvectors from a uniformdistribution.
The corresponding sample variance-covariance matrix and mean were calculated. A 99%
confidence region of an ellipse with two eigenvectors given by the very sample variance-
covariance matrix and mean to form a bivariate normal distribution are also super imposed
on to the 500 random samples in Fig. 2-11. Assuming another set of 500 samples are from
e1
e2
53
a bivariate normal distribution of the same covariance matrix and sample mean, then 99%
of them should be located inside the elliptical confidence region. By design, these two sets
of samples have the identical eigenvector-eigenvalue pairs. Once data reduction is required
by the first eigenvector approximation, the total error of the normal sample set by the first
eigenvector approximation is less that of uniform sample set since the total error is
proportional to their corresponding areas.
When the calculated eigenvectors with their corresponding eigenvalues explaining
small variances, which are considered as insignificant, are overlooked, the overlooked
eigenvectors may still be important for reconstruction due to not knowing how the
samples are distributed. The corresponding eigenvalues can not provide correct
information for data (dimensionality) reduction; that is, the estimation of the true
dimensionality based on the information of the total explained variance calculated by Eq.
(2-26) is not objective. Therefore, the information given by the calculation of percent
variance is less informative by non-normal multivariate distributions for the decision of
data (dimensionality) reduction.
It is required to check on two necessary conditions for the normality of a sample
set. First is the normality of the marginal distributions (spectral information measured at
each sampled wavelength) since all linear combinations of normal distributions are also
normal (Johnson and Wichern, 1992b). The normality of a univariate, often a marginal,
distribution can be inspected by the Q-Q plot or gamma plot for a bivariate distribution
54
(Johnson and Wichern, 1992c). The degree of normality can be specified by the
correlation coefficient of the Q-Q plot and the Chi-squire distance for univariate and
bivariate distributions, respectively. The normal plot procedure provided in MATLAB was
utilized for visual inspection. A virtual normal marginal distribution of one thousand
samples was generated by linear combinations of the spectral absorption coefficients of
three linearly independent colorant vectors. If an examined set is normally distributed then
the corresponding normal plot is a straight line.
As an example, Fig. 2-12 depicts the normal plot of absorption coefficients at 500
nm. The coefficients of linear combinations were generated from a normal random number
generator. The normal plot of the absorption coefficients at each sampled wavelength
appears to be straight lines as expected. Non-normal population of one thousand samples,
whose normal plot of absorption coefficients at 500 nm is shown as Fig. 2-13, was also
obtained by linear combinations of the same colorant vectors as that of Fig. 2-12 whose
coefficients came from an exponential distribution. The non-normal marginal distribution
reveals curvature in the normal plot. The shape of the sample set with non-normal
marginal distributions can not be ellipsoidal in a higher dimensional space by the
implications of the curvatures of marginal distributions in the normal plot. It can be
expected that most errors yielded of the reconstructed population by PCA are from those
samples deviating from the estimated straight line of their corresponding normal plot.
55
2 4 6 8 1 0 1 2 1 4
0 . 0 0 10 . 0 0 3
0 . 0 1 0 . 0 2
0 . 0 5
0 . 1 0
0 . 2 5
0 . 5 0
0 . 7 5
0 . 9 0
0 . 9 5
0 . 9 8 0 . 9 9
0 . 9 9 70 . 9 9 9
A b s o r p t io n c o e f f ic ie n t s a t 5 0 0 nm
Pro
ba
bili
ty
Fig. 2-12: Normal probability plot of absorption coefficients at 500 nm from the virtualnormal population obtained by linear combinations of three linearly independent dyes.
0 5 1 0 1 5 2 0 2 5 3 0
0 .0 0 10 .0 0 3
0 .0 1 0 .0 2
0 .0 5
0 .1 0
0 .2 5
0 .5 0
0 .7 5
0 .9 0
0 .9 5
0 .9 8 0 .9 9
0 .9 9 70 .9 9 9
A b s o r p t io n c o e f f ic ie nts a t 5 0 0 nm
Pro
ba
bili
ty
Fig. 2-13: Normal probability plot of absorption coefficients at 500 nm of the exponentialpopulation obtained by linear combinations of three linearly independent dyes.
56
The second condition is determined by whether two dimensional scatter plots of
any measurements of any two random variables generate an elliptical appearance. For a set
of spectral measurements using sampled between 400 nm and 700 nm at 10 nm interval,
there are four hundred and forty-three ( C(31,2)-31=434) scatter plots to be examined. An
elliptical appearance of a two dimensional scatter plot reveal the existence of bivariate
normality. Figure 2-14 is depicted as an example by scatter plotting the absorption
coefficients at 550 nm versus absorption coefficients at 500 nm of the virtual normal
sample set. The elliptical appearance of the two dimensional scatter plot suggests that
absorption coefficients at 550 nm and 500 nm are bivariate normally distributed.
If the first and second conditions are met then univariate and bivariate normality
are generated. Whether or not it implies the multivariate normality of an examined sample
set, it is difficult to conclude inductively. Without the satisfaction of the first and second
conditions, the existence of multivariate normality of the examined sample set should be
denied. Non-normal samples can be further transformed to a set of representations with
more degrees of normality by logarithmic, power, or polynomial transformations (Johnson
and Wichern, 1992d).
57
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
Absorption coefficients at 500 nm
Abs
orpt
ion
coef
ficie
nts
at 5
50 n
m
Fig. 2-14: The two-dimensional scatter plot of absorption coefficients at 550 nm vs. 500nm from the virtual normal population obtained by linear combinations of three
independent dyes.
F. SPECTRAL PRINTING MODELS
Conventional CMYK halftone printing processes are the most popular color
reproduction processes for their relative inexpensive cost and reproduction speed. High
speed printing presses generate more than 10,000 prints per hour. A priceless artwork can
be massively reproduced in such a fashion at a relatively low cost. Since the goal of this
research is to build a six-color output system. A halftone printing process is a natural
choice.
58
In this section, the underlying physical phenomena of the color formation of a
halftone printing process will be discussed. Mathematical description for the halftone color
formation such as Murray-Davies and Neugebauer equations as well as the dot-gain effect
introduced by Yule-Nielsen will be shown.
Basic Assumptions
A halftone cell is designed by laying down a grid of pixels. The overall reflectance
factor over a halftone cell is theoretically the average result of the reflectance factor of the
pixels which are turned on, termed as on-pixels, and the reflectance factor of the pixels
which are off, termed as off-pixels, modulated by the percentage of on-pixels and the
percentage of off-pixels. A halftone cell with NxN addressibility is depicted in Fig. 2-15,
where N = 2.
Fig. 2-15: The outline of a 2x2 halftone cell.
The calculations for spectral reflectance of a halftone cell equation are based on the
following underlying assumptions:
1. The spectral reflectance factor of the primary colorant varies proportionally to
that of the primary at 100% area coverage.
i = 1 i = 2
i = 3 i = 4
59
2. Spectral reflectance factor is additive within the halftone cell.
3. Human eye cannot resolve the halftone cell.
These three assumptions lead to the spectral reflectance factor Rλ of a halftone cell,
conceptually, being the sum of the spectral reflectance factor of each primary color at
100% area coverage modulated by their corresponding fractional dot area coverage inside
the halftone cell. In practice, the analytical description is more complicated than this
simple concept. The analytical description of a halftone color formation depends on the
type of halftone cell and the alignment of dots. Generally, the so called dot-on-dot type of
halftone device can be analytically modeled by Murray-Davies theory, and the traditional
halftone dot placement by rotated-screen or stochastic dot formation can be analytically
described by Neugebauer theory together with Demichel’s probability model for dot
overlap.
Murray-Davies Theory
Figure 2-16 shows a halftone cell printed by a single ink on an opaque support can
be covered with different dot area coverage. It shows the notion of how the spectral
reflectance factor over a halftone cell varies with respect to the percentage of the area
covered by a single color ink. By the assumptions of additivity inside a halftone cell, the
spectral reflectance factor Rλ of a halftone cell printed on paper by a single color ink, can
be estimated based on the Murray-Davies equation,
R aR a R paperλ λ λ= + −, ,( )100% 1 , (2-32)
60
where a is the fractional dot area of a single color ink, Rλ,100% is the spectral reflectance
factor of the color ink at 100% dot area coverage, and Rλ,paper is the spectral reflectance
factor of the paper support (Murray, 1936).
11% area coverage 56% area coverage 100% area coverage
Fig. 2-16: The 3x3 halftone cell covered by three different dot areas.
The accuracy of the estimated spectral reflectance factor of a halftone cell, predicted by
the Murray-Davies equation, relies on linearity for each ink printed on a paper support.
Linearity is defined as that, for each primary, the reflectance factors should be summed up
according to their fractional dot areas. That is, if the fraction dot areas of a primary are
summed up to a percent, where 0 ≤ a ≤ 100, then the resultant reflectance factor is
summed up to a percent of the reflectance factor of the primary at 100% area coverage.
Due to the failure of linearity, the Murray-Davies equation can not provide adequate
prediction of the estimated spectral reflectance factor inside a halftone cell. The physical
phenomena that limit linearity are known as mechanical and optical dot-gain. To model
both mechanical and optical dot-gain, the Yule-Nielsen n-factor was employed to modify
61
the Murray-Davies equation (Yule and Nielsen, 1951). The n-factor modified Murray-
Davies Equation is shown in Eq. (2-33), where terms in the Eq. (2-33) are similarly
defined as those in Eq. (2-32).
R a Rnpapern n
λ λ λ= + −[aR ( ) ],/
,/
100%1 11 , (2-33)
Dot Gain
There are two sources of dot gain, mechanical and optical dot gain. Mechanical
dot gain is caused by printing ink submerging and spreading when delivered onto a paper
substrate causing the physical dot size to be changed. Figure 2-17 depicts the appearance
of mechanical dot gain at a microscopic level.
Ideal dot shape non-ideal dot shape aftermechnical dot gain
Fig. 2-17: Non-ideal halftone dot shapes vary due to the mechanical dot gain effect.
Optical dot gain effect is depicted in Fig. 2-18, known as the Yule-Nielsen effect, and
originates from light flux that penetrates the ink film, enters the paper substrate, scatters,
and finally emerges from non-ink area. Simultaneously, light flux can enter the paper
62
substrate and get scattered, penetrate the ink film, and finally emerge from the ink film.
These two physical phenomena cause printing models to predict smaller or larger dot size.
Paper Paper
Light
Dot
Fig. 2-18: The cause of optical dot gain.
Several dot gain models have been published to take into account that the dot gain
effect is a function of the size of halftone dots, scattering, and the surface reflection of ink
films (Arney, Engeldrum, and Zeng, 1995; Wedin and Kruse, 1995; Arney, Arney, and
Engeldrum, 1996). Arney et al., (1996) further determined the paper spread function, or
the modulation transfer function, by Fourier analysis.
Although the Yule-Nielsen n-factor is an empirical parameter, several research
efforts have shown that it is capable to maintain high accuracy of model fitting. Yule-
Nielsen n-factor is still widely employed for its simplicity (Viggano, 1985;
Balasubramanian, 1995; Iino and Berns, 1997; 1998a; 1998b; Pearson, 1980; Pope, 1989).
Neugebauer Theory
The Murray-Davies equation is only used for prediction of a single-color halftone
cell. It can not predict the spectral reflectance over a multiple-color halftone cell. A
multiple-color halftone cell is defined as an area covered by various printer primary inks.
63
Wide varieties of colors are produced by modulating the area coverage of printer primaries
in a halftone cell on an opaque medium such as paper. The spectral Neugebauer equation
can be used to predict spectral reflectance factor over a multiple-color halftone cell
(Neugebauer, 1937).
For a CMYK printing process, channels of white, cyan, magenta, yellow, red,
green, blue, three-color black, black, black-cyan, black magenta, black yellow, black-red,
black-green, black-blue, and four-color black are known as the sixteen Neugebauer
primaries. Neugebauer theory assumes that each channel of Neugebauer primary is linear
and color halftone dots are placed at random. Thus, the spectral reflectance factor over a
halftone cell, printed by a CMYK printer, is calculated by the sum of the fractional dot
area of each channel multiplied by the spectral reflectance of the corresponding
Neugebauer primary at 100% dot area coverage. The spectral Neugebauer equation is
defined as the following:
R a Rii
iλ λ==
∑1
16
, , (2-34)
where Rλ is the spectral reflectance factor of a halftone cell printed on a paper support,
Rλ,i is the spectral reflectance factor of the ith Neugebauer primary, and ai is the fractional
dot area coverage of the ith Neugebauer primary.
Fractional dot area coverage, ai, of the ith Neugebauer primary is calculated by
employing the Demichel’s dot-overlap model (Demichel, 1924; Yule, 1967; Kang, 1997).
The Demichel model is based on the assumption that dots are delivered on the substrate in
64
a random fashion. Hence, the overlap area of two inks is nothing more than the joint
probability of these two inks. There are 24 = 16 combinations to predict the Neugebauer
primaries in the four-color case. Non-overlapping portions are simply the CMYKW
primaries, two-color overlaps are defined as RGB secondary primaries, three-color
overlaps are tertiary primaries, and four-color overlap is essentially the unique four-color
black. The fractional dot areas for sixteen Neugebauer primaries are calculated as below:
White : a1 = (1 - c) (1 - m) (1 - y) (1-k)Cyan : a2 = c (1 - m) (1 - y) (1-k)Magenta : a3 = m (1 - c) (1 - y) (1-k)Yellow : a4 = y (1 - c) (1 - m) (1-k)Red : a5 = y m (1 - c) (1-k)Green : a6 = c y (1 - m) (1-k)Blue : a7 = c m (1 - y) (1-k)3-Color Black : a8 = c m y (1-k) (2-35)Black : a9 = k (1 - c) (1 - m) (1 - y)Black-Cyan : a10 = c k (1 - m) (1 - y)Black-Magenta : a11 = m k (1 - c) (1 - y)Black-Yellow : a12 = y k (1 - c) (1 - m)Black-Red : a13 = y m k (1 - c)Black-Green : a14 = c y k (1 - m)Black-Blue : a15 = c m k (1 - y)4-Color Black : a16 = c m y k
where c, m, y, and k are fractional dot areas of printer primaries within a halftone cell.
As the result of linearity failure caused by mechanical dot-gain, optical dot-gain, or both,
the spectral Neugebauer equation can not accurately predicte spectral reflectance over a
halftone cell. The spectral Neugebauer equation together with the Yule-Nielsen n-factor
65
again can be employed to improve the model accuracy which is shown as equation (2-36),
where terms in Eq. (2-36) are similarly defined as that of (2-34).
R a Rii
in n
λ λ==
∑[ ],/
1
161 , (2-36)
66
III. LINEAR COLORANT MIXING SPACES
As discussed in the previous chapter of theoretical background, researchers
frequently perform PCA using reflectance factor of color samples requiring reproduction.
Often, the number of significant dimensions (basis vectors) exceeds the number of physical
parameters, for example, a photographic system requires more than three dimensions for
spectral reconstruction. This contradicts the knowledge that photographic materials are
manufactured by three known dyes. Apparently, the color synthesis by PCA using
reflectance factor as the representation for photographic materials is not optimal. This
justification also applies for all analysis and synthesis using PCA on surface colors of all
types. Hence, a transformation to account for the real physical dimensions of a set of
measurements as well as to agree with the process of an opaque coloration is desirable.
Mathematical transformation based on Kubelka-Munk turbid media theory is an obvious
choice. The resultant space after Kubelka-Munk transformation is termed as Φ space for
simplicity through the rest of this dissertation. Upon our numerous attempts using Φ space
for opaque color analysis and synthesis, PCA failed to predict the reconstructed accuracy
with constrained number of dimensions. For example, a set of opaque mixtures was
created by mixing six opaque paints. Six eigenvectors of the set of opaque mixtures in Φ
space frequently fail to spectral reconstruct the low absorptive samples. Such phenomena
which are not realizable for physical materials are profound by the negative spectral
67
components as a result of six-eigenvector reconstruction. Since Kubelka-Munk
transformation is highly nonlinear, the consequence of inverse transforming the negative
spectral components is not interpretable.
Our primary goal is to employ or derive a transformation to ensure that the
dimensionality of the linear representation of a set of color samples agrees with their
physical dimensionality. During the process of searching and deriving the optimal
transformation, the fundamentals of PCA were thoroughly reviewed. It was discovered
that the multivariate normality of a sample set contributed to the efficiency of data
(dimensionality) reduction. This lead us to utility the multivariate normality as a beneficial
factor in designing the transformation.
A. REFLECTANCE AND ABSORPTION SPACES
Both reflection and absorption occur when opaque objects are exposed to
electromagnetic radiation. If there is no thermal or other energy loss then the total
reflected and absorbed energy should be identical to the total incident energy.
Quantitatively, there is a relationship between reflectance and absorption according to the
law of conservation of energy. Kubelka and Munk initiated the derivations of this
relationship (Kubelka and Munk, 1931; Kubelka, 1948). They found reflectance factor is a
function of the ratio of absorption coefficient, K, and scattering coefficient, S. The ratio,
K/S, is again denoted as Φ for simplicity and (K/S)λ as well as Φλ are denoted as their
68
spectral extension. It is also known that the ratio of the absorption coefficient and
scattering coefficient is approximately linear with respect to concentration (Allen, 1966;
Allen, 1980; Shah and Gandhi,1990). Colorant formulation can be performed by linear
combinations of (K/S)λ of colorants used for synthesis, i. e.,
(K/S) c (k/s),mixture i ,ii
n
λ λ==∑
1
, (3-1)
where c represents the concentration, k and s are the absorption and scattering coefficients
of a colorant normalized to unit concentration. Therefore, the absorption and scattering
properties of materials also provide a useful platform for color scientists to perform color
analysis and synthesis. Again, the vector space of Eq. (3-1) is denoted as Φ space for
compactness of terminology. Although Φλ has the advantageous linear property with
respect to concentration, interestingly, there is relatively little research concerned with
estimating spectral information by applying linear modeling techniques in Φ space.
Φ space related research by Ohta determined the three eigenvectors from
measurements of spectral densities of a photography material (Ohta, 1973). He then tried
to linearly transform the three eigenvectors to a set of all-positive vectors as the estimation
of the underlying real dye spectra. Once the basis density spectra are statistically
uncovered, any spectral density measured from the photographic material can be
synthesized with high colorimatric and spectral accuracy. Reconstructed density spectra
can be further transformed to the representations of spectral transmission or reflectance
69
factors. Berns and Shyu extensively carried out this process to estimate the spectral
transmission and reflectance factors of four photographic films (Berns and Shyu, 1995). In
order to validate the accuracy of their dye estimates based on PCA, a tristimulus matching
algorithm was used to achieve an exact match for CIE illuminant D50 and the 1931
observer (Allen, 1980). CIELAB color difference for illuminant A was used as an
metamerism index. Smaller indices indicated better spectral reconstruction. Accuracy for
the four photographic films tested, averages and maximum metamerism indices varied
from 0.1 to 0.3 ∆E*ab and 0.4 to 1.3 ∆E*
ab, respectively.
By the inspiration of these two researches, Φ space as opposed to reflectance
space might be a better alternative space to approximate natural scenes since Φ is
approximately linear with respect to concentration. The superiority between two spaces
depends on whether or not the spectral properties (reflectance factor or Φ) forming a
scene whose dimensionality distribute in each space agree with their physical
dimensionality. A linear colorant mixing space with larger degree of multivariate normality
can be further approximated with lower dimensionality. This is a beneficial factor when
research applications are frequently confined to use limited number of primary colorants
for color synthesis. This research project will be focusing on the comparisons among
spaces based on these two factors.
70
B. TRANSFORMATION BETWEEN REFLECTANCE AND ΦΦΦΦSPECTRA
The transformation between reflectance factor and Φ is often based upon Kubelka-
Munk turbid media theory. Equations (3-2) and (3-3) are used for opaque materials such
as acrylic paints and textiles, where the R λ,∞ is the spectral reflectance factor of an opaque
material.
Rλ λ λ λ,∞ = + − +1 22Φ Φ Φ (3-2)
Φ λ λ λ= − ∞ ∞( ) /, ,1 22R R (3-3)
Equations (3-4) and (3-5) are used for transparent color layer in optical contact with an
opaque support such as photographic paper,where Rλ, g is the spectral reflectance factor of
an opaque support and X is the thickness of the transparent colorant layer. Equation (3-5)
is the inverse transformation of Eq. (3-4) by assuming that the thickness, X, is unity.
lim ,S
gXR R e
→
−=0
2λ λ
λΦ (3-4)
lim . ln( ),
Sg
R
R→= −
00 5Φλ
λ
λ(3-5)
Kubleka-Munk turbid media theory is based on a two-flux assumption, that is, the
light in the colorant layer only become scattered upward or downward. No other
directional scattering is assumed. Hence, the Kubleka-Munk transformation itself is an
approximation of coloration processes (Van De Hulst, 1980; Nobbs, 1985). Accuracy is
71
quite reasonable for materials with optical characteristics modeled by Eqs. (3-4) and (3-5).
However, as this research progressed, it was discovered that the transformation for
opaque materials does not always describe the optical properties of mixtures formed by
the corresponding coloration. In retrospect, this leads to the violation of the two flux
assumption. A real material most frequently scatters light in all directions which causes the
failure of Eqs. (3-2) and (3-3). Furthermore, measurement by spectrophotometers with
non-optimal aperture sizes causes failure to the linear assumption in Φ space (Tzeng and
Berns, 1998a). Concentration is no longer linear with respect to Φλ. Consider a
spectrophotometer measuring the surface of a multicolor object. Its field of view may
cover several color surfaces as shown in Fig. 3-1.
Fig. 3-1: The possible field of view of a spectrophotometer.
In this case, the reading of the spectrophotometer is the result of spatial averaging inside
its field of view. This implies the additive operation has already been performed in
Spectrophotometer Sensor plane Field of view
Color object
72
reflectance space. Since the Kubelka-Munk opaque transformation is nonlinear, the
additivity of colorant vectors is, therefore, not well defined in Φ space.
In dealing with the failure of Kubelka-Munk turbid media theory, many more
theories utilizing multi-flux methods in solving radiation transfer problem have been
published by a number of authors for improving the predicting accuracy (Mudget and
Richards, 1971; Mudget and Richards, 1972; Maheu, Letoulouzan and Gouesbet, 1984;
Mehta and Shah, 1986a; Mehta and Shah, 1986b). However, these complex models,
despite their improved correlation with the true optics of colorant mixtures, usually
required considerable parameter optimization in order to result in acceptable accuracy. It
seems reasonable to directly derive an empirical transformation. The main concerns for the
derivation of an empirical space are: transforming a non-normal population to a near
normal population since the normality is a beneficial factor for data (dimensionalty)
reduction, obtaining a new colorant vector space with reduced dimensionality that
corresponds to the physical dimensionality of a given sample set, and the vector addition
and scalar multiplication in new vector space should approximately describe the process of
subtractive opaque coloration. Consider the subtractive opaque colorant mixing, the more
colorants that are added for coloration, the darker the resultant mixture is. A vector space
formed by adding reflectance factors is not realizable for opaque coloration. Thus, an
empirical equation was derived based on these restrictions.
73
The new near-normal as well as reduced dimensionality space, Ψ, and its inverse
transformation were determined and described by
Ψλ λ= −�
a R12 (3-6)
and
R aλ λ= −( )� Ψ 2 , (3-7)
where Ψλ represents the new linear vector representation of an opaque colorant and
�
�
a≅ 1 which is empirically determined from a set of samples requiring reproduction. The
process of optimizing �
a and the MATLAB programs are attached in Appendix B.
Determination of �
a is to perform the transformation by Eq. (3-6) such that the resultant
set of Ψλ is of the requested dimensionality upon color reproduction, i.e., the Ψλ for
photographic material should be three dimensional. The use of square root transformation
of the spectral reflectance factors improves normality (Johnson and Wichern, 1992) and
the offset term is required to account for a subtractive opaque coloration.
C. DIMENSIONALITY REDUCTION: NORMAL VERSUS NON-NORMAL POPULATIONS
The proof for multivariate normality used as a beneficial property for sample
reproduction requiring data (dimension) reduction is to construct both multivariate normal
and non-normal populations with known dimensionality and employ a fewer number of
eigenvectors for spectral reconstruction. The metric for judging the reconstruction
74
accuracy should be the RMS error of reflectance factors, Rλ. If the Ψ or Φ space is used
for linear modeling, then minimizing the RMS error of Ψλ or Φλ is not equivalent to
minimizing the RMS of Rλ since the transformations is nonlinear for the Ψ or Φ spaces. In
order to compare the reconstruction accuracy in terms of RMS error, the comparison for
the superiority between a normal and a non-normal set linearly modeled by PCA is
convenient using reflectance space. Hence, the simulation discussed next will use
reflectance space for color mixing though it is not realizable by subtractive coloration, and
the colorimetric error and metameric index can be shown. Colorimetric accuracy of
eigenvector reconstruction is specified by ∆E*94 color difference equation (CIE Technical
Report, 1995) under CIE standard illuminant D65 and 1931 standard observer. Spectral
accuracy is indicated by metameric index based on parameric correction (Fairman, 1987)
for D65 and the 1931 observer followed by a color difference calculation for illuminant A.
The parameric correction is equivalent to modulating colorant concentration such that an
exact colorimetric match is achieved for D65 and the color difference under A indicated
the degree of metamerism. The higher the metamerism index, the larger the degree of
spectral mismatch.
The multivariate normal sample set is generated by six linearly independent
colorant vectors modulated by coefficients which are randomly sampled from a beta
distribution with parameters α = 5 and β = 5 (Dougherty, 1990). Hence, the resultant
population, which is the linear combination of six bell-shape marginal distributions, is
75
approximately multivariate normally distributed. On the contrary, the same six colorant
vectors were modulated by coefficients which come from six beta distributions with six
different combinations of α and β to create a non-normal distribution. Sample sets of five
hundred six-colorant mixtures, one multivariate normal and one non-normal, were created
computationally. Normality is conformed by the inspection of its normal plot, shown in
Fig. 3-2, revealing that the each marginal distribution is approximately normally
distributed (straight lines). Non-normality is conformed by the non-normal marginal
distributions plotted in Fig. 3-3.
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.001
0.003
0.01 0.02
0.05
0.10
0.25
0.50
0.75
0.90
0.95
0.98 0.99
0.997
0.999
Reflectance factor
Pro
bab
ility
Fig. 3-2: The normal plot of simulated reflectance factors at each sample wavelengthobtained by linear combinations of six approximately normal distributions.
76
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.001
0.003
0.01 0.02
0.05
0.10
0.25
0.50
0.75
0.90
0.95
0.98 0.99
0.997
0.999
Reflectance factor
Pro
ba
bili
ty
Fig. 3-3:The normal plot of simulated reflectance factors at each sample wavelengthgenerated by linear combinations of six non-normal distributions.
Since the two sample sets are distributed in six dimensional spaces, six eigenvectors
should ideally span the entire six dimensional sample space. This simulation is to
demonstrate that it is possible to approximate the entire six dimensional sample space by
lower dimensional subspaces if they are multivariate normally distributed. Table 3-1 shows
the colorimetric and spectral accuracy of the multivariate normal and the non-normal
sample sets reconstructed with different dimensionalities at a random simulation. By
comparing the two sets, the dimensionality can be reduced to three dimensional for the
multivariate normal sample set indicated by low RMS error and satisfactory colorimatric
77
and spectral performance. Whereas, the non-normal sample set can not be effectively
approximated below four dimensional reconstruction.
Table 3-1: The colorimetric and spectral accuracy of the multivariate normal and non-normal sample sets reconstructed with different number of dimensions, where Stdev standsfor the standard deviation and RMS representing the total root mean square error of thereconstructed reflectance spectra.
Multivariate Normal Sample set∆E*
94 Metamerism Index (∆E*94)
Dimensionality six five four three six five four threeMean 0.0 0.0 0.8 0.9 0.0 0.0 0.1 0.3Stdev 0.0 0.0 0.4 0.6 0.0 0.0 0.0 0.2Maximum 0.0 0.1 2.0 2.9 0.0 0.0 0.2 1.3Minimum 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0RMS 0.000 0.004 0.005 0.008
Non-Normal Sample setDimensionality six five four three six five four threeMean 0.0 0.1 1.5 3.1 0.0 0.0 0.1 1.5SDV 0.0 0.0 0.8 1.3 0.0 0.0 0.1 0.7Maximum 0.0 0.2 4.9 7.1 0.0 0.0 0.5 3.2Minimum 0.0 0.0 0.0 0.2 0.0 0.0 0.0 0.1RMS 0.000 0.007 0.009 0.023
D. VERIFICATIONS
The first sample set used was the ANSI IT8.7/2 reflection target measured with a
Gretag SPM 60 spectrophotometer. Since the IT8.7/2 reflection target is manufactured
with three dyes, dimensionality should be theoretically three in a vector space via a
suitable transformation. The Kubelka-Munk transformation for a transparent colorant
layer in optical contact with an opaque support, Eqs. (3-4) and (3-5), was utilized. Table
78
3-2 shows the percent variance (PV) of the significant eigenvectors calculated in both
reflectance and absorption space of the IT8.7/2 reflection target.
Table 3-2: The percent variance by eigenvector analysis in both reflectance and absorptionspace of IT8.7/2 reflection target.
Reflectance AbsorptionThe ith
EigenvectorEigenvalue Percent
Variance(PV) %
CumulativePV %
Eigenvalue PercentVariance(PV) %
CumulativePV %
1 0.9474 80.43 80.43 7.5702 74.25 74.252 0.1845 15.66 96.09 1.9038 18.67 92.923 0.0415 0.35 99.62 0.7172 7.03 99.964 0.0033 0.28 99.905 0.0006 0.05 99.95
PCA requires five eigenvectors in reflectance space to explain the same variance of
IT8.7/2 reflection target as that of absorption space with three eigenvectors. If Eq. (3-5)
transforms the measured spectral reflectance factor of the IT8.7/2 reflection target into the
representation of spectral absorption whose dimensionality is exactly three, then the
multivariate normality of IT8.7/2 reflection target in absorption space is not crucial since
three eigenvectors in absorption space already span the entire absorption space of IT8.7/2
reflection target, i.e., no data (dimensionality) reduction is required. Since the spectral
measurements in absorption space for the IT8.7/2 reflection target is approximately three
dimensional, properly modeled by Eqs. (3-4) and (3-5), the multivariate normality in
absorption space is viewed as less crucial for three-eigenvector reconstruction. On the
contrary, the dimensionality of the reflectance space of IT8.7/2 reflection target is
79
obviously beyond three dimensions. In order to achieve high colorimatric and spectral
accuracy by three-eigenvector reconstruction in reflectance space, the multivariate
normality in reflectance space for the IT8.7/2 reflection target is crucial for data
(dimension) reduction. The normality plot of reflectance factors at each sample
wavelength (marginal distribution) is shown in Fig. 3-4.
0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8
0 .003
0.01 0 .02
0 .05
0 .10
0 .25
0 .50
0 .75
0 .90
0 .95
0 .98 0 .99
0 .997
R efle ctanc e fac to r
Pro
ba
bili
ty
Fig. 3-4: The normal plot of marginal distributions of IT8.7/2 reflection target inreflectance space.
Since the multivariate-normality is less crucial for IT8.7/2 reflection target in
absorption space, the attention is focused on reflectance space. Recall the necessary
condition of multivariate normal distribution is that the marginal distribution must be
normal. By inspecting Fig. 3-4, the degree of normality is low as shown by the large
amount of curvature. It is expected that three-eigenvector reconstruction in reflectance
80
will not yield satisfactory accuracy. The performance of three-eigenvector reconstruction
and the statistical results of colorimetric and spectral metrics in both spaces are listed in
Table 3-3.
Table 3-3: The colorimetric and spectral performance of three eigenvector reconstructionin both spaces for an IT8.7/2 reflection target.
Reflectance Absorption∆E*
94 Metamerism Index ∆E*94 Metamerism Index
Mean 1.8 0.6 0.5 0.1Stdev 1.5 0.6 0.2 0.1Maximum 8.5 3.6 1.0 0.4Minimum 0.0 0.0 0.0 0.0RMS 0.014 0.006
From Table 3-3, the colorimetric and spectral accuracy is far superior in absorption space
compared with reflectance space. Tables 3-2 and 3-3 assure that Kubelka-Munk
equations, Eqs. (3-4) and (3-5), are suitable transformations to model the IT8.7/2
reflection target. The accuracy by three-eigenvector reconstruction in absorption space
agrees with the fact that the IT8.7/2 reflection target is comprised of three dyes. Modeling
the IT8.7/2 reflection target should be a three dimensional problem instead of five
dimensional. The ideal reproduction should be achieved using three eigenvectors since the
IT8.7/2 reflection target is three dimensional. Whereas, the measurement noise and
validity of Kubelka-Munk transparent transformation affect the accuracy.
The analysis for IT8.7/2 is based on the Kubelka-Munk turbid media theory while
color materials behave similarly to the underlying theoretical requirements. However,
81
when the absorption and scattering behavior of the object violates the two flux
assumption, spatial averaging occurs due to a large field of view, noise contributes to the
measurements, and objects are not completely opaque, the Kubelka-Munk opaque
transformations Eqs. (3-2) and (3-3) are no longer valid. The proposed transformation
methods, shown as Eqs. (3-6) and (3-7), are employed.
The second verification was performed on a paint target of 141 patches created by
mixing six linearly independent Galeria acrylic colors which were measured using a
Macbeth Color-Eye 7000 integrating sphere spectrophotometer with specular component
included. By knowing that this set of samples is theoretically six dimensional, the spectral
reproductions by the six significant eigenvectros in R (reflectance), Ψ, and Φ space were
performed. The colorimetric and spectral accuracy, RMS, and percent variance (PV)
explained by the six significant eigenvectros are tabulated in Table 3-4. Six-eigenvector
reconstruction in Ψ space yielded the best performance in light of metameric and RMS
error. Interestingly, the percent variance explained by six-eigenvectors in Φ space is the
highest (99.92%) among the three, whereas, the spectral reconstruction yielded the highest
RMS error, hence, the lowest colorimetric and spectral accuracy.
Table 3-4: The statistical performance by six-eigenvector reconstruction for 141 acryliccolors in R, Ψ and Φ spaces.
∆E*94 Metamerism Index (∆E*
94)Space Mean SDV Max Min Mean SDV Max Min RMS PV (%)
R 0.4 0.3 1.6 0.0 0.2 0.2 1.0 0.0 0.010 99.84Ψ 0.3 0.2 0.8 0.0 0.1 0.1 0.7 0.0 0.007 99.85Φ 1.0 1.7 6.6 0.0 0.3 0.4 1.8 0.0 0.026 99.92
82
Another verification was performed on a set of 105 color patches created by
another set of six linearly independent colorants (two Sakura poster colors and four Pentel
poster colors) using the same measuring instrument and geometry described above. The
colorimetric and spectral accuracy, RMS, and percent variance explained by the six
significant eigenvectros are shown in Table 3-5.
Table 3-5: The statistical performance by six-eigenvector reconstruction for 105 postercolors in R, Ψ ,and Φ spaces.
∆E*94 Metamerism Index (∆E*
94)Space Mean SDV Max Min Mean SDV Max Min RMS PV (%)
R 1.0 0.7 2.8 0.1 0.3 0.2 0.7 0.0 0.012 99.60Ψ 0.5 0.3 1.1 0.1 0.2 0.1 0.4 0.0 0.007 99.70Φ 0.5 0.3 1.9 0.0 0.1 0.1 0.5 0.0 0.012 99.98
This results is again showing that even though the percent variance explained by six
eigenvectors is the highest in Φ space (99.98%), the spectral reconstruction can still go
wrong due to its nonlinear inverse transformation, Eq. (3-3). Figure 3-5 shows the
nonlinear transformation is acting as an high gain amplifier at the low Φ value
corresponding to high reflectance factor. Even a tiny mismatch around the low Φ region
will be enhanced after inverse transformation causing high colorimetric error.
Furthermore, the spectral regions with low Φ components are most likely reconstructed
with error based on PCA if the number of eigenvectors is not sufficient for reconstruction.
Kubelka-Munk transformation is, thus, sensitive to the real dimensionality of a sample set
by this observation.
83
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
K/S
Ref
lect
ance
fac
tor
Fig. 3-5: The Kubelka-Munk inverse transformation, Eq. (3-3), for opaque color.
Figure 3-6 shows the spectral error is enhanced at the low Φ spectral region while
the error is suppressed at the high Φ spectral region after Kubelka-Munk inverse
transformation for opaque coloration. The proposed transformation was designed to
overcome this type of problem by first ensuring that the dimensionality in Ψ space meets
the demanded dimensionality; and second transforming from Ψ to R space, shown in Fig.
3-7, by Eq. (3-7) with less steepness does not over-amplify the spectral mismatch from Ψ
space.
84
400 450 500 550 600 650 700-0.5
0
0.5
1
1.5
2
Wavelength
K/S
400 450 500 550 600 650 7000.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength
Refle
ctanc
e fac
tor
Eq. (3-3)
Fig. 3-6: An example of enhanced spectral error after the transformation by Eq. (3-3).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Re
fle
cta
nc
e f
ac
tor
Fig. 3-7: The transformation from Ψ to R space by Eq. (3-7).
Although the multivariate normality among three spaces is not easy to conclude by
inspecting their normal plot (not shown) for these two examples, the example of IT8.7/2
ReconstructedΦ
OriginalΦ
OriginalR
ReconstructedR
85
shown above as well as the observation on the percent variance indicate that multivariate
normality allow a larger degree of data (dimensionality) reduction. The derivation for
empirical transformation to Ψ space is based on the beneficial factor of multivariate
normality.
E. CONCLUSIONS
Several linear colorant mixing spaces were discussed. Given a sample set, accurate
spectral reconstruction can take advantage of the normality of a sample set represented in
a linear space when reconstruction is limited to a lower number of dimensions. It was
shown by the numerical simulation using linear combinations of basis colorant whose
concentrations are generated from several approximately normal distributions and the
example of IT8.7/2. Spectral error essentially lies in the higher dimensions beyond the
reconstructibility with constrained dimensionality. These two examples imply that
normality has lower magnitude of higher dimensional error such that PCA confined to a
limited dimensionality can model a sample set with satisfactory accuracy.
A new transformation was empirically defined when encountering a set of sample
measurements modeled by PCA. The transformation to a reduced dimensionality and near
normal space for PCA estimation can well approximate the sample measurements. It is
also designed to account for the opaque coloration of physical materials. Examples shown
for verification are highly accurate in terms of colorimetric and spectral performance.
86
IV. COLORANT ESTIMATION OF ORIGI NAL OBJECTS
The goal of research for colorant estimation is to determine a set of six basis
colorants which are the best representation of original objects such as paintings. That is,
their spectral information can be accurately reconstructed by linear combinations of the six
estimated colorants represented in a linear colorant mixing space. Since each painting is
possibly created by different colorants, the six estimated colorants are image dependent.
The clue leading to the six estimated colorants is the six eigenvectors determined from the
corresponding spectral measurements. The relationship between the six eigenvectors and
estimated colorants is merely the linear transformation (or geometrical rotation). Based
on these observations, a constrained-rotation engine using MATLAB was devised to
perform the transformation from the eigenvectors to a set of all-positive vectors as the
estimated colorants. Once a set of reasonable colorants is uncovered, this set of colorants
can be used to synthesize the original artwork with the least metameric effect between the
reproductions and originals. This chapter will show the derivation of the relationship
between eigenvectors and the statistical primaries as the desired colorant for synthesis of
an original object. For simplicity of communication, all linear modeling techniques will
utilize the Φ space for discussions. All the algorithms derived for this module were also
applied in Ψ space.
87
A. APPROXIMATELY LINEAR COLORANT MIXING SPACE
Kubelka-Munk turbid media theory is used as the first-order approximation
transforming spectral reflectance factor, Rλ, into an approximately linear space, defined as
Φ space. The mathematical description for linear color mixing, specified as Eq. (3-1), is
re-expressed by Eq. (4-1),
Φλ λφ, ,miture i ii
k
c==
∑1
, (4-1)
where c represents concentration and (k/s)λ is replaced with φλ. As an example, a still life
painting of a floral arrangement was produced with six independent acrylic paints shown in
Fig. 4-1.
Fig. 4-1: The still life painting of a floral arrangement creating with six opaque colorants.
88
Each paint was applied on a paper stock at a thickness achieving opacity and measured
spectrally using a Gretag SPM 60 spectrophotometer. Each reflectance vector had thirty-
one components: 400 nm - 700 nm at 10 nm bandwidths and intervals. These reflectance
vectors were transformed to Φλ. Ideally, one needs thirty-one “spectral colorants” with 10
nm bandwidth absorption and scattering properties at the sampled wavelengths in order to
reconstruct the measured sample spectra. Realistically, colorants do not have such narrow
band properties. Furthermore, reproducing a color by mixing thirty-one colorants is highly
impractical for any real coloration process. Fortunately, chromatic stimuli are not
originally created by such spectral colorants; hence, their Φλ do not span the entire thirty-
one dimensional Φλ space. Instead, they are distributed in a lower dimensional Φλ
subspace. If an original painting was only painted, for example, using six independent
colorants, then, ideally, the measured set of Φλ should be distributed only in a six-
dimensional subspace of Φλ space.
B. PRINCIPAL COMPONENT ANALYSIS
Principal component analysis (PCA) can provide a measure to statistically
determine the dimensionality of the sample population. The linear combinations of the first
p eigenvectors should describe the entire set of Φλ if the original was created by p
colorants, i.e.,
89
Φλ λ, ,sample i ii
p
b e==∑
1
, (4-2)
where eλ, i is the ith eigenvector and bi is the corresponding coefficient to reconstruct a
sample. Rewriting Eq. (4-2) in matrix form:
ΦΦΦΦ = EB. (4-3)
E is the matrix of the first six eigenvectors and B is the coefficient matrix to reconstruct
the sample population, ΦΦΦΦ. Figure 4-2 shows the first six eigenvectors which were obtained
from the still life painting explaining the most sample variations (99.98%) in Φλ space.
4 0 0 5 0 0 6 0 0 7 0 0-1
-0 .5
0
0 .5
1The 1 st e ig e nve ctor.
K/S
4 0 0 5 0 0 6 0 0 7 0 0-1
-0 .5
0
0 .5
1The 2 nd e ig e nve cto r.
4 0 0 5 0 0 6 0 0 7 0 0-1
-0 .5
0
0 .5
1The 3 rd e ig enve cto r.
4 0 0 5 0 0 6 0 0 7 0 0-1
-0 .5
0
0 .5
1The 4 th e ig e nve ctor.
W a ve le ng th
K/S
4 0 0 5 0 0 6 0 0 7 0 0-1
-0 .5
0
0 .5
1The 5 th e ig e nve ctor.
W a ve le ng th
4 0 0 5 0 0 6 0 0 7 0 0-1
-0 .5
0
0 .5
1The 6 th e ig e nve ctor.
W a ve le ng th
Fig. 4-2: The six eigenvectors obtained from the still life painting.
90
C. COLORANT ESTIMATION
As shown in Fig. 4-2, the thirty-one components of the eigenvectors are often
bipolar; consequently, they are not a set of physical colorants. Furthermore, their
corresponding coefficient vectors are also bipolar not representing physical
concentrations. Real colorants should have all-positive Φλ as their vector components, and
the corresponding concentrations should be all-positive. Since an original was created by
mixing a set of existing physical colorants at different concentrations and the color mixing
operation is mathematically described by Eq. (4-1), the sampled ΦΦΦΦ are distributed in an all-
positive space. Rewriting Eq. (4-1) in matrix form:
ΦΦΦΦ = φφφφC, (4-4)
where φφφφ is the matrix of the basis colorants and C is the concentration matrix to
reconstruct the sample population, ΦΦΦΦ. Notice that Eq. (4-4) can be equated with Eq. (4-3)
in order to obtain the relationship between the eigenvectors and the φλ of the basis
colorants used for creating the original painting. Based on this observation, the
relationship between the eigenvectors and the physical basis colorants is merely a linear
transformation, or a geometric rotation. Since
ΦΦΦΦ = EB = φφφφC, (4-5)
this implies that
φφφφ = EBC- = EM, (4-6)
91
where C- stands for the pseudo-inverse of the concentration matrix and M is the
representation of the matrix product of B and C-. The linear transformation from
eigenvectors to physical basis colorants should result in two important properties. First,
the rotated eigenvectors should be a set of all-positive vectors. Second, the concentration
matrix should have all non-negative entries. These two constraints should result in
colorant spectra that are very similar or linearly related to the actual colorants.
This constrained rotation was previously performed by Ohta (1973). His research
goal was to estimate the spectral density curves of an unknown dye set for photographic
materials using only the spectra of color mixtures such as ANSI IT8 targets. A Monte
Carlo method was also used to help identify the most likely dye set. It was a three
dimensional vector transformation. This research extends the challenge to six dimensions.
In the current analysis, a constrained-rotation engine using MATLAB as the calculation
platform was devised to solve the problem. Since the ultimate goal of this research is to
identify a set of printing inks that minimize metamerism between a set of objects and their
printed reproduction, the dimensionality is limited to six, corresponding to six printing
stations. If the dimensionality of the original ΦΦΦΦ is greater than six, or if there is appreciable
spectral measurement error, residual errors will result. Hence, goodness metrics are
required. The spectral accuracy was quantified by an index of metamerism that consists of
both a parameric correction for D50 and the use of CIE94 under illuminant A. The
colorimetric accuracy is calculated using CIE94 under D50 for the 1931 observer.
92
D. JUSTIFICATION OF EIGENVECTOR RECONSTRUCTIONWITHOUT SAMPLE MEAN
In practice, the measured samples often reveal more than p dimensions due to
measurement noise and limitations in the validity of the Kubelka-Munk transformations.
Given the p limited dimensions for spectral reconstruction, one should employ Eq. (4-2)
together with the sample mean for better accuracy. That is,
Φ Φλ λ λ, , ,sample i ii
p
sample meanb e= +=∑
1
. (4-7)
The existence of a sample mean for spectral reconstruction poses several difficulties for
this research. First, the sample mean is only a statistical result which specifies the average
Φλ behavior for the set of samples. The sample mean does not represent any physical
colorant. Second, in Eq. (4-7), the sample mean is acting as an offset vector which
impedes the equality relationship in Eq. (4-5). Since the eigenvectors are the only clue
leading to a set of possible colorants, the sample mean must be excluded for maintaining
the rotation relationship between eigenvectors and the set of possible colorants which is
specified by Eq. (4-6). Finally, the confidence for excluding the sample mean is that if the
dimensionality of sample population is approximately the constrained number of
dimensions, then the sample mean approximately resides in the reconstructed sample
population. That is, the sample mean can be approximately expressed as a linear
93
combination by the limited number of eigenvectros. Henceforth, the sample mean in Eq.
(4-7) can be excluded without significant error, i.e., Eq. (4-2) will be used.
E. VERIFICATIONS
Testing the Constrained-Rotation Engine by a Virtual Sample Population
The constrained-rotation engine was first tested for a virtual sample population
with three thousand random mixtures created by linear combinations of the six acrylic-
paint spectra in Φλ space. These six spectra, plotted in Fig. 4-3, were carefully chosen and
verified to be independent colorant vectors, i.e., no one colorant vector can be expressed
as the linear combination of any other five colorant vectors.
0
0.2
0.4
0.6
0.8
1
1.2
400
420
440
460
480
500
520
540
560
580
600
620
640
660
680
700
Wavelength
Nor
mal
ized
K/S
yellow
magenta
cyan
green
blue
black
Fig. 4-3: The six acrylic paints used for the computer generated sample population.
94
Hence, the virtual sample population is ensured to be six dimensional. The corresponding
concentration vectors were randomly generated from uniform distributions. Thus, the
resulted population of linear combinations of six uniform distributions is approximately
multivariate normally distributed (convolution of six uniform distributions is approximately
normal).
Given that real sample populations can be confounded by processes and
measurements, the idea of using the computer generated sample population to test the
constrained-rotation engine is to provide a noise free sample population. This ensures that
the rotated eigenvectors with all-positive vector components as the estimated colorant
spectra should be identical or linearly related to the six acrylic-paint spectra if the
proposed vector transformation theory expressed as Eq. (4-5) is valid.
Six-eigenvector reconstruction without a sample mean vector, based on Eq. (4-2),
yielded approximately zero spectral errors, hence, zero colorimetric errors since full
dimensionalty was employed. Then, an arbitrary set of six colorant vectors were used as
the initial values for the constrained-rotation engine. The resultant all-positive
eigenvectors as the set of estimated colorant vectors are identical to the original six
acrylic-paint spectra. In addition, another set of six block spectra evenly spaced within 400
nm to 700 nm representing an initial colorant vectors was utilized and the resulted
estimated colorant vectors were also identical to the six acrylic-paint spectra. This is
surprising since the vector transformation can not be unique; multiple solutions should
95
exist. Whereas, those solution are linearly related with each other since they all are the
linear transformations of the six eigenvectors. Thus, the first test shows that the
constrained-rotation engine is able to converge to an all-positive representation of the
eigenvectors. The MATLAB program for the implementation of constrained-rotation is
attached in Appendix B.
Colorant Estimation for a Kodak Q60C Target
The second verification was performed on a Kodak Q60C, a photographic
reflection target that was a precursor to the ANSI IT8 target. Three eigenvector
reconstruction should yield low spectral and colorimetric errors corresponding to the fact
that it is manufactured using three dyes. The Kulbeka-Munk transformation for
transparent materials was used to transform reflectance factor to absorption. The spectral
and colorimetric accuracy, based on the three-eigenvector reconstruction, is shown in
Table 4-1. Ideally, this is a three dimensional problem. Whereas, the spectral and
colorimetric accuracy is confounded by the manufacturing, processing and measuring
noise, and the model accuracy limitations of Kulbeka-Munk theory.
Table 4-1: The spectral and colorimetric accuracy of the three-eigenvector reconstructionfor the Kodak Q60C.
∆E*94 Metamerism Index
Mean 0.48 0.19Stdev 0.20 0.17Max 1.12 1.00Min 0.00 0.00
96
Uncovering the set of all-positive eigenvectors as the estimated dye spectra of the
Q60C was preceded by using the first eigenvectors of cyan, magenta, and yellow ramps of
the Q60C as the initial colorant vectors. The first eigenvector of each ramp, denoted as the
"local eigenvector," is the first statistical estimation of the real dye spectrum (Berns and
Shyu, 1995). By this approach, the advantage is to get a close solution and help expedite
the rotation process. The estimated dye spectra (thick lines) and the local eigenvectors
(dotted lines) are plotted in Fig. 4-4. Since the all-positive eigenvectors representing the
estimated dyes are an exact linear transformation of the first three eigenvectors, denoted
as global eigenvectors determined from the Kodak Q60C target, the spectral and
colorimetric performance of estimated dyes is the same as that of global eigenvectors.
0
0.2
0.4
0.6
0.8
1
1.2
400
420
440
460
480
500
520
540
560
580
600
620
640
660
680
700
Wavelength
Nor
mal
ized
abs
orpt
ion
coef
ficie
nt
Fig. 4-4: The all-positive eigenvectors as the estimated dye spectra (thick line) and thelocal eigenvectors (dotted lines).
97
It was found that the local eigenvectors were not the exact transformation of the
global eigenvectors. The spectral reconstructibility by local eigenvectors was worse than
that of the estimated dyes, as expected. Furthermore, the broader absorption bandwidths
of the local eigenvectors symbolize the possible impurity contamination during the
manufacturing, processing, and measuring. In comparison, the all-positive eigenvectors as
the estimated dye spectra showing narrower absorption bandwidths may be close to the
real dye spectra based on the support of low spectral and colorimetric errors.
The testing for the proposed colorant-estimation engine favors the sense of reverse
engineering, i.e., uncovering the spectral structures of real colorants. However, for the
current research applications, it needs only one reasonable set of colorant spectra which
can be used to search through the existing ink database or for a colorant chemist to
synthesize the exact inks. Once one exact or similar set of inks is selected, spectral-based
printing process can utilize the selected ink set to fulfill the least metameric reproduction.
Hence, it is not critical for the proposed colorant estimation engine to converge to the
exact colorant spectra which were used to manufacture the colored objects.
Colorant Estimation for the Still Life Painting
Another verification for the constrained-rotation engine was performed by spectral
measurements of the still life painting mentioned previously. The painting was painted by
six independent acrylic-paints whose Φλ spectra are plotted in Fig. 4-3. One hundred and
twenty-six samples were obtained to represent the entire Φλ space of the painting whose
98
six eigenvectors plotted in Fig. 4-2 explaining 99.98% of total variation indicated that this
sample population is approximately six dimensional. The spectral and colorimetric
accuracy of the six-eigenvector reconstruction is specified in Table 4-2.
Table 4-2: The spectral and colorimetric accuracy of the six-eigenvector reconstructionfor the still life painting.
∆E*94 Metamerism Index
Mean 0.21 0.18Stdev 0.14 0.16Max 0.75 0.95Min 0.02 0.01
Initially, the colorant estimation was intended to directly rotate the six eigenvectors to one
set of all-positive representations. The resultant colorant spectra are plotted in Fig. 4-5
and show that there is a colorant (thin dotted line) with various absorption bands across
the visible spectral region and the reasonable appearance of the rest of the five colorants.
Several sets of colorant vectors were used as the initial estimation for the constrained-
rotation engine. The resulting sets of estimated colorants all possessed the similar spectral
properties. These initial attempts did not reveal the existence of a neutral colorant judged
by the lack of a flat spectrum.
99
0
0.2
0.4
0.6
0.8
1
1.2
400
420
440
460
480
500
520
540
560
580
600
620
640
660
680
700
Wavelength
Nor
mal
ized
K/S
Fig. 4-5: The six all-positive eigenvectors as the estimated colorants for the still lifepainting.
The neutral colorant with an approximately flat spectrum can be approximated by
the linear combination of the rest of the five estimated colorants. The lack of a neutral
colorant indicated that the rest of the five estimated colorants did not explain sufficient
spectral variation. Since the current research aims to uncover one neutral and five
chromatic colorants for printing processes, the approach was to constrain the assumption
of the existence of the neutral colorant. Hence, the colorant estimation for the still life
painting was proceeded by: first, estimate the neutral colorant using linear regression to fit
the perfect flat spectrum by the six eigenvectors. Second, rotate the most significant five
eigenvectors to their all-positive representations. The resultant estimated colorants should
explain a higher degree of spectral variation once the neutral dimension is constrained. The
100
spectral and colorimetric accuracy of the resultant six estimated colorants is shown in
Table 4-3 and their spectral curves (solid lines) are simultaneously plotted with the six
original colorants (astroidal lines) used for the still life painting in Fig. 4-6.
Table 4-3: The spectral and colorimetric accuracy of the six estimated colorants for thestill life painting.
∆E*94 Metamerism Index
Mean 0.22 0.21Stdev 0.16 0.18Max 0.92 1.01Min 0.02 0.01
400 500 600 7000
0.2
0.4
0.6
0.8
1
Yellow
No
rma
lize
d K
/S
400 500 600 7000
0.2
0.4
0.6
0.8
1
Magenta
400 500 600 7000
0.2
0.4
0.6
0.8
1
Cyan
400 500 600 7000
0.2
0.4
0.6
0.8
1
Green
Wavelength
No
rma
lize
d K
/S
400 500 600 7000
0.2
0.4
0.6
0.8
1
Blue
Wavelength400 500 600 7000
0.2
0.4
0.6
0.8
1
Neutral
Wavelength
Fig. 4-6: The estimated colorants (solid lines) and the original colorants (astroidal lines)used for the still life painting.
101
The constrained-rotation of the six eigenvectors obtained from the still life painting
yielded a reasonable set of estimated colorants. Judging from them, most colorants have
similar spectral properties to the original which were utilized to create the still life
painting. Whereas, the spectral property similar to green is absent in this set of estimated
colorants. Instead, the constrained-rotation process gave out a spectrum equivalent to a
yellow colorant. This can be attributed to the sampling error due to the usage of larger
aperture size of spectrophotometer which violates the additive assumption in Φλ space. It
was discussed in Chapter III that the possible field of view of a spectrophotometer may
sample at a multi-color surface. Once the spectrophotometer samples at a spot where two
or more contiguous colors are within its field of view, the reading is equivalent to the
additive result of the spectral reflectance factors confined by the spot whose spectral
reflectance is contributed by that of the two or more colors. Furthermore, the additive
operation in reflectance space undergoing a nonlinear transformation such as Eqs. (3-2)
and (3-3) results in the additive operation undefined in Φλ space. As a consequence, the
behavior of samples in Φλ space are not predictable by the linear model of Eq. (4-2). This
type of sampling error can be reduced when the spectral reflectance factor of a color
object is estimated by a high resolution CCD camera with very narrow field of views for
each pixel. The under-sampling of green and over-sampling of yellow-orange color are the
other source of errors which cause the estimated colorants not to agree with the original
colorant. Once the sample gamut is approximately uniform, i.e., each color has
102
approximately equal probability of occurrence in the sample population, this type of error
is minimized. Since the sample gamut of the still life was carefully controlled to be as
uniform as possible, the lack of green colorant is mainly caused by the violation of the
additive assumption in Φλ space due to the large field of view of the spectrophotometer.
Colorant Estimation for 105 Mixtures Using ΨΨΨΨ Space
Since the non-optimal aperture size of a spectrophotometer leads to sampling error
and a spectral image captured by a multi-spectral acquisition system is still under
development, the rational proof for the validity of the constrained-rotation mechanism is
to avoid the type of sampling error leading to the additive failure in a linear color mixing
space. Although the verification for Kodak Q60 target does not suffer this sampling error
since the measurements were performed on color patches, this section repeats the
verification with a set of six-color mixtures whose underlying primaries are known. If the
rotation results can converge or nearly go to the original primary colorants, then it not
only verifies the validity of the proposed rotation algorithms but also confirms the
effectiveness of the linear colorant mixing space utilized for analysis.
Accordingly, a set of 105 six-color mixtures, shown in Fig. 4-7, created by hand
mixing six opaque poster paints (Sakura cerulean blue No. 25, Sakura rose violet No. 22,
Pentel yellow No. 5, Pentel sap green No. 63, Pentel ultramarine N0. 25, and Pentel black
No. 28), whose spectral reflectance factor are shown in Fig. 4-8, which are the exact set
of colorants to paint the still life painting. Measurements were done by using a Macbeth
103
Color Eye 7000 spectrophotometer with SPEX measuring geometry and an integrating
sphere.
Fig. 4-7: The 105 mixtures created by hand mixing six opaque poster paints.
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1
Re
felc
tan
ce
C e ru le a n b lu e
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1R o s e v i o le t
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1Y e llo w
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1
W a v e le n g th
Re
felc
tan
ce
S a p g re e n
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1
W a v e le n g th
U lt ra m a r i n e
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1
W a v e le n g th
B la c k
Fig. 4-8: The spectral reflectance factors of the six poster colors used for creating the 105opaque mixtures.
104
Since the accuracy of Kubelka-Munk transformations for opaque colorant, Eqs (3-2) and
(3-3), did not provide sufficient accuracy, the linear colorant mixing space obtained by the
empirical transformation, Eq. (3-6), was utilized for analysis and process. The offset
vector v
a , shown in Fig. 4-9 and in Eq. (3-6), was optimized according to the 105 mixtures
in Ψ space such that the 105 mixtures in Ψ space are distributed in a near six dimensional
vector space. The optimization procedures are shown in Appendix B. Six eigenvectors,
shown in Fig. 4-10, explain 99.70% sample variation in Ψ space. The colorimetric and
spectral accuracy of six eigenvector reconstruction is listed in Table 4-4.
4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 00
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
Ψ
W a ve le n g th
Fig. 4-9: The offset vector, v
a , for transforming the 105 mixtures to Ψ space.
From Table 4-4, the colorimetric and spectral errors are low for the six-eignvector
reconstruction. This indicates that the 105 mixtures distributed in Ψ space are
approximately six dimensional.
105
4 00 5 00 6 00 7 00-1
-0 .5
0
0 .5
1The 1 s t e ig enve c to r.
Ψ
4 00 5 00 6 00 7 00-1
-0 .5
0
0 .5
1The 2 nd e ige nvec to r.
4 00 5 00 6 00 7 00-1
-0 .5
0
0 .5
1The 3 rd e ig e nvec to r.
4 00 5 00 6 00 7 00-1
-0 .5
0
0 .5
1The 4 th e ig enve c to r.
W a ve leng th
Ψ
4 00 5 00 6 00 7 00-1
-0 .5
0
0 .5
1The 5 th e ig enve c to r.
W a ve leng th
4 00 5 00 6 00 7 00-1
-0 .5
0
0 .5
1The 6 th e ig enve c to r.
W a ve leng th
Fig. 4-10: The six eigenvectors of the 105 mixtures in Ψ space.
Table 4-4: The colorimetric and spectral accuracy of the six eigenvector reconstruction forthe 105 mixtures.
∆E*94 Metamerism Index
Mean 0.22 0.21Stdev 0.16 0.18Max 0.92 1.01Min 0.02 0.01RMS 0.007
Constrained rotation by the proposed algorithms were performed on the
determined six eigenvectors derived from the 105 mixtures in Ψ space. Since the a priori
knowledge about the primary colorants is known, the spectra of the six primaries were
used as the initial vectors for rotation process. It was desired to conclude whether the
106
constrained-rotation mechanism can perform "reverse engineering" if the initial colorant
vectors were known for comparing to the estimated colorants, though it is not necessary
for this research. The six rotated all-positive eigenvectors as the statistical primaries
whose reflectance spectra are shown in Appendix G, shown in Fig. 4-11, are plotted with
original six primary colorant in reflectance space. Five chromatic colorants were
normalized to 0.9 units of reflectance factor and the black colorant was normalized to 0.1
and shifted up 0.8 to 0.9 units of reflectance factor for visual comparison.
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1
Re
felc
tan
ce
C e ru le a n b lu e vs . P r im a ry 1
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1R o s e vio le t vs . P r im a ry 2
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1Ye llo w vs . P r im a ry 3
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1
W a ve le n g th
Re
felc
tan
ce
S a p g re e n vs . P r im a ry 4
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1
W a ve le n g th
U ltra m a rine vs . P ri m a ry 5
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1
W a ve le n g th
B la c k vs . P r im a ry 6
Fig. 4-11: The reflectance factors of the six original colorants (solid lines) and statisticalprimaries (dashed lines) derived by a constrained rotation from the six eigenvectors.
From Fig. 4-11, the six statistical primaries did not converge to the original
colorants. They converged to a set of close solutions. The agreement between the six
107
original and statistical primaries are high. Although the statistical primaries are not
identical to the six originals, the probable causes leading to this discrepancy are unwanted
contamination, measuring error, and the validity of the empirical transformation by Eq. (3-
6). It was tested by a linear regression model to see if the six original primaries span the
105 mixtures in Ψ space, that is, for any vector of the 105 mixtures is a linear combination
of the six original primaries. It was found that the six original primary do not span the 105
colorants in Ψ space judged by that negative concentrations were required to synthesis a
spectrum. This implies that the six original primaries only span a partial colorant space that
does not include all of the 105 mixtures. In another words, the six original primaries do
not explain all the spectral variation of the 105 mixtures. This is strong evidence for
unwanted contamination when hand mixing the 105 mixtures. Since the statistical
primaries approximately span the entire 105 mixtures in Ψ space and the measurement
was done by a highly accurate instrument. The unwanted contamination is attributed to be
the main cause for the discrepancy. Nevertheless, the correlation between the six original
and statistical primaries is high. The statistical primaries can be used as the basis
information for synthesizing an object such as a painting created by unknown colorants.
I. CONCLUSIONS
An algorithm was developed for the colorant estimation of original objects through
vector analysis and principal component analysis. The relationship between basis colorants
108
and eigenvectors is elucidated by performing a constrained linear transformation. Since the
basis colorants used for creating original objects can be statistically uncovered with
sufficient accuracy, the color reproduction at the synthesis stage gains the maximum
capability to spectrally reconstruct a sample from the original. Therefore, metamerism
between the reproduction and original is minimized.
109
V. OPTIMAL INK SELECTION
Given a spectral image, the mission of spectral reproduction using a multiple-ink
printing process can be a challenging task. Even though the number of degrees of
freedom is increased, the spectrum corresponding to every pixel in a given spectral image
may not well be inside the spectral gamut of the multiple-ink process. Theoretically, a
given spectral image whose spectra of all pixels should be located inside the spectral
gamut of the multiple-ink printing process. Then each spectrum requiring reproduction is
described by a set mathematical functions of its printing primaries. The analytical
descriptions of a multiple-ink printing process are discussed in chapter VII. Therefore, the
set of printing primaries dominate the capability of spectral reproduction.
Given a large set of inks, the decision for choosing an optimal ink set can be a
tedious task. The combinations of candidate ink set can be a geometric figure. There are
C(n, 6) combinations of possible ink sets for n inks in storage to be chosen for a six-color
printing process. For example if n = 100, then there are 119,205,240 combinations of
choices. For a practical example, even if n = 18 (Pantone 14 basic colors in addition to
four process colors) then there still exist 18,564 combinations to choose from. To estimate
the performance of each ink set in terms of colorimetric and spectral accuracy, a spectral
printing model is needed to evaluate the spectral reconstruction. This requires
construction of 18,564 six-color printing models for the 18 ink combinations and estimate
110
the total performance model by model. The computation for trying 18,564 combinations is
unreasonable, not to mention the model building effort. It can be safely said that is a
"mission impossible." Clearly, testing each ink set is insufficient and a robust ink-selection
algorithm is required.
A. FIRST ORDER INK SELECTION BY VECTOR CORRELATION
Although there are 18,564 combinations of ink sets by selecting six inks from an
18 ink database for a six-color printing process, a large portion of combinations are
obviously impossible ink combinations for synthesis. The question boils down to this: on
what basis can these impossible ink sets be excluded analytically? For a printing system
which performs colorimetric other than spectral reproduction, it can be achieved visually
by determining all the colorimetric values of a given image inside a colorimetric gamut
spanned by a set of ink combinations. This is done by examining visualizations of a 2-D, 3-
D CIELAB, or chromaticity plots. Generally, the larger the colorimetric gamut a set of
inks can provide, the more accurate colorimetric reproduction can be accomplished.
The very reason for the current research project to derive a colorant estimation
module for a multi-spectral output system is to eliminate the impossible combinations of
ink sets analytically. The set of statistical primaries are utilized as a set of basis to remove
the impossible ink combinations. Testing the performance of the irrational ink
combinations can be avoided by well defined analytical printing models that describe the
111
color mixing behavior. Once the statistical primaries are uncovered for an input spectral
image, the process of the throughput for the proposed multiple-ink printing system
minimizing metamerism is to correlate the statistical primaries to a set of physical inks
which are the most capable of spectrally reproducing the spectral image during the
synthesis stage. Theoretically, if an exact set of inks exist in a current manufacture line or
in storage, then the use of the exact ink set will yield the ideal or closest spectral
reproduction relative to the input spectral image. Since the statistical primaries are image
dependent, the probability of an exact ink set is low. Hence, the exact spectral
reproduction can not be achieved. A compromise has to be made to balance between
colorimetric and spectral accuracy. Since the colorimetric match is the first priority for any
color application, it is necessary to trade a slight decrease in spectral accuracy in exchange
for higher colorimetric accuracy. This compromise will be discussed in a later section.
Intuitively, the use of the statistical primaries is to search for the exact or similar
inks in a given ink database. Vector correlation can be used to compare the similarity
among them. A similarity measurement for ink1 and ink2, shown as Eq. (5-1), is quantified
by the correlation coefficient, ρ, which is the cosine of the angle between a statistical
primary and an ink from a large ink database, where Ψλ,ink1 is the linear colorant vector of
ink1 and λ is a wavelength within the visible spectrum. Hence, the closer the correlation
coefficient is to unity, the higher the similarity between a statistical primary and an ink in
the database.
112
ρλ λ
λ
λλ
λλ
= =
= =
∑
∑ ∑
Ψ Ψ
Ψ Ψ
, ,
, ,
ink ink
ink ink
1 2400
700
12
400
700
22
400
700(5-1)
Since the chances of selecting inks from a large ink database which are identical to
the statistical primaries are low, candidate inks corresponding to each primary can be
selected using a threshold of an acceptable correlation coefficient, say 0.90, or the highest
twenty. Further filtration of the selected candidates is done by adopting the two candidates
with the highest chroma for each statistical primary. A larger colorimetric gamut
corresponding to a better possibility of colorimetric reproduction has been elucidated by
various literature and experiential evidences (Ostromoukhov, 1993; Boll, 1994; Stollnitz,
Ostromoukhov, and Salesin, 1998; Viggiano and Hoagland, 1998). The use of the highly
chromatic primaries for colorant mixing yields a larger colorimetric gamut which is
essentially desired when an exact spectral color reproduction cannot be accomplished.
Compromises have to be made by trading decreased spectral accuracy in exchange for
colorimetric accuracy. This is the reason for choosing candidates with the highest chroma
for balancing between colorimetric and spectral accuracy. Based on this selection method,
there are 64 (26) possible combinations of candidate ink sets. This is a significant reduction
from 18,564. Nevertheless, it is still necessary to pinpoint the exact ink set for the
application of the least metameric reproduction.
113
Now this has come to the point of which type of analytical description for printing
process should be utilized to estimate the performance of these first order selections. The
intuitive choice is the six-color printing model based on the Yule-Nielsen modified spectral
Neugebauer equation. To build 64 six-color printing models, it is required to do the
sample preparations for printing 64 sets of ramps of Neugebauer primaries and verification
targets corresponding to the 64 sets of selected ink combinations. The efforts of model
building are beyond economical consideration. Even if the sample preparations can be
replaced by computer simulation under certain assumptions, the model building efforts are
still computational costly and time inefficient. Hence, a more efficient mechanism for
estimating the colorimetric and spectral performance of the 64 selected ink sets are
desired.
B. CONTINUOUS TONE APPROXIMATION
The further removal of low proficient combinations among the 64 is judged by
scrutinizing the ink sets which are incapable of spanning the vector space of a given set of
color samples. For this task, an assumption is made that halftone reproduction can be
approximated by a continuous-tone model for subtractive color mixing (Berns, Bose, and
Tzeng, 1996; Van De Capelle and Meireson, 1997). The continuous-tone modeling
techniques used by the research program (Berns, 1993; Berns and Shyu, 1995) at the
Munsell Color Science Laboratory at Rochester Institute of Technology are mostly based
114
on Kubelka-Munk turbid media theory. However, for this application, Kubelka-Munk
theory has insufficient accuracy. Alternatively, an empirical transformation for
approximating the color mixing behavior of a halftone printing process, shown as Eq. (5-
2) whose inverse transformation is shown as Eq. (5-3), was derived as
Ψλ λ λ= −R Rpaperw w,
1 1
and (5-2)
R R paperw w
λ λ λ= −( ),
1
Ψ , (5-3)
where the Rλ,paper is the spectral reflectance factor of the paper substrate being printed on
by primary inks and 2 ≤ w ≤ ∞. The transformation of a reflectance factor to the
empirically derived space is somewhat different from Eq. (3-6) since Eq. (3-6) is derived
for opaque colorant. Whereas, they have basically the same structure, one offset vector
accounting for subtractive color mixing and a higher order power to account for the
nonlinearity. The use of R paperwλ ,
1
as the offset vector has a significant meaning. Consider
that transforming a spectrum, which is exactly Rλ,paper, to the linear color mixing space, the
result is a zero vector. This corresponds to the fact that there is not any primary presented
in the linear space. Furthermore, Eq. (5-3) transforms a zero in the linear space back to the
exact reflectance spectrum of the paper, Rλ,paper. The justifications for the use of the
proposed transformation for continuous tone approximation to be described in verification
section. Equation (5-2) transforms spectral reflectance factor to the representation for a
115
subtractive color mixing process. Hence, the synthesis, quantitatively described by Eq. (5-
4), of color mixtures is the linear combinations of the primary colorants modulated by
their corresponding concentrations
Ψλ λψ, ,mixture i ii
k
c==
∑1
, (5-4)
where ψλ is the linear operand of a primary colorant normalized to its unit concentration, c
is the corresponding concentration, and k is the number of the primary colorants.
Based on the assumption that a multiple-ink halftone printing process can be
approximated by continuous tone modeling using Eq. (5-2) and (5-3), a direct constrained
regression model using Eq. (5-4) was employed to estimate the performance of each
candidate ink set. The estimated concentration for each primary is constrained to be
positive. Positivity symbolizes the capability of a candidate ink set to span the entire
colorant vector space of the target. If a negative concentration is reported using Eq. (5-4)
without this constraint, then the corresponding ink set is only spanning the partial colorant
vector space of the target. As a consequence, the spectral reconstruction by the ink set
yields spectral error when the constraint of positivity is enforced. The final decision should
favor the ink set(s) whose spectral reconstruction for a given target achieves the higher
colorimetric and spectral accuracy. For this research project, the colorimetric accuracy is
specified by the CIE color difference equation ∆E*94 under standard illuminant D50 and
the 1931 standard observer. The spectral accuracy is quantified by the metameric index
116
which is calculated by ∆E*94 under standard illuminant A and the 1931 standard observer
based on a parameric correction.
C. VERIFICATIONS AND RESULTS
Deriving a Linear Color Mixing Space for Continuous Tone Approximation
Current research analysis is based on the validity for a linear colorant mixing space
which approximates the color mixing behavior of a multiple-ink halftone printing process
such that the vector correlation analysis and constrained regression can be performed. The
verification of deriving an empirical transformation, Eq. (5-2), for a typical halftone
printing process utilized the spectral reflectance factor of IT8/7.3 of 928 samples printed
by SWOP standard at 133 LPI screen frequency (McDowell, 1995). The SWOP specified
spectral reflectance factor of paper and process CMYK are plotted in Fig. 5-1. The
parameter in Eq. (5-2) to be optimized is w. The optimization process is to use a non-
negative least square function, nnls( ), built in MATLAB, to set up a constrained
regression model, based on Eq. (5-4), for synthesizing every spectrum of the IT8/7.3
target. The nnls( ) performs a least square match for each spectrum by constraining the
corresponding concentration for each primary to be non-negative. The w corresponding to
the highest colorimetric and spectral accuracy will be adopted. Figure 5-2 shows the
colorimetric and spectral accuracy with respect to different w values.
117
4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 00
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
W a ve le n g th
Re
flec
tan
ce
fa
cto
r
C y a nM a g e n taY e llo wB la c kP a p e r
Fig. 5-1: The SWOP specified process CMYK primaries and paper substrate.
2 2 .5 3 3 .5 4 4 .5 50
0 .2
0 .4
0 .6
0 .8
1w vs . average D e lta E 94
w
De
lta E
94
2 2 .5 3 3 .5 4 4 .5 50
2
4
6
8
10w vs . maximum D e lta E94
w
De
lta E
94
2 2 .5 3 3 .5 4 4 .5 50
0 .2
0 .4
0 .6
0 .8
1w vs . average M .I.
w
M.I.
2 2 .5 3 3 .5 4 4 .5 50
1
2
3
4
5w vs . maximum M .I.
w
M.I.
Fig. 5-2: The colorimetric and spectral error vs. w values for the empirical transformation,Eq. (5-2), where M.I. represents metamerism index.
118
It was found that colorimetric and spectral errors monotonically decease as w
approached ∞. Since the slope of decreasing of average ∆E*94 and metamerism indices are
small when w is higher than three, the change of w does not increase the average
performance significantly. It only improved the maximum errors significantly as w
increases. The adopted w was 3.5 for an empirical decision. Colorimetric and spectral
accuracy for the four SWOP primaries synthesizing 928 samples of the IT8/7.3 in the
proposed linear colorant mixing space is shown in Table 5-1. Low colorimetric and
spectral error indicated by Table 5-1 reveals that the four SWOP primaries span the 928
samples of the IT8/7.3 target in the proposed linear color mixing space. That is, every
sample is a linear combination of the CMYK primaries. The color formation for this
halftone printing process is approximately described by mixing the CMYK in the linear
colorant mixing space.
Table 5-1: The colorimetric and spectral accuracy for the four SWOP primariessynthesizing the 928 samples of IT8/7.3 target in the proposed linear colorant mixingspace.
∆E*94 Metamerism Index
Mean 0.65 0.14Stdev 0.65 0.16Max 5.08 1.41Min 0.00 0.00RMS 0.0047
Four reconstructed spectra corresponding to the highest four colorimetric and spectral
error are shown in Fig. 5-3.
119
4 00 4 50 5 00 5 50 6 00 6 50 7 000
0 .2
0 .4
0 .6
0 .8
1
Re
flec
tan
ce
4 00 4 50 5 00 5 50 6 00 6 50 7 000
0 .2
0 .4
0 .6
0 .8
1
4 00 4 50 5 00 5 50 6 00 6 50 7 000
0 .2
0 .4
0 .6
0 .8
1
W a ve leng th
Re
flec
tan
ce
4 00 4 50 5 00 5 50 6 00 6 50 7 000
0 .2
0 .4
0 .6
0 .8
1
W a ve leng th
Fig. 5-3: The four spectra reconstructed with the highest colorimetric and spectral errorsbased on the linear colorant mixing space where solid line is the measured spectrum and
the dashed line is the reconstructed spectrum .
Even though the four spectra are reconstructed with the highest colorimetric and spectral
errors, the reconstructed curves are well tracing the originally measured spectra. This
implies that the halftone printing process can be well described by a continuous-tone
approximation based on the proposed transformation. Hence, the derived new Ψ space
can be used for estimating the performance of the 64 ink sets without heavy halftone
modeling efforts.
120
Vector Correlation Analysis in ΨΨΨΨ Space
The set of 105 color patches, Shown in Fig. 4-7, was employed as the presumed
reproduction target of an arbitrary image. Its six statistical primaries, shown in Fig. 5-4,
were estimated by the module of colorant estimation discussed in Chapter IV.
4 0 0 5 0 0 6 0 0 7 0 00
0 . 2
0 . 4
0 . 6
0 . 8
1P r i m a r y 1
Re
fle
cta
nc
e
4 0 0 5 0 0 6 0 0 7 0 00
0 . 2
0 . 4
0 . 6
0 . 8
1P r i m a r y 2
4 0 0 5 0 0 6 0 0 7 0 00
0 . 2
0 . 4
0 . 6
0 . 8
1P r i m a r y 3
4 0 0 5 0 0 6 0 0 7 0 00
0 . 2
0 . 4
0 . 6
0 . 8
1P r i m a r y 4
W a v e le n g th
Re
fle
cta
nc
e
4 0 0 5 0 0 6 0 0 7 0 00
0 . 2
0 . 4
0 . 6
0 . 8
1P r i m a r y 5
W a v e le n g th
4 0 0 5 0 0 6 0 0 7 0 00
0 . 2
0 . 4
0 . 6
0 . 8
1P r i m a r y 6
W a v e le n g th
Fig. 5-4: The six statistical primaries derived from the 105 mixtures by the colorantestimation module.
Since the six statistical primaries are, theoretically, the exact rotation of the six
eigenvectors of the reproduction target in the proposed linear color space, their
colorimetric and spectral accuracy in reconstructing the 105 patches is, theoretically,
identical to the accuracy, shown in Table 4-4 of the six-eigenvector reconstruction.
Discrepancy between the accuracy reconstructed by two sets of basis vectors depends on
the numerical precision of the constrained rotation. Nevertheless, in spite of this
121
discrepancy, the six statistical primaries are treated equivalent to the six eigenvectors.
Hence, the use of the six statistical primaries is capable of reproducing the 105 six-color
mixtures with a desired accuracy. Their spectral information is the link to select a closest
set of inks from a given ink database for six-color halftone reproduction.
Pantone 14 basic and process CMYK printed coated paper, shown in Fig. 5-5,
were utilized as an ink database to perform the analysis for ink selection algorithms.
(Color names and their abbreviates are yellow (Y) , yellow 012 (Y 12), orange 021 (O
21), warm red (Wr), red 032 (R), rubine red (Rr), rhodamine red (Rh), purple (Pu), violet
(V), blue 072 (B 72), reflex blue (Rb), process blue (Prs B), green (G), black (K), process
yellow (Prs Y), process magenta (Prs M), process cyan (Prs C), and process black (Prs
K).)
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
4 0 0 5 0 0 6 0 0 7 0 00
0 .5
1
Fig. 5-5: The Pantone 14 basic colors and the process CMYK as the ink database (Colorname order corresponding to each spectrum is from left to right and top to bottom).
122
All the spectral reflectance factors for the statistical primaries and the ink database were
first transformed by Eq. (5-2) to a linear colorant mixing space. Then, correlation
coefficients of all 18 inks in the database for the statistical primaries were calculated using
Eq. (5-1). Up to two candidate inks for each statistical primary with the highest chroma
among those highly correlated inks were chosen. Their correlation coefficients and chroma
are tabulated in Table 5-2. Since the research development aims at using one black and
five chromatic inks to perform spectral reproduction, the two black inks in the ink data
base are the certain candidate inks for the continuous tone estimation. Hence, their chroma
and correlation coefficients with the sixth primaries were not tabulated in Table 5-2.
Table 5-2: The correation coefficeints and the chroma of the 18 inks with the fivechromatic statistical primaries.
Candidate Inks Primary 1 Primary 2 Primary 3 Primary 4 Primary 5 ChromaProcess Blue 0.98 68.6Process Cyan 0.98 64.1
Rhodamine Red 0.94 80.1Purple 0.88 86.8Yellow 0.98 112.6
Process Yellow 0.97 106.3Green 0.36 82.3
Blue 072 0.66 88.8Reflex Blue 0.65 75.4
There are two candidates for each statistical primary except for the primary 4 due
to the lack of similar ink existing in the 18 ink database. It was forced to chose only one
with the highest correlation coefficient with respective to primary 4 among the 18 inks.
Another situation happens when an ink in the database is simultaneously chosen as the
123
candidate for two or more primaries; that is, an ink is selected more than twice. Then this
ink is a sure candidate. Thus, the candidacy of this ink should be removed for other
primaries. This ensures that when forming ink combinations by all candidates, there are
double or triple selected inks in an ink combination, consequently, leading to a
combination of smaller gamut colorimetrically and spectrally.
Colorimetric and Spectral Performance by Continuous Tone Approximation
Thirty-two (2x2x2x1x2x2) ink sets were formed by 11 candidate inks. Their
colorimetric and spectral accuracy were estimated based on the constrained regression
model using Ψ space. Since the validity of continuous tone approximation has been
verified for the IT8/7.3 printed by SWOP standard, it is generalized to any of the printing
process meeting the SWOP specification. Three ink sets were designated as the optimal
ink sets for reproducing the 105 mixtures based on their highest spectral accuracy
specified by the metamerism index. Their spectral and colorimetric accuracy are listed in
Table 5-3 and their ink combinations are described in Table 5-4.
Table 5-3: The spectral and colorimetric accuracy of the three optimal ink sets.
Metamerism Index ∆E*94
Ink Set Mean Stdev Max Min Mean Stdev Max Min RMS23 0.70 0.51 1.86 0.05 2.26 0.99 4.35 0.37 0.02824 0.73 0.52 1.88 0.05 2.35 0.96 4.20 0.37 0.02819 0.73 0.53 1.86 0.04 2.40 1.17 4.75 0.20 0.028
124
Table 5-4: The Pantone color names of the three optimal ink sets.
Ink Set Primary 1 Primary 2 Primary 3 Primary 4 Primary 5 Primary 623 Process
CyanRhodamine
RedYellow Green Blue 72 Process
Black24 Process
CyanRhodamine
RedYellow Green Blue 72 Black
19 ProcessCyan
RhodamineRed
ProcessYellow
Green Blue 72 ProcessBlack
The performance among the three ink sets are not significantly different. Set 23
and set 24 are only different in the use of the sixth primary. It is concluded that the use of
process black and the black is approximately invariant with the resultant performance. In
addition, set 23 and set 19 are different by the use of the third primary. It is concluded that
the use of yellow and process yellow is also approximately invariant with performance.
Three reconstructed spectra for a sample of 105 mixture corresponding to the maximum
error predicted by the three sets are plotted in Fig. 5-6. The three reconstructed spectra
are nearly identical. Based on this observation, the three optimal ink sets approximately
span the same colorant mixing space.
Three worst performed ink sets, whose colorimetric and spectral accuracy in
predicting the 105 mixtures are shown in Table 5-5, are specified in Table 5-6.
Performances of the three worst performing ink sets are about identical to each other. The
same justification applied for the three optimal ink sets can also be applied for the three
worst performers. Their reconstructed spectra for the sample, shown in Fig. 5-6, are
depicted in Fig. 5-7.
125
4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 00
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
W a ve le ng th
Re
flec
tan
ce
fa
cto
r
M e a s u re dS e t 2 3S e t 2 4S e t 1 9
Fig. 5-6: The three spectral reconstructions of a sample corresponding to the maximumprediction error by the three optimal ink sets.
Table 5-5: The spectral and colorimetric accuracy of the three worst performing ink sets.
Metamerism Index ∆E*94
Ink Set Mean Stdev Max Min Mean Stdev Max Min RMS26 0.93 0.74 2.68 0.03 3.73 1.34 7.03 0.48 0.03714 0.93 0.73 2.68 0.05 3.69 1.36 7.03 0.43 0.0379 0.96 0.73 2.67 0.05 3.73 1.38 6.69 0.43 0.037
Table 5-6: The ink combinations of the three worst performing ink sets.
Ink Set Primary 1 Primary 2 Primary 3 Primary 4 Primary 5 Primary 626 Process
CyanPurple Process
YellowGreen Reflex
BlueProcessBlack
14 ProcessBlue
Purple Yellow Green ReflexBlue
ProcessBlack
9 ProcessBlue
Purple ProcessYellow
Green ReflexBlue
Black
126
400 450 500 550 600 650 7000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
W avelength
Re
flect
an
ce fa
cto
r
M easuredSe t 26Se t 14Se t 9
Fig. 5-7: The three spectral predictions of the sample, shown in Fig. 5-6, by the threeworst performed ink sets.
Finally, two predictions for a sample by set 23 and set 26, plotted in Fig. 5-8, are
shown. By the numerical results, the colorimetric and spectral accuracy of set 23 is higher
than that of set 26. By Fig. 5-8, the shape of the spectrum reconstructed by set 23 whose
RMS error is 0.041 is closer to the measured spectrum than that of the reconstruction by
set 26 whose RMS error is 0.050. The implication is that the color space spanned by set
23 is closer to the color space of the 150 mixtures than the color space spanned by set 26.
Since the decision of choosing optimal ink sets is based on the metamerism index, the
above justification was made to correlate the effectiveness of metamerism index to a set of
127
ink combinations, which is the optimal selection. The implementation of ink selection
subsystem is attached in C.
4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 00
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
W a ve le ng th
Re
fle
cta
nc
e f
ac
tor
M e a s u re dS e t 2 3S e t 2 6
Fig. 5-8: The two reconstructed spectra by set 23 and set 26 for the sample used as theexample in Fig. 5-6.
D. CONCLUSIONS
An optimal ink set selection algorithm was proposed and justified. Its primary goal
is to bridge between multi-spectral acquisition systems and multiple-ink output systems for
the least metameric color reproduction. It serves the purpose of removing the redundancy
or the deficiency of large ink combinations of a given ink database. This research also
proposed applying a linear color mixing space for a continuous tone approximation to a
halftone printing process. It dramatically reduced large scale of modeling efforts in
128
estimating the validity of optimal ink selection. The ink selection algorithm was comprised
of vector correlation analysis follow by a constrained regression analysis. The proposed
approach was able to remove a large number of ink combinations from an ink database. It
pinpointed the optimal ink combination for a spectral-based halftone color reproduction
system minimizing metamerism.
129
VI. SPECTRAL REFLECTANCE PREDICTION OF INKOVERPRINT USING KUBELKA-MUNK TURBID M EDIATHEORY
Consider the microscopic structure of ink on paper as delivered by a typical
halftone printing process, shown in Fig. 6-1; a three-color halftone print is shown for
demonstration. The total spectral reflectance factor over a square area, which is assumed
to be the area of interest, is a summation of the individual spectral reflectance factors of
each color inside the area.
Fig. 6-1: The microscopic structure of color formation by a halftone printing processwhere Rλ,color represents the spectral reflectance factor of a color appearing in the square
area.
As we can see, the colors appearing inside the area of interest for reproduction are not
only the primary colors, white, primary one (P1), primary two (P2), and primary three (P3),
R Pλ , 2
R P Pλ , 1 2
R P Pλ , 2 3
R p p pλ , 1 2 3
R Pλ, 1
R P Pλ, 1 3
R Pλ , 3
R whiteλ ,
130
but also the overprints of the primaries, primary one on primary two (P1P2), primary one
on primary three (P1P3), primary two on primary three (P2P3), and the three-primary
overprint (P1P2P3). These colors are usually called the Neugebauer primaries. Intuitively,
the total spectral reflectance factor, Rλ,mix, over the area of interest is the linear sum,
known as the spectral Neugebauer equation, of each spectral reflectance factor of the
Neugebauer primaries modulated by their corresponding probability of occurrences. Yule
and Nielsen further introduced an empirical n-factor to modify the spectral Neugebauer
equation in order to account for light scattering within the paper, usually referred to as
"optical dot gain." The Yule-Nielsen modified spectral Neugebauer equation for a three-
color (P1, P2, and P3) printing process is defined as
R R a R a R
a R a R a R a R
a a a a a a a R
mix P Pn
P Pn
P Pn
P P P Pn
P P P Pn
P P P Pn
P P P P P Pn
P P P P P P P P P P P P
λ λ λ λ
λ λ λ λ
λ
, ,/
,/
,/
,/
,/
,/
,/
,
[a
( )
= + + +
+ + + +
− − − − − − −
1 1 2 2 3 3
1 2 1 2 1 3 1 3 2 3 2 3 1 2 3 1 2 3
1 2 3 1 2 1 3 2 3 1 2 3
1 1 1
1 1 1 1
1 whiten n1/ ]
, (6-1)
where a (indexed by P1, P2, P3, P1P2, P1P3, P2P3, and P1P2P3) represents the fractional dot
area of a Neugebauer primary.
In order to use the Yule-Nielsen modified spectral Neugebauer equation, the
spectral reflectance factor of each ink overprinted at 100% dot area coverage (known as
secondary, tertiary, quaternary primaries, and so forth) are required as a priori knowledge
for predicting the fractional dot area coverage of given spectra from an input spectral
image. There are 2j-j-1 overprints for a j-color halftone printing system. For example, a
seven-color halftone printing process needs to print and measure 120 (27-7-1) overprints.
131
As the number of colors used for halftone printing increases linearly, the number of
overprints increases exponentially. Hence, an analytical method for predicting the spectral
properties of overprints can avoid the necessity of exhaustively printing and measuring
each overprint upon using different ink and paper materials.
A. PREVIOUS RESEARCH
Previous research for this task had been performed by Allen (1969). He proposed a
three-layer model using Kubelka-Munk turbid media theory for translucent inks printed on
top of a highly scattering support (Kubelka and Munk, 1931; Kubelka, 1948). Somehow,
Allen abandoned the complex model proposed in 1969 in favor of a simpler approach
(Allen and Hoffenberg, 1973). Basically, they applied a thin ink film on a Mylar film and
backed the film with both black and white supports in optical contact in order to determine
two surface reflectance measurements over the printed Mylar film. The two optical
constants known as absorption, K, and scattering, S, coefficients of an arbitrary ink were
numerically estimated by using the weight of the ink film to calibrate the thickness.
According to Kubelka-Munk theory, the surface reflectance factor is a function of K, S,
the thickness of the ink film alone, and the reflectance factor of the background.
Van De Capelle and Meireson (1997) took a different approach in which the
surface reflectance factor of an ink printed on an opaque support is a function of three
parameters, conceptually similar to the absorption and scattering coefficients and an
132
additional interaction term. The determination of these three parameters requires printing
an arbitrary ink onto white, gray, and black surfaces for setting up three simultaneous
equations in order to solve for the three unknowns.
Another type of approach was recently exercised by Stollnitz, et al. (1998). It was
claimed that the surface reflectance factor of multiple ink layers sitting on top of an
opaque support is a function of the transmitance factor of each ink layer, multiple internal
reflections at each interface, and the reflectance factor of support. The scattering of each
ink layer was not considered.
Finally, Viggiano and Hoagland (1998) used the additivity of ink density to predict
ink overprints.
Unfortunately, the colorimetric and spectral accuracy for the Allen, Stollnitz, and
Viggiano studies were not disclosed. The approach by Van De Capelle has been patented
by Barco Co. Thus, it was of interest to explore whether Kubelka-Munk theory could
predict these overprints with sufficient colorimetric and spectral accuracy for spectral-
based color reproduction.
B. TECHNICAL APPROACH
The famous Kubelka-Munk basic equation (1931) is shown in Eq. (6-2):
RR a b b S X
a R b b S Xg
gλ
λ λ λ λ λ
λ λ λ λ λ
=− −
− +1 ,
,
[ coth( )]
coth( ), (6-2)
133
where λ is a wavelength within the visible spectrum, Rλ,g is the spectral reflectance factor
of an opaque support, Kλ is the absorption coefficient, Sλ is the scattering coefficient, X is
the thickness of the layer of colorant, aλ is equal to 1+(K/S)λ, bλ is equal to [(aλ)2 - 1]1/2,
and coth( ) is the hyperbolic cotangent function. The determination of K and S is carried
out by drawing down or printing, for example, a thin ink film on black and white contrast
paper, depicted as Fig. 6-2. Contrast paper whose surface is covered by a transparent
plastic layer or resin coating is normally used. This layer prevents the ink from submerging
into the paper fiber. The process utilizing contrast paper with plastic or resin coating is
similar to that of Allen and Hoffenberg’s preparation using Mylar film back-coated by
black and white paints.
Rλ,Pk Rλ,Pw
Rλ,wRλ,k
Fig. 6-2: An ink film applied on black and white contrast paper.
134
Four spectra can be attained by this technique to set up the two nonlinear
equations using Eq. (6-2) by assuming the ink thickness is unity and homogenous where
Rλ,Pw and Rλ,Pk are the spectral reflectance factor of an primary ink (P) printed over white
and black support, respectively, and Rλ,w and Rλ,k are the spectral reflectance factor of the
white and black areas of the contrast paper, respectively. First, the surface reflectance
factor, Rλ,Pw, of the ink printed on top of the white background is described by
RR a b b S
a R b b SPw
w P P P P
P w P P Pλ
λ λ λ λ λ
λ λ λ λ λ,
, , , , ,
, , , , ,
[ coth( )]
coth( )=
− −− +
1 , (6-3)
where X is assumed to be unity. Second, the surface reflectance factor, Rλ,Pk, of the same
primary ink printed on top of black background is described by
RR a b b S
a R b b SPk
k P P P P
P k P P Pλ
λ λ λ λ λ
λ λ λ λ λ,
, , , , ,
, , , , ,
[ coth( )]
coth( )=
− −− +
1 . (6-4)
Notice that the difference between Eqs. (6-3) and (6-4) is the term of Rλ,g in Eq. (6-2)
which is substituted with the spectral reflectance factor of the white support in Eq. (6-3)
and substituted with the spectral reflectance factor of the black support in Eq. (6-4). Thus,
Eqs. (6-3) and (6-4) construct a nonlinear system with two equations and two unknowns,
Kλ and Sλ. To solve this system of nonlinear equations, a numerical method based on the
techniques of operational research can be employed to estimate these two optical
constants. This is repeated for each ink of interest.
Once all the optical constants related to the assumed thickness are numerically
determined, the prediction of overprints depends on the thickness of inks actually printed
135
on a medium, such as the SWOP specified standard paper. The effective thickness for each
ink can be estimated using Eq. (6-2) by the known optical constants for each ink,
measured surface reflectance factors of each ink printed on a specific paper, and the
measured reflectance factor, Rλ,paper, for the specific paper. The equation for estimating the
thickness, typical of a printing or proofing process, of an primary ink (P) is set up by
RR a b b S X
a R b b S XP
paper P P P P P
P paper P P P Pλ
λ λ λ λ λ
λ λ λ λ λ,
, , , , ,
, , , , ,
[ coth( )]
coth( )=
− −− +
1 , (6-5)
where Rλ,P is the spectral reflectance factor of a primary ink printed on top of a specific
paper with spectral reflectance factor, Rλ,paper, and XP is the effective thickness. XP again
can be solved by a numerical method. Once Kλ, Sλ, and X for each ink are estimated, the
prediction of ink overprints is simply a recursive calculation using Eq. (6-2). That is,
taking a three-ink-layer overprint as an example, given that all sets of characteristic
parameters for all ink layers were estimated, the prediction of the topmost spectral
reflectance factor requires the knowledge about the spectral reflectance factor of the
second layer which are predicted based on the a priori knowledge about the known
spectral reflectance factor of the bottom layer.
136
Paper
Fig. 6-3: The diagram of a three-ink-layer overprint.
Figure 6-3 is shown to conceptualize this process where R Pλ, 3is the spectral reflectance
factor of primary three printed on paper with R paperλ, , R P Pλ, 2 3is the spectral reflectance
factor of primary two printed on top of the primary three, and R P P Pλ, 1 2 3is the spectral
reflectance factor of primary one printed on the topmost layer. Analytically, the estimation
of spectral reflectance factor for all three ink layers can be described by Eq. (6-6) for the
bottom layer, Eq. (6-7) for the inner layer, and Eq. (6-8) for the topmost layer,
respectively:
RR a b b S X
a R b b S XP
paper P P P P P
P paper P P P Pλ
λ λ λ λ λ
λ λ λ λ λ,
, , , , ,
, , , , ,
[ coth( )]
coth( )3
3 3 3 3 3
3 3 3 3 3
1=
− −− +
, (6-6)
RR a b b S X
a R b b S XP P
P P P P P P
P P P P P Pλ
λ λ λ λ λ
λ λ λ λ λ,
, , , , ,
, , , , ,
[ coth( )]
coth( )2 3
3 2 2 2 2 2
2 3 2 2 2 2
1=
− −− +
, (6-7)
K S X
K S X
K S X
P P P
P P P
P P P
λ λ
λ λ
λ λ
, ,
, ,
, ,
1 1 1
2 2 2
3 3 3
R Pλ , 3
R paperλ ,
R P Pλ, 2 3
R P P Pλ , 1 2 3
137
RR a b b S X
a R b b S XP P P
P P P P P P P
P P P P P P Pλ
λ λ λ λ λ
λ λ λ λ λ,
, , , , ,
, , , , ,
[ coth( )]
coth( )1 2 3
2 3 1 1 1 1 1
1 2 3 1 1 1 1
1=
− −− +
. (6-8)
The accuracy of this process was defined using the CIE94 color difference
equation calculated under standard illuminant D50 and the 1931 standard observer (CIE,
1995). The spectral accuracy is quantified both by root-mean-square (RMS) error in units
of reflectance factor and the CIE94 color difference equation calculated under standard
illuminant A and the 1931 standard observer as the metamerism index (M.I.) after
parameric correction (Fairman, 1987).
C. EXPERIMENTAL
For the convenience of verifying the technical approach described above, the
DuPont Water Proof system was used to print six primaries, which are cyan, magenta,
yellow, red, green, and blue, on six pieces of contrast paper shown in Fig. 6-4. Twenty-
five overprints, shown in Fig. 6-5, were generated using at most three-primary
combinations. Among the 25 overprints, 14 are two-color overprints and 11 are three-
color overprints. The printing order corresponds to the color order putting cyan at the
bottom-most layer, magenta on cyan, yellow on magenta, red on yellow, green on red, and
blue on the top-most layer. These samples were measured using a Gretag Spectrolino by
averaging five measurements for each color. Due to the different refractive indices among
air, ink, and support, the Saunderson correction was employed to correct for refractive-
index discontinuity at each interface (Saunderson, 1942; Allen, 1987).
138
Fig. 6-4: The six primaries printed on contrast paper.
Fig. 6-5: Twenty-five overprints printed on coated paper.
139
D. RESULTS
Equations (6-3) and (6-4) were used to set up the system of nonlinear equations
and solved for the two unknowns by assuming the thickness for each ink is unity. The
estimated Kλ and Sλ, attached in Appendix H, of the six primaries are plotted in Fig. 6-6.
400 500 600 7000
0.5
1
1.5
2
2.5
3Cyan
K o
r S
400 500 600 7000
0.5
1
1.5
2
2.5
3Magenta
400 500 600 7000
0.5
1
1.5
2
2.5
3Yellow
400 500 600 7000
0.5
1
1.5
2
2.5
3Red
Wavelength
K o
r S
400 500 600 7000
0.5
1
1.5
2
2.5
3Green
Wavelength
400 500 600 7000
0.5
1
1.5
2
2.5
3Blue
Wavelength
Fig. 6-6: The spectral absorption (solid line) and scattering (dashed line multiplied by tentimes) curves of the six primaries.
Thickness was then estimated for each primary printed on the coated paper shown in Fig.
6-5, using Eq. (6-2). Due to the high colorimetic and spectral accuracy for predicting
140
primaries, the difference spectra between measured and predicted primaries printed on the
coated paper are plotted in Fig. 6-7 and their colorimetric and spectral accuracy as well as
their statistical thicknesses are shown in Table 6-1. Thickness is a ratio related to the ink
thickness of each primary printed on contrast paper.
400 500 600 700-0.05
0
0.05
Del
ta R
Cyan
400 500 600 700-0.05
0
0.05Magenta
400 500 600 700-0.05
0
0.05Yellow
400 500 600 700-0.05
0
0.05
Del
ta R
Wavelength
Red
400 500 600 700-0.05
0
0.05
Wavelength
Green
400 500 600 700-0.05
0
0.05
Wavelength
Blue
Fig. 6-7: The difference spectra between measured and predicted primaries.
Table 6-1: The colorimetric accuracy, spectral accuracy, and the statistical thickness forthe six primaries.
Cyan Magenta Yellow Red Green Blue∆E*
94 0.8 0.3 0.1 0.4 0.2 0.8Metamerism Index 0.0 0.0 0.0 0.0 0.0 0.2RMS Error 0.005 0.002 0.004 0.004 0.001 0.004Thickness 0.956 0.958 0.975 1.026 0.962 0.987
141
According to Table 6-1, the prediction of each primary is of high spectral accuracy as
indicated by the near zero metamerism index and the low RMS error. Thus, the first
verification ensures the success using of Kubelka-Munk theory to predict the translucent
material backed by an opaque support. With the knowledge of optical constants, Kλ and
Sλ, and the effective thickness of ink deposited by a typical printing process, the estimation
of spectral reflectance factor can be accomplished whenever the paper support is changed
under the assumption that there is no interaction between ink and paper (coated paper is
preferred). Since the accuracy of the first prediction is high, the interaction between ink
and paper is considered insignificant. Second, the prediction of overprints is based on the
assumption that no chemical or physical interaction at interfaces of each ink layer. In our
experiment, the statistical colorimetric and spectral performance of predictions for the 25
overprints, shown in Table 6-2, is considered high judged by the low average and standard
deviation (Stdev) of metamerism indices whose histogram is shown in Fig. 6-8. It indicates
that almost all the overprints are predicted with high accuracy since most of estimated
metamerism indices of the 25 overprints are concentrated around 0.3 ∆E*94 units. Figures
6-9 and 6-10 are shown as examples of good predictions and predictions with relatively
low accuracy in terms of their metamerism indices. However, the spectral predictions of
these “relative low accuracy” samples are considered acceptable judged by their low
colorimetric and spectral error. The predicted spectral curves correspond well to the
142
measured spectral curves indicated by the difference spectra. The implementation of ink
overprint prediction is shown in Appendix D.
Table 6-2: The colorimetric and spectral accuracy of the 25 overprints.
∆E*94 Metamerism index
Mean 0.9 0.3Stdev 0.5 0.3Maximum 2.1 1.2Minimum 0.2 0.0RMS Error 0.004
0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .40
1
2
3
4
5
6
7
8
9
1 0
M e ta m e ris m Ind e x
Fre
qu
en
cy
Fig. 6-8: Histogram of the metamerism indices for prediction of the 25 overprints.
143
400 500 600 700-0.05
0
0.05
De
lta R
Red on Magenta
400 500 600 700-0.05
0
0.05Yellow on Magenta
400 500 600 700-0.05
0
0.05Red on Yellow
Wavelength
De
lta R
400 500 600 700-0.05
0
0.05Green on Yellow
Wavelength
Fig. 6-9: The difference spectra of four overprints best predicted with high accuracy.
400 500 600 700-0.05
0
0.05
De
lta R
G reen on Red
400 500 600 700-0.05
0
0.05Blue on Y ellow
400 500 600 700-0.05
0
0.05Blue on Cyan
Wavelength
De
lta R
400 500 600 700-0.05
0
0.05Blue on Y ellow on Cyan
Wavelength
Fig. 6-10: The difference spectra of four overprints predicted with relatively low accuracy.
∆E*94 0.2
M. I. 0.0∆E*
94 0.4M. I. 0.1
∆E*94 0.3
M. I. 0.1∆E*
94 0.4M. I. 0.1
∆E*94 2.1
M. I. 0.4∆E*
94 0.9M. I. 0.5
∆E*94 1.4
M. I. 0.7∆E*
94 1.2M. I. 1.1
144
E. DISCUSSIONS
The errors contributed to the prediction using the technical approach described
herein can be attributed to three types of error. The first is the uniformity of the paper
support to be printed. The second is the homogeneity of the ink thickness delivered by the
printing process. The last is the accuracy of the determination of the two optical constants,
Kλ and Sλ. Recall that Kλ and Sλ are solved numerically. In the process of solving for Kλ
and Sλ, the constraints of the positivity of these two optical constants should be
superimposed in numerical estimation to account for the absorption and scattering
properties of real material, i.e., the negativity of Kλ and Sλ is not realizable. Another
realistic consideration for the numerical estimation is that the term aλ in Eq. (6-2) is a
function of the ratio of absorption to scattering coefficient, (K/S)λ, at a sampled
wavelength. When estimated Sλ is zero, the numerical error of division by zero happens.
The process required setting the lower boundary of Sλ to be nonzero to prevent the
numerical error of the division by zero in addition to that Eq. (6-2) can be confounded by
the numerical result of zero divided by zero. This limitation causes the over-prediction for
the reflectance factor when both the estimated Kλ and Sλ are low and close to zero. The
strongest evidence is the prediction for cyan ink printed on the coated paper. Figure 6-11
indicates that the estimated low absorption (solid line) and scattering (dashed line) for
cyan ink happens from 440 nm to 470 nm. Its corresponding prediction of spectral
145
reflectance factor is over-predicted (the difference spectrum is obtained by the predicted
subtracted from the measured) at the same spectral region due to this numerical limitation.
4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 0
0
0 .2
0 .4
0 .6
0 .8
1
1 .2
1 .4
1 .6
1 .8
2E s t i m a te d K & S
W a ve le n g th
K o
r S
4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 0-0 .0 2
-0 .0 1 5
-0 .0 1
-0 .0 0 5
0
0 .0 0 5
0 .0 1
0 .0 1 5
0 .0 2D i f fe re n c e S p e c t ru m
W a ve le n g th
De
lta
R
Fig. 6-11: The estimated two optical constants for cyan ink and corresponding differencespectrum in units of reflectance factor.
F. CONCLUSIONS
An algorithm for predicting overprints by a proofing device was exercised and
verified. The errors contributing to this verification process are the uniformity of a coated
paper to be printed, homogeneity of the printed ink thickness, and numerical limitations
when estimating the two optical constants. Nevertheless, high colorimetric and spectral
accuracy was achieved by this approach using Kubelka-Munk turbid media theory.
146
VII. SPECTRAL-BASED SIX-COLOR SEPARATIONMINIMIZING METAMERISM
Given an input image, the task in the synthesis stage of its color reproduction using
halftone printing device is to determine the ink amount corresponding to each primary to
be delivered onto a paper substrate. This task is so called "color separation". The
determined ink amount in terms of fractional areas are stored as color separation records
corresponding to each primary ink by digital storage or conventional high contrast
lithographic film. The reproduction is accomplished by transferring the ink information in
each color-separation record digitally by an inkjet printer or conventionally by a printing
press. This chain completes the synthesis stage of a halftone printing process.
Conventionally, color separation schemes often utilize empirical ink tables or
multi-dimensional look up tables (CLUT). Such approaches require quite a few data
samples and measurements. On the contrary, analytical models frequently require relatively
small number of samples to build device profile. This is especially efficient whenever the
printing process is subject to a change of ink and paper material. Most analytical models
are based on the Murray-Davies and Neugebauer equations. The former is utilized as the
analytical description for the color formation of a single ink printed on a substrate.
Whereas, the latter is employed as the analytical description for the color formation of
multi-color printing processes.
147
Four-color modeling techniques based on the Neugebauer equation for halftone
printing process have long been disclosed and well established (Pobboravsky and Pearson,
1972; Balasubramanian, 1995; 1996; 1998; Viggiano, 1985). The quality of this type of
color synthesis is based on the accuracy of colorimetric reproduction. It suffers from the
problem of metamerism due to a lack of degrees of freedom. The use of more than four
inks for the halftone printing process not only increases the colorimetric gamut but also
increases the "spectral gamut" of the color mixtures produced by a halftone device.
Spectral gamut is defined as the set of spectra which is spanned by a set of basis primaries
used for halftone synthesis. Theoretically, the larger the spectral gamut, the higher the
probability matching a sample from the input image spectrally. Based on the economy and
production convenience as described in the introduction, this research extends the
challenge to build an analytical model for a six-color halftone printing process.
A. SIX-COLOR YULE-NIELSEN MODIFIED SPECTRALNEUGEBAUER EQUATION
An analytical estimation for the synthesis of a reflectance spectrum using six
primary inks can employ the Yule-Nielsen modified spectral Neugebauer equation
generalized to a six-primary expression, shown as Eq. (7-1),
R a Rii
in n
λ λ==∑[ ],
/
1
641 , (7-1)
148
where terms are similarly defined as that of Eq. (2-36). Notice that Eq. (7-1) is extended
to a linear sum of 64 Nuegebauer primaries. The corresponding 64 coefficients as the
fractional dot areas can be expressed again by the Demichel probability model extended to
a six-primary case shown as Eq. (7-2),
White : a1 = (1 - P1) (1 - P2) (1 - P3) (1 - P4) (1 - P5) (1 - P6)
Six Primaries
P1 : a2 = P1 (1 - P2) (1 - P3) (1 - P4) (1 - P5) (1 - P6)P2 : a3 = (1 - P1) P2 (1 - P3) (1 - P4) (1 - P5) (1 - P6)P3 : a4 = (1 - P1) (1 - P2) P3 (1 - P4) (1 - P5) (1 - P6)P4 : a5 = (1 - P1) (1 - P2) (1 - P3) P4 (1 - P5) (1 - P6)P5 : a6 = (1 - P1) (1 - P3) (1 - P3) (1 - P4) P5 (1 - P6)P6 : a7 = (1 - P1) (1 - P3) (1 - P3) (1 - P4) (1 - P5) P6
Secondaries (fifteen two-color overprints)
P1P2 : a8 = P1 P2 (1 - P3) (1 - P4) (1 - P5) (1 - P6)P1P3 : a9 = P1 (1 - P2) P3 (1 - P4) (1 - P5) (1 - P6)P1P4 : a10 = P1 (1 - P2) (1 - P3) P4 (1 - P5) (1 - P6)P1P5 : a11 = P1 (1 - P2) (1 - P3) (1 - P4) P5 (1 - P6)P1P6 : a12 = P1 (1 - P2) (1 - P3) (1 - P4) (1 - P5) P6
P2P3 : a13 = (1 - P1) P2 P3 (1 - P4) (1 - P5) (1 - P6)P2P4 : a14 = (1 - P1) P2 (1 - P3) P4 (1 - P5) (1 - P6)P2P5 : a15 = (1 - P1) P2 (1 - P3) (1 - P4) P5 (1 - P6)P2P6 : a16 = (1 - P1) P2 (1 - P3) (1 - P4) (1 - P5) P6
P3P4 : a17 = (1 - P1) (1 - P2) P3 P4 (1 - P5) (1 - P6)P3P5 : a18 = (1 - P1) (1 - P2) P3 (1 - P4) P5 (1 - P6)P3P6 : a19 = (1 - P1) (1 - P2) P3 (1 - P4) (1 - P5) P6
P4P5 : a20 = (1 - P1) (1 - P2) (1 - P3) P4 P5 (1 - P6)P4P6 : a21 = (1 - P1) (1 - P2) (1 - P3) P4 (1 - P5) P6
P5P6 : a22 = (1 - P1) (1 - P2) (1 - P3) (1 - P4) P5 P6
Tertiaries (twenty three-color overprints)
149
P1P2P3 : a23 = P1 P2 P3 (1 - P4) (1 - P5) (1 - P6)P1P2P4 : a24 = P1 P2 (1 - P3) P4 (1 - P5) (1 - P6)P1P2P5 : a25 = P1 P2 (1 - P3) (1 - P4) P5 (1 - P6)P1P2P6 : a26 = P1 P2 (1 - P3) (1 - P4) (1 - P5) P6
P1P3P4 : a27 = P1 (1 - P2) P3 P4 (1 - P5) (1 - P6)P1P3P5 : a28 = P1 (1 - P2) P3 (1 - P4) P5 (1 - P6)P1P3P6 : a29 = P1 (1 - P2) P3 (1 - P4) (1 - P5) P6
P1P4P5 : a30 = P1 (1 - P2) (1 - P3) P4 P5 (1 - P6)P1P4P6 : a31 = P1 (1 - P2) (1 - P3) P4 (1 - P5) P6
P1P5P6 : a32 = P1 (1 - P2) (1 - P3) (1 - P4) P5 P6, (7-2)P2P3P4 : a33 = (1 - P1) P2 P3 P4 (1 - P5) (1 - P6)P2P3P5 : a34 = (1 - P1) P2 P3 (1 - P4) P5 (1 - P6)P2P3P6 : a35 = (1 - P1) P2 P3 (1 - P4) (1 - P5) P6
P2P4P5 : a36 = (1 - P1) P2 (1 - P3) P4 P5 (1 - P6)P2P4P6 : a37 = (1 - P1) P2 (1 - P3) P4 (1 - P5) P6
P2P5P6 : a38 = (1 - P1) P2 (1 - P3) (1 - P4) P5 P6
P3P4P5 : a39 = (1 - P1) (1 - P2) P3 P4 P5 (1 - P6)P3P4P6 : a40 = (1 - P1) (1 - P2) P3 P4 (1 - P5) P6
P3P5P6 : a41 = (1 - P1) (1 - P2) P3 (1 - P4) P5 P6
P4P5P6 : a42 = (1 - P1) (1 - P2) (1 - P3) P4 P5 P6
Quaternaries (fifteen four-color overprints)
P1P2P3P4 : a43 = P1 P2 P3 P4 (1 - P5) (1 - P6)P1P2P3P5 : a44 = P1 P2 P3 (1 - P4) P5 (1 - P6)P1P2P3P6 : a45 = P1 P2 P3 (1 - P4) (1 - P5) P6
P1P2P4P5 : a46 = P1 P2 (1 - P3) P4 P5 (1 - P6)P1P2P4P6 : a47 = P1 P2 (1 - P3) P4 (1 - P5) P6
P1P2P5P6 : a48 = P1 P2 (1 - P3) (1 - P4) P5 P6
P1P3P4P5 : a49 = P1 (1 - P2) P3 P4 P5 (1 - P6)P1P3P4P6 : a50 = P1 (1 - P2) P3 P4 (1 - P5) P6
P1P3P5P6 : a51 = P1 (1 - P2) P3 (1 - P4) P5 P6
P1P4P5P6 : a52 = P1 (1 - P2) (1 - P3) P4 P5 P6
P2P3P4P5 : a53 = (1 - P1) P2 P3 P4 P5 (1 - P6)P2P3P4P6 : a54 = (1 - P1) P2 P3 P4 (1 - P5) P6
P2P3P5P6 : a55 = (1 - P1) P2 P3 (1 - P4) P5 P6
P2P4P5P6 : a56 = (1 - P1) P2 (1 - P3) P4 P5 P6
P3P4P5P6 : a57 = (1 - P1) (1 - P2) P3 P4 P5 P6
Quinaries (six five-color overprints)
150
P1P2P3P4P5 : a58 = P1 P2 P3 P4 P5 (1 - P6)P1P2P3P4P6 : a59 = P1 P2 P3 P4 (1 - P5) P6
P1P2P3P5P6 : a60 = P1 P2 P3 (1 - P4) P5) P6
P1P2P4P5P6 : a61 = P1 P2 (1 - P3) P4 P5 P6
P1P3P4P5P6 : a62 = P1 (1 - P2) P3 P4 P5 P6
P2P3P4P5P6 : a63 = (1 - P1) P2 P3 P4 P5 P6
Hexary (one six-color overprint)
P1P2P3P4P5P6 : a64 = P1 P2 P3 P4 P5 P6
where P1, P2,…, P6 represent the six-primary inks.
Although Eq. (7-1) theoretically describes the synthesis of a desired color
spectrum, whether or not an actual six-color halftone printing process which is capable of
abiding by this analytical description is not known. The failure is due to the ink-trapping
limitation which is defined as the capability in terms of percentage of ink amount for
successive inks printed on top of the preceding ink. At the stage of ink-trapping, the
preceding inks are still wet. Several practical observations have reported that ink-trapping
lithography off-set printing is limited to 300~400% of total ink amount printed over a
fixed area. Hence, it is very likely that the validity of Eq. (7-1) is confined by this physical
limitation.
From Eq. (7-1), a given reflectance spectrum requires fitting by the linear
combination of 64 basis spectra† (Neugebauer primaries). The redundancy among those 64
basis spectra is worthwhile checking since the linear independency among them is not a
151
certainty. It can be reasoned by an example of six-color halftone printing process using
cyan, magenta, yellow, green, orange, and black inks. Consider that the quinary and
hexary as well as some quaternary whose spectra are flat are neutral colors. It is highly
probable that those Neugebauer primaries can be approximated by linear combinations of
others. If so, then the redundancy of those Neugebauer primaries is high. Their
contribution to the synthesis of a spectrum is insignificant.
Based on the physical limitation of a certain printing process, the use of six-color
Yule-Nielsen modified spectral Neugebauer equation seems impractical. In addition, the
speculation on the basis spectra indicated that the existence of five or six color overprints
may not be significant for spectral reproduction. Thus, an alternative approach which
takes advantage of more degrees of freedom as well as abiding by the physical limitation
of a particular printing process is desired.
B. AN ALTERNATIVE APPROACH USING SIX-COLORHALFTONE PRINTING PROCESS MINIMIZING METAMERISM
The current research project proposes a paradigmatic algorithm in balancing
between the spectral gamut and physical limitation due to ink-trapping failure. These
modules for the current research development are sequentially outlined in Fig. 7-1.
† We don’t want to call the Neugebauer primaries as the basis vectors since they are not necessary linearlyindependent. Recall that basis vectors are defined as the set of vectors which are not only linearlyindependent but also span the entire vector space.
152
Subdivisionof Six-ColorModeling
Forward Four-ColorHalftone SpectralPrinting Models
Proper Four-ColorSub-Model Selection
Backward PrintingModels for Six-ColorSeparation MinimizingMetamerism
Fig. 7-1: The structure chart for the development of six-color separation minimizingmetamerism.
Subdivision of Six-Color Modeling
The task of analytical modeling for a six-color halftone printing process is
subdivided into k four-color modeling processes in order to comply with the ink-trapping
limitation. The number, k, depends on the division algorithm and will be specified later in
this section. The six-color printing model can be viewed as the super-model of the k four-
color printing sub-models. The spectral gamut of the six-color printing model is assumed
to be a good approximation to the spectral gamut of an input image by the ink-selection
algorithm. If the ink-selection algorithm suggests an optimal inkset, for an arbitrary
example, comprised by cyan (C), magenta (M), yellow (Y), green (G), orange (O), and
153
black (K) inks and a division algorithm recommends the use of one black and three
chromatic inks to synthesize each pixel of a given spectral image, then there are ten four-
color printing sub-models to be constructed. They are CMYK, CMGK, CMOK, CYGK,
CYOK, CGOK, MYGK, MYOK, MGOK, and YGOK.
Forward Four-Color Halftone Spectral Printing Models
Since the modeling of six-color halftone printing process is subdivided into ten
four-color modeling processes, a modeling process for CMYK halftone printing will be
utilized for discussion. Then, it is generalized for any four-color printing process using
different primaries.
1. The first-order forward model
The mission for a forward model is to obtain an accurate estimation of a
synthesized spectrum given a set of requested (or theoretical) dot areas "dialed in" by a
printer. Owing to mechanical and optical dot gain, the effective dot size printed on a
substrate is different from the theoretical dot size. Hence, the first task is to determine the
Yule-Nielsen n-factor, the second is to relate the theoretical dot area to the effective dot
area through a mathematical function or look up table (LUT), and finally, the estimated
spectrum is estimated by the Yule-Nielsen modified spectral Neugebauer equation. The
chain of first-order forward modeling process based on Yule-Nielsen modified spectral
Neugebauer equation is outlined in Fig. 7-2.
154
cmyk(theoretical dot areas)
f(c, m, y, k)
Nλ,n (c’, m’, y’, k’)
c’m’y’k’(effective dot areas)
Rλ,synthesis
Fig. 7-2: The structure of a general forward halftone printing model where f( ) is amathematical function or LUT describing the dot-gain effect and Nλ,n( ) is the function of
Yule-Nielsen modified spectral Neugebauer equation.
In order to complete the forward printing model, the transfer function relating the
theoretical to effective dot areas, f( ), and the Yule-Nielsen n-factor need to be uncovered
for a particular halftone printing process. The n-factor is utilized to compensate for the
non-linearity using the Neugebauer equation. Hence, the use of n-factor accounts for both
the mechanical and optical dot-gain behavior. Solving for n-factor corresponding to a
halftone printing system usually comes first followed by a determination of theoretical to
effective dot area transfer function, f( ). This modeling procedure only requires a sample
preparation of primary ramps of different fractional dot area and overprints of all
combinations of them.
155
From the primary ramps, the color formation can be described by the Yule-Nielsen
modified spectral Murray-Davies equation, specified as Eq. (2-33). By inverting Eq. (2-
33), the effective dot area of each patch can be estimated. The inversion of Eq. (2-33) in
terms of fractional dot area is shown as
aR R
R Rkn
papern
npapern=
−−
λ λ
λ λ
, %/
,/
,/
,/
1 1
100%1 1 , (7-3)
where a is the solved effective dot area corresponding to a primary printed at theoretical
dot area to be k% (0 ≤ k ≤ 100 ) whose spectral reflectance factor is Rλ,k%, Rλ,100% is the
spectral reflectance factor of the primary printed at 100% ink coverage, and Rλ,paper is the
spectral reflectance factor of the paper substrate. Notice that a = k% if n=1, i.e, there is no
dot-gain effect or the effective dot area is equal to the theoretical dot area.
With the solved fractional dot area, a, the predicted spectral reflectance factor for
each patch can be estimated by Eq. (2-33) given a known n-factor. Hence, the optimality
of n-factor determines the predicting accuracy of the estimated reflectance spectra. The
optimization of n-factor can be carried out by the procedure outlined in Fig. 7-3. Figure 7-
3 suggests the use of all the patches of primary ramps to step through all values of n-factor
with increment of ∆n and find an optimal n value corresponding to the smallest average
∆E*94 of the predictions of all patches.
156
INV(YNMD) YNMD
n = 1
R kλ, %
if ∆E*94 is minimum
�
, %R kλ
Yesn
no
n = n + ∆na
Fig. 7-3: The algorithm structure of determining the Yule-Nielsen n-factor whereINV(YNMD) stands for the inverse function, Eq. (7-3), of n-factor corrected spectral
Murray-Davies equation.
Once the n-factor is optimized, the corresponding solved effective dot area, a, for each
patch of each primary ramp is used to build the transfer function, f( ), by higher order
polynomials or nonlinear interpolation by cubic spline functions. A set of transfer
functions, depicted in Fig. 7-4, determined from a printing process is shown as an
example.
157
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Theoretical dot area
Effe
ctiv
e do
t are
a
cyanmagentayellowblack
Fig. 7-4: A set of theoretical to effective dot area transfer functions determined from aCMYK halftone printing process.
2. Second order improvement (modeling for ink- and optical-trapping)
The first-order forward printing model does not take ink trapping and optical
interaction into account. Its model accuracy is confined by these two physical effects. It
assumes that 100% ink-trapping capability and the optical dot gain of a multi-layer ink
overprint is the same as that of single primary layer. Since the percentage of ink trapping
varies for each printing process, the 100% assumption is not optimal. Furthermore, the
optical dot gain is obviously different among multi-layer overprints. This concept, termed
as optical-trapping, can be found in the literature published by Iino and Berns (1998).
Both ink trapping and optical trapping caused the first-order prediction to be too dark for
a sample; that is, the effective dot areas by printing the successive ink on top of the
preceding ink are smaller than expected. Hence, by overestimating the effective dot areas
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for synthesizing a sample requiring reproduction, the resultant reflectance spectrum by the
first-order forward printing model is under predicted.
Iino and Berns proposed the use of a correction factor, q, to rectify the effective
dot area estimated by the theoretical to effective dot area transfer function, f( ), for the
first-order printing model. This correction factor originated from an observation on the
dot gain effect of a primary ramp at the existence of another primary. Dot-gain, g, can be
quantitatively defined as the difference between effective dot area, aeff, and theoretical dot
area, atheo, shown as Eq. (7-4),
g = aeff - atheo . (7-4)
Figure 7-5 shows the dot gain curves (g vs. theoretical dot area) of the CMYK ramps
given in Fig. 7-4.
0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 10
0 .05
0 .1
0 .15
0 .2
0 .25
0 .3
0 .35
0 .4
Theoretica l do t a rea
Do
t-g
ain
in
un
its
of
fra
ctio
nal
do
t a
rea
c yanma gentayello wblack
Fig. 7-5: The dot-gain functions of the CMYK ramps given in Fig. 7-4.
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Iino and Berns hypothesized each primary ink has its inherent dot-gain function
which can be obtained from its corresponding primary ramp. They further hypothesized
that the dot-gain functions with respect to a primary varies their extent but not shape
given the existence of other primaries. For example, the family of dot-gain curves of a
cyan ramp given that the magenta ink is present at 0%, 25% 50%, and 75% fractional dot
areas is plotted in Fig. 7-6.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Theoretical dot area
Do
t-ga
in in
un
its o
f fr
actio
nal
do
t a
rea
mag=0%mag=25%mag=50%mag=75%
Fig. 7-6: The family of dot-gain curves of a cyan ramp when magenta ink presents at 0%,25%, 50%, and 75% fractional dot areas.
Thus, when magenta ink is present at 0%, the corresponding curve whose extent is the
highest describes the dot-gain behavior of the cyan primary alone, denoted as gc. When
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magenta ink presents at nonzero dot area, the dot-gain of the cyan ramp is reduced. The
percentage of reduction is represented by a scalar, q, which depends on the existence of
other primary inks, where 0 ≤ q ≤ 1. Hence, quantitative description of global dot-gain int
the presence of another primary, g, is the product of q and the dot-gain, gp, of a primary
ramp, i.e.,
g = qgp = q(aeff, priamry - atheo,primary), (7-5)
where aeff, priamry and atheo,primary are the effective and theoretical dot areas of a primary,
respectively.
Intuitively, the correction scalar, q, for a primary ink is a function of the theoretical
dot area of the second ink. It was assumed that the dot-gain variance for an arbitrary
primary printed at 50% theoretical dot area is maximized although variance of dot-gain
curves shown in Fig. 7-6 seems to peak around 45%. (The location of peak variance is
related to the dot shape formed by a specific halftone screening process. Usually, the
elliptical, diamond dot shape, and the dot shape formed by FM screening peaks around
45% theoretical dot area, whereas, the round and square dot shape peak around at 50%
theoretical dot area.) Hence, to model this correction scalar, it is required to print a
primary at 50% theoretical dot area and varying the second ink from 0% to 100% by
assuming the dot-gain variance peaks at 50% theoretical area for a particular halftone
printing process. Let the gi=50%,j be the dot gain of a primary i printed at 50% theoretical
dot area given that the second primary j is present at a known theoretical dot area,
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gi=50%,j = aeff,i=50%,j - 0.5 , (7-6)
where aeff,i=50%,j is the effective dot area estimated using the inverse Yule-Nielsen modified
spectral Neugebauer equation of the primary i at 50% theoretical dot area given that the
second primary j is present at a known theoretical dot area. Further, the dot gain, gi=50%, of
the primary i alone printed on a paper substrate at 50% theoretical dot area is the
difference of the corresponding effective dot area, aeff,i=50% and atheo,i, i.e.,
gi=50% = aeff,i=50% - atheo,i , (7-7)
where atheo = 0.5. Thereby, the correction scalar, q, is the ratio of gi=50%,j to gi=50%, i.e.,
q = gi=50%,j / gi=50% = (aeff,i=50%,j - atheo,i) / (aeff,i=50% - atheo,i) . (7-8)
Since q varies with the theoretical dot area of the second primary j, atheo,j, q is defined as a
function of atheo,j, that is, q = fi_j(atheo,j). Table 7-1 is the example published by Iino and
Berns (1998). They fixed cyan ink at 50% theoretical dot area and overlapping the
magenta ink at 0%, 25%, 50%, 75% and 100%.
Table 7-1: The effective dot areas and the correction scalar, q, of cyan fixed at 50%theoretical dot area by overlapping the secondary magenta ink at various theoretical dotareas (Iino and Berns, 1998).
TheoreticalDot area Effective Dot area Correction scalar1 2 3 4 5
Cyan Magenta Cyan Magenta q0.500 0.000 0.629 0.000 1.0000.500 0.250 0.613 0.320 0.8760.500 0.500 0.597 0.605 0.7520.500 0.750 0.592 0.809 0.7130.500 1.000 0.594 1.000 0.729
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Using Eq. (7-6) the dot-gain of the cyan ink when the secondary magenta ink exists,
gc=50%,m, is the difference between the column 3 and column 1. Since the dot-gain of cyan
ink at 50% printed at paper alone (magenta ink is not present) by Eq. (7-7) is 0.129
(0.629 - 0.5). Hence, by Eq. (7-8), the correction scalar, q, listed in column 5, of the cyan
ink at 50% when the magenta ink is present is gc=50%,m / 0.129.
Iino and Berns further generalized the dot-gain correction scalar, q, for tertiary,
and quaternary overprint of a four-color halftone process. They hypothesized that the dot-
gain correction scalar is the product of correction scalar of each combination of secondary
overprint. For example if i, j and s primaries coexist, the correction scalar for primary i, qi,
is equal to the product of fi_j(atheo,j) and fi_s(atheo,s). For the CMYK halftone printing
process, the global correction scalar for each primary is defined as
qc = fc_m(atheo,m) fc_y(atheo,y) fc_k(atheo,k)qm = fm_c(atheo,c) fm_y(atheo,y) fm_k(atheo,k) . (7-9)qy = fy_c(atheo,c) fy_m(atheo,m) fy_k(atheo,k)qk = fk_c(atheo,c) fk_m(atheo,m) fk_y(atheo,y)
Hence, final effective dot area is the sum of the theoretical dot area and dot-gain, i.e.,
aeff = atheo + g = atheo + qgp = atheo + q(aeff,primary - atheo,primary). (7-10)
3. Alternative second order improvement
The algorithms proposed by Iino and Berns resulted in significant and impressive
improvement for CMYK processed. Whereas, these algorithms implemented for processes
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using primaries other than CMYK (e.g., CMGK, CMOK, MYOK,…, etc.) do not
significantly improve or sometimes worsen the accuracy relative to the first-order printing
model. Whether or not the dot-gain characteristic of the process using other than CMYK
does not agree with the Iino and Berns' assumptions, it is difficult to conclude. Since our
goal is to use various four-color processes to enhance the capability of spectral
reproduction, it is necessary to uncover other algorithms which are capable of modeling
the ink- and optical-trapping of any primary combination for the second order
improvement.
Conceptually, the proposed modification is not too different from the algorithms
proposed by Iino and Berns. The difference is in the determination of the correction scalar,
q. The alternative approach (using the Iino and Berns' notation) is to look at the dot-gain
effect of a primary ramp given that the secondary is present at 50% theoretical dot area.
The use of secondary at 50% theoretical dot area is hypothesized to have the average
influence on the dot-gain characteristic of the primary ramp. Hence, dot gain of a primary j
at theoretical dot area p% given that the existence of the secondary i at 50% theoretical
dot area† is
gi=50%,j=p% = aeff,i=50%,j=p% - p/100 . (7-11)
By Eq.(7-7), the dot gain of the primary j printed alone (secondary i is not presented) at
p% theoretical dot area is
† The intention of assigning j to be the primary and i to be the secondary is to be consistent with the Iinoand Benrs' notation.
164
gj=p% = aeff,j=p% - atheo,j , (7-12)
where atheo,j = p/100. Therefore, the correction scalar, q, is the ratio of gi=50%,j=p% to gj=p%,
i.e,
q = gi=50%,j=p% / gj=p% , (7-13)
Similarly, q varies with the theoretical dot area of the primary j, atheo,j, q is again defined as
a function of atheo,j, that is, q = gi_j(atheo,j). Table 7-2 shows an example of the determined q
scalar and dot gains of primary (magenta) given that the secondary (cyan) is present where
the estimation of effective dot area is performed by inverting the Yule-Nielsen modified
spectral Neugebauer equation.
Table 7-2: The determined q scalars by proposed modification and the dot-gain of theprimary (magenta) given that the secondary (cyan) is present.
Theoretical Dot area Effective Dot area Correction scalar1 2 3 4 5
Cyan Magenta Magenta Magenta q(with (withoutCyan) Cyan)
0.500 0.000 0 0 1.0000.500 0.250 0.503 0.546 0.8560.500 0.500 0.808 0.813 0.9840.500 0.700 0.919 0.931 0.9460.500 0.900 0.979 0.990 0.872
By Eq. (7-13), the correction scalar, q, column 5, is obtained by (column 3 - column 2) /
(column 4 - column 2).
165
To generalize the dot gain correction scalar, q, for tertiary and quaternary
overprints of a four-color halftone process, it uses the similar hypothesis to the Iino and
Berns that the dot-gain correction scalar is the product of the correction scalar of each
combination of secondary (two-color) overprint. For example if i, j and s primaries
coexist, the correction scalar for primary j, qj , is equal to the product of gi_j(atheo,j) and
gs_j(atheo,j). For the CMYK halftone printing process, the global correction scalar for each
primary is defined as
qc = gm_c(atheo,c) gy_c(atheo,c) gk_c(atheo,c)qm = gc_m(atheo,m) gy_m(atheo,m) gk_m(atheo,m) . (7-14)qy = gc_y(atheo,y) gm_y(atheo,y) gk_y(atheo,y)qk = gc_k(atheo,k) gm_k(atheo,k) gy_k(atheo,k)
Iino and Berns' algorithms are different from the proposed alternative algorithms.
Conceptually, Iino and Berns' algorithms suggested that, for example, for cyan and
magenta ink mixtures, whose theoretical dot areas are m% and k%, respectively, the dot
gain of the cyan is normalized to the dot-gain locus of the cyan when the magenta is
present at k%. Similarly, the dot gain of the magenta is normalized to the dot gain locus of
the magenta when the cyan is present at m%. Whereas, the proposed algorithms
recommend that the optimal dot gain of the cyan is normalized to the dot-gain locus of the
magenta presenting at 50% theoretical dot area whenever the magenta is present at any
theoretical dot area other than zero.
166
To be more specific, when cyan and magenta are present at 50% and 25%
theoretical dot areas, respectively. The dot gain of the cyan estimated by the first-order
model is the point a3 in Fig. 7-7. After Iino and Berns' modification the estimated dot gain
of the cyan subjected to the optical trapping is at point b3. Whereas, the suggested dot
gain of the cyan subjected to the optical trapping by the proposed algorithms is at point c3.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Theoretical dot area
Do
t-g
ain
in u
nits
of
fra
ctio
na
l dot
are
a
mag=0%mag=25%mag=50%mag=75%
Fig. 7-7: The dot-gain loci of a cyan ramp when a magenta ink is present at differenttheoretical dot areas where the locus goes through a3, b3, and c3 are the dot gains esimated
by the first-order model, Iino and Berns' algoritms, and the proposed algorithms,respectively.
Iino and Berns modeled the optical trapping by defining the dot gain correction scalar, q =
fc_m(atheo,m), exemplified by Fig. 7-7, to be the ratio of the dot gain value of the cyan at
a1
a2 a3
a4
c1
c2 c3
c4
b3
d3
167
50% theoretical dot area when the magenta is present at atheo,m to the dot gain value of the
cyan at 50% theoretical dot area when the magenta is not present. Whereas, the proposed
algorithm modeled the optical trapping by defining the dot gain correction scalar, q =
gc_m(atheo,c), to be the ratio of the dot gain value of the cyan at atheo,c theoretical dot area
when the magenta is present at any theoretical area other than zero to the dot gain value
of the cyan at atheo,c theoretical dot area when the magenta is not present. The difference
between the two second-order improvements is shown by Fig. 7- 8.
0 0 .2 0 .4 0 .6 0 .80
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
at h e o , m
f cm
T h e c o rr e c ti o n s c a la r b y Ii n o a n d B e r n s
0 0 .2 0 .4 0 .6 0 .80 .5
0 .5 5
0 .6
0 .6 5
0 .7
0 .7 5
0 .8
0 .8 5
0 .9
0 .9 5
1
at h e o , c
gm
c
T h e c o rr e c ti o n s c a la r b y th e p ro p o s e d a lg o r i th m
Fig. 7- 8: The functions of dot gain corrrection scalar by Iino and Berns (left) and theproposed (right) algorithms.
a3/a3
b3/a3
c3/a3
d3/a3
c1/a1
c2/a2
c3/a3
c4/a4
168
The examples in Fig. 7- 8 pictorially show how the correction scalars associated with the
two second-order improvements were obtained. Notice that the fc_m, shown as an
monotonic function, is not necessary an monotonic function since the dot gain loci of real
situation is not necessary as regularly shaped and spaced as that shown in Fig. 7-7.
4. Modeling by matrix transformation
One straightforward approach to model a halftone printing process is to directly
relate the theoretical dot area to the effective dot area of the printed sample through a
matrix transformation. This requires extensive sampling of the colorimetric or spectral
gamut to achieve high accuracy. Ideally, the set of effective dot areas of the representative
sampling on a particular printing process is equal to the product of a transformation and
the set of corresponding theoretical dot areas, i.e.,
aeff = M atheo , (7-15)
where aeff and atheo are the vector-matrix representations of effective and theoretical dot
areas, respectively and M is a transformation matrix. Realistically, the effective dot area
obtained by inverting Yule-Nielsen modified spectral Neugebauer equation is highly
nonlinearly related to the theoretical dot area. Thus, the theoretical dot area needs to
undergo a transformation to a representation which is linearly related to the effective dot
area. Thereafter, the transformation matrix, M , can be derived by the pseudo-inverse
matrix operation.
169
The most frequently used process in finding a transformation is through polynomial
regression. Its process is to represent the theoretical dot area by a higher order
polynomial. For example, assuming the set of theoretical dot areas for a CMYK printing
process is represented as c, m, y, and k, in vector-matrix form, the corresponding matrix
of a second order polynomial representation is [c m y k c2 m2 y 2 k2 cm cy ck my mk
yk]. Hence, the transformation matrix, M , can be obtained by
M = pinv([c m y k c2 m2 y 2 k2 cm cy ck my mk yk]) aeff , (7-16)
where pinv( ) stands for the pseudo-inverse function.
Proper Four-Color Sub-Model Selection
At the color separation stage, it is necessary to decide a set of four inks (one black
and three chromatic inks) associated with a four-color sub-model for synthesizing an input
spectrum abiding by the ink limiting scheme. Conventionally, the selection can be done by
locating a colorimetric value requiring reproduction inside a colorimetric gamut spanned
by a set of candidate inks. The decision is based on whether the set of candidate inks
whose colorimetric gamut includes the colorimetric value requiring reproduction. This
requires a printing model to populate the colorimetric gamut boundary of the set of
candidate inks and an inclusion algorithm to determine whether the colorimetric value is
interior to the colorimetric gamut.
This conventional method is not suitable for our challenge to accomplish spectral
reproduction, since the higher dimensional spectral gamut is impossible to visualize. The
170
use of inclusion algorithms is computationally intensive for determining an interior point
inside a spectral gamut spanned by a set of candidate inks. If there exists a linear color
mixing space that approximates the color formation of the employed halftone printing
process, then it is possible to select the most significant set of four inks through regression
or select the most significant set of four inks with the minimum reconstructing error in the
linear color mixing space. It is of great interest to derive a linear color mixing space for a
halftone printing process utilized by the experiment.
1. Deriving a linear color mixing transformation for a halftone printing process
Although the derivation of the linear color mixing space for halftone printing process has
been shown in Chapter V, it is described here again for the completeness of this chapter.
First, it should agree with subtractive color mixing. Second, the dimensionality should
coincide with the number of primaries to span the spectral gamut in the resultant mixing
space. The transformation and its inverse transformation for halftone color are empirically
derived as
Ψλ λ λ= −R Rpaperw w,
1 1
and (7-17)
R R paperw w
λ λ λ= −( ),
1
Ψ , (7-18)
respectively, where the Rλ,paper is the spectral reflectance factor of the paper substrate
being printed on by primary inks and 2 ≤ w ≤ ∞. The transformation of reflectance factor
to the empirically derived space is somewhat different from Eq. (3-6) since Eq. (3-6) is
171
derived for opaque colorants. Whereas, they have basically the same structure, one offset
vector accounting for subtractive color mixing and a higher order power to account for
the nonlinearity. The use of R paperwλ ,
1
as the offset vector has a significant meaning.
Consider that transforming a spectrum, which is exactly Rλ,paper, to the linear color mixing
space, the result is a zero vector. This corresponds to the fact that there is not any primary
presented in the linear space. Furthermore, Eq. (7-18) transforms a zero in the linear space
back to the exact reflectance spectrum of the paper, Rλ,paper.
2. Selecting the most significant four primaries from the defined six-colors inkset
The next step is to set up a regression model using the linear color mixing space to
determined the suitable set of three chromatic primaries by constraining on the absolute
existence of black ink for a given input spectral image pixel by pixel. One can utilize any
statistical software with a stepwise option to remove the two least significant chromatic
inks out of five. Whereas, for a robust process, it is suggested to try all the ten
combinations of a set of four primaries, which are partitioned by the proposed method
described in the section of sub-division of six color modeling, for a given pixel. The set
with the least spectral error reconstruction is the candidate for the particular pixel.
Backward Printing Models for Six-Color Separation Minimizing Metamerism
The last component for the proposed six-color separation algorithm, which
minimizes the metamerism between a given input spectral image and its spectral
reproduction, is to estimate the set of theoretical dot areas corresponding to each pixel.
172
The estimated theoretical dot areas will be stored as six separation records corresponding
to each primary ink. To accomplish this, a backward printer model needs to be derived.
Since it is not possible to analytically invert the forward printing model, the most feasible
approach is to numerically invert the forward printing model. Thus, the backward six-
color printing model is comprised of the proposed six-color forward spectral halftone
printing model and a optimization module using the Simplex or Newton-Raphson iterative
method to estimated a set of optimal theoretical dot areas which minimizes metamerism
pixel by pixel between an input spectral image and it reproduction. For this prototypical
research, the constrained optimization function in the optimization toolbox of MATLAB is
utilized for the numerical engine. The structure of the six-color backward spectral printing
model is depicted in Fig. 7-9.
In Fig. 7-9, those modules inside the dashed box construct the proposed six-color
forward spectral halftone printing model. Outside the dashed box, the process is handled
by a numerical optimization engine. The process of spectral-based six-color separation
starts from a given reflectance spectrum depicted as an input module located at the lower
left corner. The proper four inks for reproducing an input spectrum are determined by the
four-ink selector. Then, the optimization initializes a set of theoretical dot areas, which are
the concentrations estimated in the linear color mixing space, corresponding to the
selected four inks in the throughput processes for estimating the reconstructed spectrum,
that is the output of the forward model and located at the lower right corner. The decision
173
module in the diamond box located in the lower center then compares the input and output
spectra at this iteration stage. If the spectral reconstruction satisfies the error criteria then
output the set of theoretical dot areas at current iteration stage and terminate the iteration.
If not, the following module located near the input module decides and feeds back the
modification of the set of theoretical areas for the next iteration.
f(c, m, y,k) Nλ,n(c’, m’, y’, k’)c’m’ y’k’
Second OrderImprovement
f(c, m, g,k) Nλ,n(c’, m’, g’, k’)c’m’ g’k’
f(c, m, o,k) Nλ,n(c’, m’, o’, k’)c’m’g’k’
f(c, y, g, k) Nλ,n(c’, y’, g’, k’)c’y’g’k’
f(c, y,o, k) Nλ,n(c’, y’, o’, k’)c’y’o’k’
f(c, g,o, k) Nλ,n(c’, g’,o’, k’)c’g’o’k’
f(m,y,g, k) Nλ,n(m’, y’, g’, k’)m’y’g’k’
f(m,y,o, k) Nλ,n(m’, y’, o’, k’)m’y’o’k’
f(m,g,o, k) Nλ,n(m’, g’, o’, k’)m’g’o’k’
f(y,g,o, k) Nλ,n(y’,g’, o’, k’)y’g’o’k’
Four-InkSelector
Rλ,sample
cmyk
cmgk
cm k
cygk
cyok
cgok
mygk
myok
mgok
ygok
InitialTheoreticalDot Area
(optional)
Rλ,estimated
ChangeTheoreticalDot Area
No
Output the CurrentTheoretical Dot Area
Yes
Minimum Spectral Error orMinimum Four Tristimulus Error
Six-Color Forward Spectral Halftone Printing Model
Effective Dot Areas
Fig. 7-9: The structure of the six-color backward spectral printing model using cyan,magenta, yellow, green, orange and black ink as printing primaries.
174
Initially, the optimization criteria is to minimize the spectral error. If the spectral
minimization yields unsatisfactory accuracy, then the optimization pursues to balance
between spectral and colorimetric accuracy. Because the six-color forward spectral
printing model is composed by ten forward four-color sub-models, the optimization
criteria for balancing between spectral and colorimetric accuracy is to minimize the total
error of four tristimulus values. The four tristimulus values are obtain by the tristimulus
values under standard viewing illumant such as D50 for printing industry. Then, the fourth
value is the X value calculated under second standard viewing illumant such as A, based
on the research by Allen (1980).
C. EXPERIMENTAL AND VERIFICATION
In order to verify the proposed algorithm in building a six-color spectral halftone
printing model, the current research utilized the DuPont Waterproof® system to represent
a halftone printing process. The standard cyan (C), magenta (M), yellow (Y), black (K),
green (G), and orange (O) designed for DuPont Waterproof® system were advised to be
used as the six printing primaries for a stable proofing process. The choice, which was an
arbitrary decision, of these colors was not defined by the optimal inkset selection
algorithm since a spectral image was not given. Although the six primaries were not
defined by the optimal inkset selection algorithm, the development of the six-color spectral
printing model is independent from any of previous colorant estimation, optimal inkset
175
selection, and overprint prediction modules. Thus, this research assumes that the usage of
the six primaries is independent from the proposed algorithms developed for a general
spectral-based six-color printing system. The spectral reflectance factors of the printed six
primaries and the paper substrate (100# Vintage Gloss Text) are plotted in Fig. 7-10.
4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 00
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
W a ve le ng th
Re
flec
tan
ce
fa
cto
r
C ya nM a gen taYe llowG ree nO rangeB la c kS ubs tra te
Fig. 7-10: The reflectance spectra of the printed six primaries and substrate.
Sample Preparation
Usually, modeling of halftone printing process requires a set of ramps including
primary secondary, tertiary, quaternary,…, and so forth and a verification target
representing the device gamut. Since this research aims at developing a six-color printing
system, the preparation for each printed sample patch utilized FM screening to avoid
moiré patterns. The screen frequency was chosen at 175 LPI resolution or equivalent (the
176
resolution of stochastic screening can not be represented in units of LPI) to account for
the most common condition for practical use.
1. Preparation for ramps
Each ramp was printed at 5%, 10%, 15%, 20%, 30%, 40%, 50%, 60%, 70%,
80%, 90%, and 100% theoretical dot areas. For a four-color halftone printing process,
there are secondary (two-ink overprints), tertiary (three-ink overprints), and quaternary
(four-ink overprints). Although this research used six primary inks for halftone
reproduction, the approach to reproduce an input color (spectrally) is limited to use three
chromatic and one black inks for synthesis. Therefore, five-ink and six-ink overprints were
not generated in the six-color printing process. The overprints are categorized by their
number of inks and listed as follows. There were fifteen two-ink overprint combinations
(C(6,2)) selecting two inks out of six. They were CM, CY, CG, CO, CK, MY, MG, MO,
MK, YG, YO, YK, GO, GK, and OK. There were twenty three-ink overprint
combinations (C(6,3)) selecting three inks out of six. They were CMY, CMG, CMO,
CMK, CYG, CYO, CYK, CGO, CGK, COK, MYG, MYO, MYK, MGO, MGK, MOK,
YGO, YGK, YOK, and GOK. There were ten four-ink overprint combinations (C(5,3))
selecting three chromatic inks out of five plus one black due to the constraint mentioned
previously. They are CMYK, CMGK, CMOK, CYGK, CYOK, CGOK, MYGK, MYOK,
MGOK, and YGOK.
2. Preparation of the verification target (5x5x5x5 combinatorial design for mixtures)
177
Again, the constraint is to use black and three chromatic inks to reproduce each
pixel from a spectral image. The six-color halftone printing model can be viewed as a
super set of several sets of four-color halftone models. In this case, there were ten subsets
of four-color halftone models. They are CMYK, CMGK, CMOK, CYGK, CYOK,
CGOK, MYGK, MYOK, MGOK, and YGOK. There were 625 combinations for each
subset, the 5x5x5x5 combinatorial design of mixtures modulated by five different
fractional dot areas. To printing the 625 mixtures for CMYK sub-model, twenty-five
samples were printed on paper by varying the theoretical dot areas of cyan and magenta
inks at the 0% of yellow and 0% of black ink. This printing procedure was repeated 25
times by varying the yellow and black inks at five different fractional dot areas ( 0%, 25%,
50%, 70%, and 90%). Since there are ten sets of four-ink combinations to compose the
proposed six-color model, the total number of samples was 6,250 for the 5x5x5x5
combinatorial design.
3. Sample measurements
Samples were measured using a Gretag Spectrolino, whose sampling geometry is
0/45, with automatic station to obtain reflectance spectra. The adopted spectral range was
from 400 nm to 700 nm at 10 nm intervals. The spectral data of ramps (total 612 patches)
were obtained by the average of four measurements on each patch. Due to the large
number of patches of the verification target (total 6,250), the spectral data of each patch
were only measured once.
178
4. Accuracy metric
The colorimetric accuracy was specified by the CIE94 color difference equation
for standard illuminant D50 and 1931 standard observer. The spectral accuracy was
specified by a metamerism index which is quantified by the CIE94 color difference
equation for standard illuminant A and 1931 standard observer after a parameric
correction.
Determining the Yule-Nielsen n-Factor
Performing the estimation using the algorithm outlined in Fig. 7-3, the n-factor was found
to be 2.2 for this set of six-primary ramps. The mean colorimetric accuracy, specified in
units of ∆E*94, in predicting all 72 samples of the six primary ramps against different n
values is plotted in Fig. 7-11. With n = 2.2, the colorimetric and spectral accuracy is listed
in Table 7-3 and the reconstructed spectra for each primary ramp are plotted in Fig. 7-12.
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30.2
0.4
0.6
0.8
1
1.2
1.4
1.6
n-factor
Me
an
De
lta E
94
und
er
D5
0
Fig. 7-11: The mean prediction accuracy of all 72 samples of the six primary ramps asfunction of n-factor.
179
Table 7-3: The colorimetric and spectral accuracy of n = 2.2 in predicting all 72 samplesof the six primary ramps where Stdev stands for the standard deviation and RMSrepresents the root-mean-square error in unit of reflectance factor.
∆E*94 Metamerism Index
Mean 0.27 0.04Stdev 0.28 0.05Max 1.42 0.20Min 0.00 0.00RMS 0.005
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1C yan ra m p
Re
fle
cta
nc
e
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1M a g e nta ra m p
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1Ye llo w ra m p
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1G re e n ra m p
W a ve le ng th
Re
fle
cta
nc
e
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1W a rm re d (o ) ra m p
W a ve le ng th
4 0 0 5 0 0 6 0 0 7 0 00
0 .2
0 .4
0 .6
0 .8
1B la ck ra m p
W a ve le ng th
Fig. 7-12: The measured and the predicted reflectance spectra by Eqs. (7-3) and (2-33)using n = 2.2 where the solid lines are measured spectra and the dashed lines are the
predicted spectra.
180
First, Fig. 7-12 provides a visual confirmation for the spectral prediction using n = 2.2.
Second, from Table 7-3 the mean and maximum colorimetric errors in predicting the
primary ramp are 0.27 and 1.4 unit of ∆E*94, respectively. Their low mean and maximum
metamerism index as well as low RMS error indicate that the determined Yule-Neilsen n-
factor was optimal for the primary ramp prediction. This n-factor (n = 2.2) will be used as
the optimal parameter to the construction of the six-color spectral halftone printing model
for this particular printing process.
Accuracy of the First-Order Six-Color Forward Printing Model
By adopting n = 2.2, the theoretical to effective dot area transfer function, f( ), for
each primary ramp was attained by discretely sampling at the effective dot area forming
continuous transfer functions using cubic spline interpolation. The theoretical to effective
dot area transfer functions for the six primaries are plotted in Fig. 7-13.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Theoretical dot area
Effe
ctiv
e do
t are
a
CyanMagentaYellowGreenOrangeBlack
Fig. 7-13: The theoretical to effective transfer functions of the six primaries.
181
The performance of the first order forward halftone printing model in predicting the 6,250
samples of the verification target is specified in Table 7-4 and the histogram of the
colorimetric accuracy is plotted in Fig. 7-14.
Table 7-4: The colorimetric and spectral accuracy of the first order forward model inpredicting the 6,250 samples of the verification target.
∆E*94 Metamerism Index
Mean 1.95 0.22Stdev 1.51 0.22Max 9.7 1.58Min 0.00 0.00RMS 0.008
0 1 2 3 4 5 6 7 8 9 1 00
5 0 0
1 0 0 0
1 5 0 0
2 0 0 0
2 5 0 0
D e lta E 9 4
Fre
qu
en
cy
Fig. 7-14: Histogram of the colorimetric performance using the first-order forwardprinting model in predicting the 6,250 samples of the verification target.
182
Observing the accuracy with an average ∆E*94 =1.95 and the histogram of color
differences for the verification target, the first impression of this model is that it is quite
capable to describe the color formation of the Waterproof® system since the most of
samples (approximately 4,000 samples) can be matched under two units of ∆E*94.
Nevertheless, there are still about 824 samples which were predicted with colorimetric
error higher than four units of ∆E*94. As Iino and Berns suggested, these prediction errors
are mainly due to the overestimating effective dot areas, thereby, under predicting the
spectral reflectance factors. On account of the non ideal ink and optical trapping, the
effective dot areas were actually smaller than that estimated by the first-order
approximation since it does not these the trapping factors into account. Since the printing
device used for this verification generates samples similar to Matchprint, the ink-trapping
ability is assumed to be ideal; thus the influence is mainly referred to as optical-trapping.
The visual evidence of overestimating the effective dot areas leading to the under
prediction of reflectance factors can be provided by the vector plot of the L* vs. a*, shown
in Fig. 7-15. Due to the large number of data (6,250), it only shows the vector plots of the
100 predicted samples with the highest colorimetric errors. As pictured by Fig. 7-15, the
trend of the predicted L* indicates the over estimation of effective dot areas. Another
visual aid, Fig. 7-16, shows that the four estimated samples with large prediction errors
have been reproduced with lower reflectance factors.
183
0
10
20
30
40
50
60
70
-50 -40 -30 -20 -10 0 10 20 30 40 50
L*
a*
Fig. 7-15: The vector plot of the 100 predicted samples with the highest colorimeric errorsby the first-order forward model. The vector tail represents the measured coordinate and
the vector head represents the prediction.
4 00 4 50 5 00 5 50 6 00 6 50 7 000 .02
0 .04
0 .06
0 .08
0 .1
0 .12
0 .14
Re
flec
tan
ce
fa
cto
r
4 00 4 50 5 00 5 50 6 00 6 50 7 000
0 .02
0 .04
0 .06
0 .08
0 .1
M e a s uredPre dic te d
4 00 4 50 5 00 5 50 6 00 6 50 7 000
0 .05
0 .1
0 .15
0 .2
W a ve le ng th
Re
flec
tan
ce
fa
cto
r
4 00 4 50 5 00 5 50 6 00 6 50 7 000 .02
0 .04
0 .06
0 .08
0 .1
0 .12
0 .14
0 .16
W a ve le ng th
Fig. 7-16: Four example spectra showing under prediction by the first order model.
184
Second-Order Modification (By Iino and Berns' Suggestion)
Iino and Berns' algorithms were implemented to estimate the colorimetric and
spectral performance in predicting the verification target of 6,250 mixtures. This second-
order printer model required determining the dot-gain correction scalar, q, as a function of
theoretical dot area, mentioned previously. The database for modeling q was obtained
from some of the samples in the verification target with the exact theoretical dot area
combinations. For example, to determine the fc_m described in Eq. (7-9), the five samples
with the theoretical dot area combinations for cyan = 50% and magenta = 0%, 25%, 50%,
70%, and 90% were selected and cubic spline was employed to interpolate the data in
between. The sample of cyan = 50% and magenta = 100% was not included at the
beginning of designing the verification target. Hence, the corresponding fc_m(am=100%) was
extrapolated. Since the designed verification target does not include this sample, the
effectiveness or the validity of the extrapolation will not affect the verification accuracy.
There are 12 fi_j functions determined for each four-color sub-model. Three of the
example fi_j functions, fc_m, fc_y, and fc_k, determined for the CMYK sub-model is shown
in Fig. 7-17. In Iino and Berns' article, the fi_j functions are second-order polynomials
since those function were found to be monotonically decreasing by their experiment.
Whereas, the fi_j functions, shown in Fig. 7-17, were not monotonic. This indicates that
the two-dimensional dot gain surface of two-ink mixtures were not smooth and
monotonically decreasing for the Waterproof® samples. It was suspected that this
185
discrepancy is not caused by different printing systems but the halftone screening shcemes.
That is, the regular dot pattern generated by the conventional rotated-screen (utilized by
Iino and Berns) has smoother dot gain characteristic, whereas, the stochastic dot pattern
(utilized by this research) has relatively irregular dot gain behavior.
0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
0 .85
0 .9
0 .95
1
Theore tica l do t a re a
q
fcm
fcy
fck
Fig. 7-17: The three example functions of dot-gain correction scalar determined forCMYK sub-model based on Iino and Berns' algorithms.
After the determination of all q functions the correction scalar for each primary,
qc, qm, qy, qg, qo, and qk, were calculated by Eq. (7-9) for each theoretical dot area
combination. Colorimetric and spectral accuracy, shown by Table 7-5, was estimated and
the histogram of colorimetric errors is shown in Fig. 7-18.
186
Table 7-5: The colorimetric and spectral accuracy in predicting the verification of 6,250sample by the algorithms suggested by Iino and Berns.
∆E*94 Metamerism Index
Mean 1.93 0.32Stdev 1.41 0.32Max 9.79 2.41Min 0.00 0.00RMS 0.008
0 1 2 3 4 5 6 7 8 9 1 00
2 0 0
4 0 0
6 0 0
8 0 0
1 0 0 0
1 2 0 0
1 4 0 0
1 6 0 0
1 8 0 0
2 0 0 0
D e lta E 9 4
Fre
qu
en
cy
Fig. 7-18: Histogram of the colorimetric error in units of ∆E*94 for the 6,250 samples
predicted by Iino and Berns' algorithms.
At first glance, the performance of this second-order modification seems not
significantly different from that of the first-order printing model due to the similar average
and maximum values. However, in predicting the large number of samples, the standard
187
deviation and histogram of ∆E*94 should be examined and compared. The Iino and Berns
model performed with a smaller standard deviation of colorimetric errors. Furthermore, it
improves the first-order printing model by lowering the number of high colorimetric errors
above 4 units of ∆E*94 from 824 to 507 samples. This can be visually confirmed by
comparing the histograms in Figs. 7-14 and 7-18. The vector plot , shown in Fig. 7-19, of
L* vs. a* for the exact same 100 samples, plotted in Fig. 7-15, indicates that this second-
order modification significantly reduced the lightness errors relative to the first- order
model.
0
10
20
30
40
50
60
70
-40 -30 -20 -10 0 10 20 30 40
L*
a*
Fig. 7-19: The vector plot of L* vs. a* for the 100 samples used as examples in Fig. 7-15predicted by Iino and Berns' algorithms.
188
The spectral predictions of the four samples used as examples are Fig. 7-16 is
depicted in Fig. 7-20. Compared with Fig. 7-16, the spectral predictions are significantly
improved. Based on this analysis, it is concluded that the Iino and Bern algorithms
significantly corrects the under prediction of mixtures due to optical-trapping. However,
given that the average predicted colorimetric error is similar to that of the first- order
prediction, the other alternative second-order improvements need to evaluated. Hopefully,
one of these will yield high colorimetric and spectral accuracy.
400 450 500 550 600 650 7000
0.05
0.1
0.15
Re
flect
anc
e fa
cto
r
400 450 500 550 600 650 7000
0.02
0.04
0.06
0.08
0.1
MeasuredPredicted
400 450 500 550 600 650 7000
0.05
0.1
0.15
0.2
Wavelength
Re
flect
anc
e fa
cto
r
400 450 500 550 600 650 7000
0.05
0.1
0.15
0.2
Wavelength
Fig. 7-20: The spectral predictions of the four example samples used as examples in Fig.7-16 by the Iino and Berns' algorithms.
189
Alternative Second-Order Modification (Proposed Algorithm)
This proposed alternative second-order modification attacks the (ink and) optical
trapping from the stand point of modeling the optical-trapping of primary-secondary
interactions under the assumption of average influence of a secondary ink presenting at
50% theoretical dot area. To determine the gm_c described in Eq. (7-14), the five samples
with the theoretical dot area combinations for magenta = 50% and cyan = 0%, 25%, 50%,
70%, and 90% were selected and cubic spline was employed to interpolate the data in
between. The sample of magenta = 50% and cyan = 100% was not included at the
beginning of designing the verification. Hence, the corresponding gm_c(ac=100%) was set to
one since there will be no dot-gain for cyan at 100% theoretical dot area. There are 12 q
functions determined for each four-color sub-model. Three of the example q functions,
gm_c, gy_c, and gk_c, determined for the CMYK sub-model are shown in Fig. 7-21. One
logical consideration using Eq. (7-14) is worth mentioning. Consider a set of the
theoretical dot areas with ac = t%, am = 0%, ay = 0%, ak = 0%, the corresponding
correction scalar for each primary should be 0 ≤ qc ≤ 1, qm = 1, qy = 1, and qk = 1 where
0 ≤ t ≤ 100. That is, when a sample is exactly a primary color, there is no correction scalar
needed for its dot gain determined by the first-order theoretical to effective dot area
transfer function, f( ). Whereas, if Eq. (7-14) is utilized without modification then the qc in
Eq. (7-14), being a product of gm_c, gy_c, and gk_c, will yield correction when cyan appears
alone. This error also is encountered for two or three primary mixtures. To be more
190
specific, Table 7-6 shows some samples' theoretical dot areas and their correction scalars
for each ramp using Eq. (7-14) without logical operator correction.
0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .90 .7
0 .7 5
0 .8
0 .8 5
0 .9
0 .9 5
1
T he o re ti c a l d o t a re a
q
gm c
gy c
gk c
Fig. 7-21: The three example functions of dot-gain correction scalar determined forCMYK sub-model based on proposed algorithms.
Table 7-6: The theoretical dot areas and their correction scalar by Eq. (7-14) withoutlogical correction.
ac am ay ak qc qm qy qk
0 0 0 0 1 1 1 10.25 0 0 0 0.260 1 1 10.5 0 0 0 0.470 1 1 10.7 0 0 0 0.645 1 1 10.9 0 0 0 0.477 1 1 10 0.25 0 0 1 0.346 1 1
0.25 0.25 0 0 0.260 0.346 1 10.5 0.25 0 0 0.470 0.346 1 10.7 0.25 0 0 0.645 0.346 1 10.9 0.25 0 0 0.477 0.346 1 1
191
It can be seen that the first five samples are pure cyan at different dot areas. Their
correction scalars, qcs are not unity as expected, i. e., no correction is needed for its dot
gain. Hence, a logical modification is required to rectify this error. The logical
modification was derived by
g'i_j = gi_j.*{[(1./g i_j).*(~atheo,i)]+(~~atheo,i)} , (7-19)
where .* and ./ are the vector-matrix componentwise multiplication and division,
respectively, and ~ is a logical "not" operator (MATLAB, 1996). It will be shown by
example when applying the logical modification, Eq. (7-19), to the correction scalar, q, for
each ramp. It turns out that the logical modification is needed only when the secondary is
present. Let i be m and j be c, thus c is assumed as primary and m is assumed as
secondary.
Case 1: if atheo,m is zero then ~atheo,m = 1 and ~~atheo,c = 0. This dot area combination
indicates the samples are cyan ramp along. Using Eq. (7-19) results in
g'm_c(atheo,c) = gm_c(atheo,c).*{[(1./g m_c(atheo,c)).*(~atheo,m)]+(~~atheo,m)}= gm_c(atheo,c).*{[(1./g m_c(atheo,c)).*(1)]+(0)}= gm_c(atheo,c).*[1./gm_c(atheo,c)] = 1.
Case 2: If atheo,m is nonzero then ~atheo,m = 0 and ~~atheo,c = 1. Using Eq. (7-19) results in
g'm_c(atheo,c) = gm_c(atheo,c).*{[(1./g m_c(atheo,c)).*(~atheo,m)]+(~~atheo,m)}= gm_c(atheo,c).*{[(1./g m_c(atheo,c)).*(0)]+(1)}= gm_c(atheo,c).*(0 + 1) = gm_c(atheo,c).
That is, the dot-gain correction scalars are not changed when a secondary is present.
192
Table 7-7 shows the theoretical dot areas and the corresponding correction scalar
for each ramp of the same samples shown in Table 7-6 after logical correction. After
logical modification the first five samples, which are cyan primaries printed at different
areas, have the correct corresponding correction scalars, qc, of unity. The rest of
correction scalars for none-appearing secondaries are all one as expected. In addition, the
sixth through tenth samples, which are cyan and magenta mixtures, whose correction
scalars for the cyan ramp are different from the correction scalars for the cyan ramp in
Table 7-6, neither for the magenta ramp. Hence, the logical error is dramatically affecting
the accuracy of the dot-gain correction scalar, q, for the proposed algorithms. (It was
investigated to see if Iino and Berns' algorithms also required logical modification for dot-
gain correction scalar. It was found that the dot-gain correction scalar for each primary by
Eq. (7-9) was invariant with this logical modification.)
Table 7-7: The theoretical dot areas and their correction scalar by Eq. (7-14) with logicalcorrection.
ac am ay ak qc qm qy qk
0 0 0 0 1 1 1 10.25 0 0 0 1 1 1 10.5 0 0 0 1 1 1 10.7 0 0 0 1 1 1 10.9 0 0 0 1 1 1 10 0.25 0 0 1 1 1 1
0.25 0.25 0 0 0.848 0.856 1 10.5 0.25 0 0 0.960 0.856 1 10.7 0.25 0 0 0.992 0.856 1 10.9 0.25 0 0 0.977 0.856 1 1
193
The colorimetric and spectral accuracy of the proposed algorithms in predicting
the 6,250 samples is listed in Table 7-8 and the histogram is plotted in Fig. 7-22.
Table 7-8: The colorimeric and spectral accuracy of the proposed algorithms in predictingthe verification target of 6,250 sample mixtures.
∆E*94 Metamerism Index
Mean 1.44 0.22Stdev 1.16 0.25Max 7.59 1.75Min 0.00 0.00RMS 0.006
0 1 2 3 4 5 6 7 80
5 0 0
1 0 0 0
1 5 0 0
2 0 0 0
2 5 0 0
D e lta E 9 4
Fre
qu
en
cy
Fig. 7-22: Histogram of the colorimeric accuracy of the proposed algorithms in predictingthe verification target of 6,250 sample mixtures.
194
Table 7-8 reveals that the average and maximum colorimetric errors are 1.44 and 7.59
∆E*94, respectively. In addition, it also indicates that the majority of sample mixtures can
be reproduced with lower colorimeric errors by the proposed algorithms given the tight
standard deviation. There are only 251 as opposed to 507 (by Iino and Berns'
modification) or 824 (by the first order prediction) samples predicted with higher than four
units of ∆E*94. Colorimetrically, this is a significant improvement. The spectral accuracy
specified by the low average metamerism index indicates that the verification target is
spectrally well predicted for all three models discussed so far. Performance of spectral
prediction for all three models is not significantly different judging by the statistical results
of the metamerism index. The L* vs. a* and the four predicted sample spectra used for the
previous example are plotted in Figs. 7-23 and 7-24, respectively.
0
10
20
30
40
50
60
70
-40 -30 -20 -10 0 10 20 30 40
L*
a*
Fig. 7-23: The vector plot of L* vs. a* for the 100 samples used as examples in Fig. 7-15predicted by the proposed algorithms.
195
At first glance of Figs 7-23 and 7-24, the proposed algorithms tend to over predict the
spectral reflectance factors or under estimate the effective dot areas for low lightness
mixtures. Although the colorimetric and the spectral performance of the proposed
algorithm is considered as superior, the appearance of the spectral matches shown by Fig.
7-24 is disappointing.
400 450 500 550 600 650 7000
0.05
0.1
0.15
0.2
Re
flect
anc
e f
ac
tor
400 450 500 550 600 650 7000
0.05
0.1
0.15
400 450 500 550 600 650 7000
0.05
0.1
0.15
0.2
W avelength
Re
flect
anc
e f
ac
tor
400 450 500 550 600 650 7000
0.05
0.1
0.15
0.2
W avelength
Fig. 7-24: The spectral prediction of the four samples used as examples in Fig. 7-16 by theproposed algorithms.
By comparing at the L* vs. a* vector plots of Figs 7-21 and 7-13, it was found that
the prediction trend by the proposed algorithms is approximately opposite to that of the
first-order prediction. Thus, it was hypothesized that the accurate estimation of effective
196
dot areas linearly lies between the estimation by the proposed algorithms and by the first
order prediction. With this observation, the final estimation of effective dot area, aeff,est is
modified by a linear combination of aeff,proposed and aeff,1st, i. e.,
aeff,est = taeff,proposed + (1-t)aeff,1st , (7-20)
where t is a linear scalar and 0 ≤ t ≤ 1, aeff,proposed is the effective dot area estimated by the
proposed algorithm, and aeff,1st is the effective area estimated by the first order printing
model. Equation (7-20) can be expressed as Eq. (7-21) based on Eq. (7-10),
aeff,est = t(atheo + q(aeff,primary - atheo,primary)) + (1-t)aeff,1st . (7-21)
It was found that best spectral prediction by Eq. (7-20) occurred when t = 0.62. The
colorimetric and spectral performance of the proposed algorithms modified by Eq. (7-20)
in predicting the verification target is shown in Table 7-9 and the histogram of the
colorimetric error is plotted in Fig. 7-25. The implementation for the six-color forward
printing model based on the proposed algorithm is shown in Appendix E.
Table 7-9: The colorimeric and spectral accuracy of the proposed algorithms modified byEq. (7-20) in predicting the verification target of 6,250 sample mixtures.
∆E*94 Metamerism Index
Mean 0.90 0.10Stdev 0.75 0.10Max 6.06 1.19Min 0.00 0.00RMS 0.004
197
0 1 2 3 4 5 6 70
50 0
10 00
15 00
20 00
25 00
30 00
D elta E 94
Fre
qu
enc
y
Fig. 7-25: Histogram of the colorimeric accuracy of the proposed algorithms modified byEq. (7-20) in predicting the verification target of 6,250 sample mixtures.
The modification by Eq (7-20) yielded excellent accuracy improvement in predicting the
6,250 samples judged by the statistical results of colorimetric and spectral error. It also
rectifies the problem of over prediction of spectral reflectance factors for low L* samples.
Visual evidence is provided by the vector plot, shown in Fig. 7-26, of L* vs. a*, L* vs. b*,
and b* vs. a* for the 300 samples whose colorimetric error is predicted higher than 2.41
units of ∆E*94.
198
Fig. 7-26: The vector plot of L* vs. a*, L* vs. b*, and b* vs a* for the 300 samples whosecolorimetric error is predicted higher than 2.41 units of ∆E*94.
Observing Fig. 7-26, lightness errors of the reproduction are dramatically reduced.
There are only three to four samples with large lightness errors and two to three with large
0
10
20
30
40
50
60
70
80
90
100
-60 -40 -20 0 20 40 60
L*
a*
0
10
20
30
40
50
60
70
80
90
100
-60 -40 -20 0 20 40 60
L*b*
-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60
b*
a*
199
chromatic errors. The rest of sample are predicted with low colorimetric error. Recall that
Fig. 7-26 represents the 300 samples predicted with higher colorimetric error. Thus, the
proposed algorithm together with the modification Eq. (7-20) provided the best
performance second-order printing model for this research project. The spectral prediction
for the four samples used as examples in Fig. 7-16, shown in Fig. 7-27, reflects this
superiority.
400 450 500 550 600 650 7000
0 .05
0 .1
0 .15
Re
flec
tan
ce
fa
cto
r
4 00 450 500 550 600 650 7000
0 .02
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Mea suredPredic ted
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Fig. 7-27: The spectral prediction by the proposed algorithms modified by Eq. (7-20) forthe four samples used as examples in Fig. 7-16.
Modeling by Matrix Transformation
A modeling method was evaluated by directly relating a set of theoretical dot areas
of a multi-ink mixture to the corresponding set of effective dot areas estimated by
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inverting the Yule-Nielsen modified spectral Neugebauer equation. This requires a large
sample database to represent major sample behavior. Hence, its accuracy relies on the size
of database. Since the theoretical dot area is nonlinearly related to the effective dot area
due to the mechanical dot-gain, optical dot-gain, and ink and optical trapping, a direct
linear transformation using Eq. (7-15) is not possible. Hence, this research project
implemented a higher-order polynomial regression using Eq. (7-16) seeking the matrix
relationship between the theoretical and effective dot areas. It was found that the order of
the polynomial matrix should be higher than four to achieve the similar accuracy predicted
by the proposed algorithms. The process of the six-color modeling by matrix method is
similar to the structure of the six-color forward printing model depicted in Fig. 7-9. The
only difference is that the ten theoretical to effective dot area transfer functions, f( )s, and
the second order improvements are replaced with ten matrices for each four-color sub-
model. The fourth order polynomial matrix, as the input to the matrix transformation, for
each four-color sub-model was implemented with 27 terms to account for any of
nonlinearity between the theoretical and effective dot area.
This research project investigated the matrix method in predicting the verification
of 6,250 samples. Its colorimetric and spectral accuracy is shown Table 7-10 and the
histogram of the predicted colorimetric errors is plotted in Fig. 7-28.
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Table 7-10: The colorimetric and spectral performance of the verification target predictedby the matrix method.
∆E*94 Metamerism Index
Mean 1.66 0.20Stdev 1.39 0.21Max 14.08 3.04Min 0.02 0.00RMS 0.012
0 2 4 6 8 10 12 140
500
1000
1500
2000
2500
3000
3500
4000
D e lta E 94
Fre
qu
en
cy
Fig. 7-28: Histogram of the colorimeric accuracy of the matrix method in predicting theverification target of 6,250 sample mixtures.
From Table 7-10 and Fig. 7-28, the performance of the matrix method is not a favorable
approach relative to the proposed algorithms in predicting the verification target.
Furthermore, it requires a relatively large sample database to explain the color mixing of a
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halftone printing process. Furthermore, the estimated effective dot area obtained by the
matrix method sometime is a negative value leading to a non-rational synthesis by Yule-
Nielsen modified spectral Neugebauer equation. Hence, the matrix method is not adopted
by the research development of six-color separation minimizing metamerism.
Six-Color Separation Minimizing Metamerism
It was desired to verify the validity of the backward printing model which is
comprised by a four-ink selector, a forward printing model, and a numerical engine. The
proposed algorithms modified by Eq, (7-20) were utilized as the forward printing model
since it has the best accuracy for the Waterproof® system. This research project chose the
Gretag Macbeth Color Checker as a separation target and tried to reproduce the spectrum
for each color with minimal metamerism. Owing to the original spectral image (the Gretag
Macbeth Color Checker) was unavailable, the use of CMYGOK as printing primaries was
not a decision from the optimal inkset selection module. Hence, some of color spectra may
be outside the spectral gamut of the developed six-color spectral printing model using the
Waterproof® system. Nevertheless, the objective of this section is to verify the spectral
reproducibility simultaneously balancing with the colorimetric accuracy of the proposed
six-color modeling process.
As mentioned previously, given a reflectance factor of a color sample, the task of
color separation is to determine a set of theoretical dot areas (or sometime called request
dot areas), corresponding to the printing primaries, for a printer to synthesize the desired
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spectrum. The estimated 24 sets of theoretical dot areas with respect to the 24 colors in
the Color Checker, the corresponding 24 colorimetric errors in units of ∆E*94, the
corresponding 24 metamerism indices, and the corresponding RMS error in units of
reflectance factor are shown in Table 7-11 and the statistical results in predicting the 24
colors using the proposed six-color printing model is listed in Table 7-12.
Table 7-11: The predicted theoretical dot areas, colorimetric and spectral errors of the 24colors in Macbeth color checker where M.I. represents the metamerism index.
Color Name ac am ay ag ao ak ∆E*94 M.I. RMS
Dark Skin 0.000 0.040 0.121 0.000 0.200 0.346 0.01 0.09 0.007Light Skin 0.000 0.140 0.227 0.000 0.000 0.093 0.02 0.11 0.031Blue Sky 0.305 0.106 0.000 0.000 0.000 0.133 0.41 0.86 0.041Foliage 0.026 0.000 0.308 0.099 0.000 0.310 0.11 0.34 0.013Blue Flower 0.278 0.209 0.012 0.000 0.000 0.022 0.24 0.69 0.051Blue Green 0.144 0.000 0.000 0.103 0.000 0.000 0.94 0.17 0.044Orange 0.069 0.000 0.460 0.000 0.364 0.021 0.20 0.56 0.021Purplish Blue 0.555 0.323 0.000 0.000 0.000 0.000 1.48 1.63 0.037Moderate Red 0.000 0.479 0.262 0.020 0.000 0.031 0.03 0.16 0.024Purple 0.461 0.499 0.171 0.000 0.000 0.050 0.07 0.73 0.079Yellow Green 0.000 0.000 0.621 0.120 0.032 0.000 0.32 0.65 0.026Orange Yellow 0.000 0.000 0.648 0.031 0.240 0.001 0.36 1.01 0.041Blue 0.921 0.418 0.000 0.000 0.000 0.000 5.57 1.55 0.028Green 0.000 0.000 0.297 0.250 0.025 0.074 0.02 0.17 0.021Red 0.059 0.763 0.515 0.000 0.000 0.040 0.35 0.39 0.071Yellow 0.000 0.000 0.820 0.015 0.078 0.008 0.17 0.39 0.021Magenta 0.092 0.448 0.004 0.000 0.000 0.000 0.36 0.49 0.115Cyan 0.562 0.047 0.000 0.039 0.000 0.086 0.33 0.89 0.032White 0.000 0.000 0.000 0.000 0.000 0.000 1.60 0.35 0.062N8 0.005 0.000 0.000 0.001 0.000 0.088 0.15 0.30 0.054N6.5 0.012 0.000 0.000 0.001 0.000 0.189 0.29 0.57 0.043N5 0.021 0.000 0.000 0.000 0.000 0.317 0.24 0.45 0.027N3.5 0.034 0.005 0.000 0.000 0.000 0.466 0.28 0.52 0.013Black 0.207 0.110 0.155 0.000 0.000 0.601 0.46 0.90 0.005
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Table 7-12: The statistical results in predicting the 24 colors using the proposed six-colorprinting model.
∆E*94 Metamerism Index
Mean 0.58 0.58Stdev 1.14 0.41Max 5.57 1.63Min 0.02 0.11RMS 0.045
First, the overall colorimetric accuracy in predicting the spectral reflectance of the
24 colors with average 0.57 and maximum 5.57 units of ∆E*94 is satisfactory. There are 20
out of 24 colors predicted with colorimetric errors below 0.5 units of ∆E*94. Furthermore,
the Blue and White are the out of gamut colors. Excluding these out of colorimetric gamut
colors, the statistical results of the colorimetric and spectral performance for 22 in gamut
colors is listed in Table 7- 13.
Table 7- 13: The colorimetric and spectral performance of the 22 in gamut colors.
∆E*94 Metamerism Index
Mean 0.31 0.55Stdev 0.33 0.36Max 1.48 1.63Min 0.01 0.11RMS 0.045
Since the primary goal for this research development is to minimize the metamerism
between the original and its color reproduction, special attention is drawn to metamerism.
The metamerism index represents the color difference under the second reference
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illuminant (A) while the color difference between the original and reproduction is zero
under the first standard illuminant (D50) after parameric correction (Fairman, 1987).
Thus, the average, 0.55, and maximum, 1.63, in units ∆E*94 indicates that the metamerism
is not severe at all for this reproduction. There are 21 out of 24 colors which were
predicted under one unit of metamerism index. This implies that the spectral mismatch
between the original and reproduction is small such that their color matches are
approximately invariant and constant across illuminants. The four best and four worst
predicted spectra in terms of metamerism index are plotted in Figs. 7-29 and 7-30,
respectively.
400 450 500 550 600 650 7000
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1
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Dark Skin
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L ight Skin
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G reen
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B lue G ree n
Fig. 7-29: The four best predicted spectra in terms of metamerism index of the 24 colors.
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400 450 500 550 600 650 7000
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Purplish Blue
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Blue
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Cyan
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Orange Yellow
Fig. 7-30: The four worst predicted spectra in terms of metamerism index of the 24 colors.
The implementation of the spectral-based six-color separation algorithm is shown in
Appendix F.
D. SPECTRAL PERFORMANCE COMPARISONS FOR THREE-,FOUR-, AND SIX-COLOR PRINTING PROCESSES
In order to substantiate the superiority of the spectral performance by adopting
more number of degrees of freedom, two three-color (CMY) continuous tone printing
devices, Fujix Pictrography 3000 and Kodak Professional 8670 PS thermal printer, and
one four-color (CMYK) printing processes, DuPont Waterproof®, were utilized to
formulate the color reproductions of the Macbeth Color Checker computationally.
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Kubelka-Munk transformations, Eqs. (2-9) and (2-10), for transparent material in
optical contact with opaque support were used to describe the color synthesis for the two
continuous tone printers. Two sets of one thousand samples of 10x10x10 combinatorial
mixtures in units of monitor RGB were printed by the two three-color printers. The three
eigenvectors derived in the absorption space for each printer were used as the pseudo-
primaries for color synthesis (recall that the eigenvectors are theoretically linearly related
to that actual primary colorants, described in Chapter IV).
The proposed algorithm for modeling a four-color halftone printing process was
employed to describe the color synthesis of DuPont Waterproof® system using CMYK.
Exact colorimetric matches not including the out of colorimetric gamut colors
under illuminant D50 and 1931 observer for the Macbeth Color Checker were obtained by
all the three- and four-color printing processes. The color difference in unit of ∆E*94 under
illuminant A and F2, shown in Table 7-14, were calculated for the prediction by Fujix,
Kodak, Waterproof® CMYK, and Waterproof® CMYKGO printing processes whose
predicted spectral reflectance factors are attached in Appendices L, M, and N,
respectively. It can be seen that the average color difference under illuminant A for the
Waterproof® CMYK process was relatively large, whereas, that of rest of the three
printing processes were small. In addition, the average color differences under illuminant
F2 for the Fujix, Kodak, Waterproof® CMYK processes were relatively large, whereas,
the color difference predicted by the Waterproof® CMYKGO was the smallest.
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Table 7-14: The color difference of the predicted Macbeth Color Checker under illuminantA and F2 by four different printing processes, where the color names corresponding to thebold faced entries are the out of colorimetric gamut colors of each device.
Fujix Kodak WaterproofCMYK
WaterproofCMYKGO
∆E*94
underA
∆E*94
underF2
RMS ∆E*94
underA
∆E*94
underF2
RMS ∆E*94
underA
∆E*94
underF2
RMS ∆E*94
underA
∆E*94
underF2
RMS
Dark Skin 0.78 1.39 0.057 0.23 2.72 0.043 2.47 2.26 0.029 0.09 0.04 0.007Light Skin 0.05 1.24 0.041 0.21 2.30 0.066 0.76 1.39 0.034 0.11 0.89 0.031Blue Sky 0.64 0.77 0.060 1.41 1.78 0.041 1.77 1.23 0.047 0.86 0.63 0.041Foliage 0.36 1.80 0.050 0.59 2.94 0.048 2.67 1.79 0.034 0.34 0.90 0.013
Blue Flower 1.00 1.65 0.058 1.72 2.30 0.082 0.81 1.27 0.05 0.69 1.21 0.051Blue Green 0.98 1.07 0.060 1.16 1.68 0.0660.65 1.70 0.055 0.17 0.98 0.044
Orange 1.59 1.85 0.080 1.70 2.16 0.0892.23 2.49 0.079 0.56 0.45 0.021Purplish Blue 0.23 1.13 0.039 1.08 1.64 0.044 1.60 1.53 0.038 1.63 1.57 0.037Moderate Red 0.05 0.36 0.033 0.24 1.34 0.059 0.28 0.56 0.023 0.16 0.45 0.024
Purple 0.84 3.33 0.050 1.42 3.68 0.095 0.79 2.90 0.078 0.73 2.85 0.079Yellow Green 0.23 0.37 0.052 0.23 0.95 0.063 1.79 1.56 0.064 0.65 0.67 0.026Orange Yellow 1.69 2.14 0.070 1.90 2.50 0.0902.23 2.64 0.081 1.01 1.39 0.041
Blue 0.15 1.87 0.050 0.99 2.05 0.0461.25 2.25 0.031 1.55 2.66 0.028Green 0.31 0.59 0.048 0.33 0.93 0.039 1.81 2.00 0.049 0.17 0.93 0.021Red 0.63 1.32 0.037 0.77 1.23 0.062 0.44 1.53 0.074 0.39 1.41 0.071
Yellow 0.46 0.60 0.031 0.73 1.06 0.0710.91 1.02 0.042 0.39 0.53 0.021Magenta 0.63 2.17 0.087 0.72 1.82 0.106 0.53 2.09 0.118 0.49 1.95 0.115
Cyan 2.70 1.54 0.052 2.92 2.03 0.051 0.50 0.92 0.034 0.89 0.67 0.032White 0.99 0.41 0.106 0.04 0.25 0.028 0.35 0.18 0.062 0.35 0.18 0.062
N8 0.51 1.00 0.068 0.16 1.69 0.037 1.64 1.27 0.067 0.30 0.39 0.054N6.5 0.87 1.35 0.070 0.26 2.70 0.046 2.98 2.16 0.067 0.57 0.55 0.043N5 0.82 1.60 0.064 0.16 3.40 0.045 3.61 2.55 0.052 0.45 0.43 0.027
N3.5 0.95 1.80 0.051 0.20 3.76 0.035 3.92 2.51 0.032 0.52 0.30 0.013Black 0.90 1.82 0.034 0.27 3.61 0.022 3.18 1.68 0.014 0.90 0.53 0.005
Mean 0.77 1.38 0.81 2.10 1.63 1.73 0.58 0.94Stdev 0.59 0.69 0.73 0.94 1.09 0.69 0.41 0.73Max 2.70 3.33 2.92 3.76 3.92 2.90 1.63 2.85Min 0.05 0.36 0.04 0.25 0.28 0.18 0.09 0.04
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Observing the individual color match predicted by these three- and four printing processes,
most colors such as blue, foliage, purple, neutral colors, and so forth were highly
metameric. Whereas, the predictions by the proposed six-color spectral printing process
was the least metameric results. Hence, the color reproductions by the Fujix, Kodak,
Waterproof® CMYK are of high degree of metamerism and the degree of metamerism
synthesized by the proposed six-color spectral printing process using Waterproof®
CMYKGO can be minimized.
E. CONCLUSIONS
Although the six-color Yule-Nielsen modified spectral Neugebauer equation
should be employed to construct a six-color printing model, the physical printing limitation
such as ink-trapping failure confines the existence or stability of real five-color and six-
color overprints which are the Neugauer primaries. Since a six-color Yule-Nielsen
modified spectral Neugebauer uses up to six-color overprints as its basis spectra for
spectral reconstruction, stable five-color and six-color overprints are necessary for
accurate reconstruction of a given spectral reflectance factor. This especially is a problem
when attempting to derive a more than six-color printing model. There are 97 (27 -31)
five-color, six-color, and seven-color overprints for a seven-color printing process. Even
the sample preparation of these Neugebauer primaries will be a critical issue. Although the
analytical prediction of those overprints were presented in Chapter VI, the physical
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feasibility is still questionable due to the ink-trapping limitation. Hence, the proposed
approach by this research development seems a reasonable paradigm. The modeling for a
k-color printing process is suggested to subdivide into several four-color sub-models.
Their union of their spectral gamut can be a good approximation of the spectral gamut
spanned by the all combinations of k primaries since the color of the more than four
overprints tend to be neutral and low lightness.
Spectral-based six-color separation algorithms were derived in building a six-color
forward and backward spectral printing model. Four six-color forward spectral printing
models were compared for their performance in terms of colorimetric and spectral
accuracy using Waterproof®, a halftone proofing system. They are the first-order forward
printing model comprised of a theoretical to effective dot area transfer function, f( ), and
the Yule-Nielsen modified spectral Neugebauer equation, the second order modification
suggested by Iino and Berns, an alternative second order improvement by the proposed
algorithms, and the printing process modeled by matrix transformation method. The
proposed algorithms together with the modification by Eq. (7-20) were the favorite
forward printing model for its highest performance both in terms of high colorimetric and
spectral accuracy.
A backward spectral printing model was constructed by a four-ink selector, the
adopted six-color forward spectral printing model based on the proposed algorithms, and
a numerical engine. The performance of this backward printing model was tested in
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estimating the theoretical dot areas of 24 colors of the Gretag Macbeth Color Checker.
The colorimetric and spectral accuracy, shown in Table 7-12, of the predictions
corresponding to the estimated theoretical dot areas by the proposed backward model is
high. One thing worth mentioning is that the four-selector is able to select a proper set of
four inks from the defined six inks for reproducing the Gretag Macbeth Color Checker.
The validity for the hypothesis of continuous tone approximation and the use of Eq. (7-
17), derived empirically, are reassured by the accuracy of the color separation performed
on the Color Checker.
An analysis was made by comparing the spectral performance of three- and four
printing processed to the proposed six-color technology. It was shown that the degree of
metamerism for the prediction by the three- (Fujix Pictrography 3000 and Kodak
Professional 8670 PS thermal printer) and four-color (Dupont Waterproof® using CMYK)
printing processes were relatively high.
Based upon all of these analyses, it is concluded that the proposed color separation
algorithms are capable of minimizing metamerism between the original and its
reproduction while maintaining high colorimetric accuracy.
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VIII. MULTIPLE-INK DIRECTING PRINTING
The last stage of this dissertation is to utilize the implemented spectral-based color
separation algorithm in building a multi-spectral output system. This requires predefining a
multiple-ink halftone printing process to be used as an output device since forward and
backward halftone printing models are device dependent. The Waterproof® system was
utilized to verify the research development of the six-color spectral halftone printing
models and a set of six color separation records for the Gretag Macbeth Color Checker
was calculated corresponding to the six primaries for this printing process. The final
verification for the research development in building a multi-spectral output system is to
output the six color separation records, shown in Table 7-11, by employed printing
process and observe the colorimetric and spectral accuracy of the reproduction.
A. VERIFICATIONS
The six color separation records of the Gretag Macbeth Color Checker were
output by the DuPont Waterproof® system using the specified six primaries. Four six-color
printed Checker were measured five times for each by the Gretag Spectrolino (since it was
utilized for the samples to model the six-color printing process) and averaged to account
for the print to print variation and spatial uniformity. The colorimetric and spectral
accuracy for each individual color is shown in Table 8-1. The statistical colorimetric and
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spectral performance for original vs. reproduction and prediction vs. reproduction are
listed Table 8-2.
Table 8-1: The colorimetric and spectral accuracy of original vs. reproduction andprediction vs. reproduction for the Gretag Macbeth Color Checker.
Origianl vs. Reproduction Prediction vs. ReproductionColor Name ∆E*
94 M.I. RMS ∆E*94 M.I. RMS
Dark Skin 1.64 0.51 0.013 1.64 0.60 0.010Light Skin 1.57 0.14 0.038 1.57 0.27 0.017Blue Sky 2.09 0.68 0.038 1.73 0.49 0.010Foliage 1.70 0.49 0.016 1.71 0.34 0.009Blue Flower 2.21 0.20 0.056 2.07 0.79 0.020Blue Green 1.26 0.21 0.041 0.98 0.08 0.011Orange 1.76 0.33 0.019 1.80 0.39 0.020Purplish Blue 2.53 1.51 0.037 1.49 0.66 0.012Moderate Red 1.54 0.12 0.028 1.54 0.12 0.011Purple 1.87 1.08 0.083 1.81 0.50 0.008Yellow Green 1.50 0.52 0.022 1.27 0.32 0.016Orange Yellow 2.81 1.06 0.046 3.02 0.29 0.028Blue 5.80 1.39 0.028 0.74 0.37 0.007Green 1.05 0.29 0.020 1.04 0.23 0.008Red 0.91 0.56 0.078 0.86 0.17 0.008Yellow 1.52 0.32 0.027 1.60 0.19 0.020Magenta 0.96 0.80 0.123 0.83 0.18 0.012Cyan 1.55 1.74 0.034 1.72 0.38 0.011White 1.45 0.40 0.059 0.26 0.05 0.004N8 1.44 0.16 0.054 1.42 0.47 0.022N6.5 2.89 0.25 0.050 2.90 0.95 0.033N5 3.08 0.48 0.033 3.09 1.01 0.024N3.5 2.00 0.11 0.014 2.01 0.67 0.009Black 1.55 0.63 0.005 1.45 0.58 0.004
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Table 8-2: The statistical colorimetric and spectral accuracy corresponding to thepredicted and reproduced Gretag Macbeth Color Checker.
Origianl vs. Reproduction Prediction vs. Reproduction∆E*
94 Metamerism Index ∆E*94 Metamerism Index
Mean 1.94 0.58 1.61 0.42Stdev 1.01 0.46 0.69 0.26Max 5.80 1.74 3.09 1.01Min 0.91 0.11 0.26 0.05RMS 0.048 0.016
Ideally, the colorimetric and spectral error between the prediction and the actual
reproduction should be zero. Actual average colorimateric and spectral error between the
prediction and the reproduction were 1.61 ∆E*94 and 0.42 unit of metamerism index,
respectively. This discrepancy is due to the print to print variation and calibration status of
the printing process. (The proposed six-color spectral printing model was established one
month before the Gretag Macbeth Color Checker was printed.) Nevertheless, this
accuracy is still satisfactory. Figure 8-1 shows the spectral reflectance factors of six colors
arbitrarily chosen for demonstration. It can been seen that the spectral reflectance factors
of reproduction was systematically lower than the prediction. This was attributed to the
print to print variation and calibration status of the printing process. The reproduced
Gratag Macbeth Color Checker is bounded at the end of this chapter and the original,
predicted, and reproduced spectral reflectance factors are shown in Appendices I, J, and
K, respectively.
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400 500 600 7000
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B lue flower
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1Ora nge
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1P urp lish b lue
Origina lPredictedReproduc tion
400 500 600 7000
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W ave length
Yellow green
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W ave length
Neutra l 8
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W ave length
Neutra l 6 .5
Fig. 8-1: The original, predicted, and reproduced spectral reflectance factors of the sixGretag Macbeth Colors.
B. CONCLUSIONS
The six color separation records in terms of the theoretical dot areas, shown in
Tabel 7-11, for the Gretag Macbeth Color Checker were output by the DuPont
Waterproof® system using the cyan, magenta, yellow, black, green, and orange, the
standard DuPont Waterproof® primaries, which were utilized to established the six-color
spectral printing model. Curve shapes of the reproduced reflectance spectra were
approximately parallel to the spectra predicted by the proposed six-color printing model.
216
The discrepancy is due to the print to print variation and calibration status of the physical
printing device. Nevertheless, the similarity of the spectral curves between the reproduced
and predicted spectra indicates that the proposed six-color printing model is capable of
predicting the color formulated by the actual Dupont Waterproof® printing process.
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IX. CONCLUSIONS DISCUSSIONS AND SUGGESTIONS FORFUTURE RESEARCH
Research for bridging a multi-spectral image acquisition system and a spectral-
based output system has been described. It includes colorant estimation, optimal ink-
selection, spectral reflectance factor prediction for ink overprints, spectral-based six-color
separation minimizing metamerism, and verification from direct printing. Results show a
promising future for the feasibility of the proposed algorithms in terms of colorimetric and
spectral accuracy.
Since linear modeling techniques are heavily employed for the first-order analysis
for this research project, a linear colorant mixing space was desired. Kubelka-Munk turbid
media theory has frequently been used to represent opaque colorant mixtures in a linear
vector space. It was observed for the colorant mixtures used in this research that the
accuracy of spectral prediction based on Kubelka-Munk theory was not satisfactory.
Furthermore, excessively large fields of view for conventional spectrophotometry result in
additivity failure when spectral data are transformed to Kubelka-Munk representations.
As a consequence, an empirical transformation was derived, as described in
chapter III, resulting in improved prediction accuracy for a set of opaque colorant
mixtures requiring reproduction. The proposed transformation was designed according to
the coloration process of subtractive colorant mixing and the exact or reduced
dimensionality of a set of mixtures in the linear colorant mixing space after the proposed
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transformation. The former was concerned with the observation that the more colorants
used for coloration, the darker the resultant colorant mixture. The latter accounted for the
underlying number of colorants used for creating the set of colorant mixtures. The
accuracy of the proposed transformation was high.
In the analysis stage, a colorant estimation algorithm, described in chapter IV,
statistically decomposed an input spectral image according to the six directions explaining
major variation of the spectral image in a linear colorant mixing space. These directions
are known as the directions of six significant eigenvectors. Hence, each pixel of the
spectral image represented in a linear colorant mixing space is a linear combination of the
six eigenvectors corresponding to the input spectral image. Since the six eigenvectors did
not represent physical colorant spectra judged by their bipolar appearance across the
visible spectrum, a constrained rotation algorithm was applied to transform them to a set
of all-positive vector representations leading to non-negative coordinates for describing
every pixel in the input spectral image. The six all-positive eigenvectors, as the statistical
primaries, symbolized a set of six primary colorants utilized for creating the original object
corresponding to the captured spectral image. The non-negative coordinates associated
with each pixel account for a typical coloration process, i.e. no negative concentrations,
can be used for color synthesis. Based on these two physical constraints, the proposed
colorant estimation algorithm was able to converge to a set of reasonable basis colorants
219
whose prediction accuracy were identical to that of the six eigenvectors derived for the set
of input spectra requiring synthesis.
Once the set of statistical primaries corresponding to a given image were
uncovered, the optimal ink-selection module correlated the statistical primaries to an
existing ink database in order to attain the most similar ink set for a halftone printing
process using multiple inks. Since the number of six-ink combinations generated from a
large ink database is enormous, the effort of inspecting the performance of every
combination was prohibitive. The obviously infeasible ink combinations, such as those sets
formed by the inks of same hue, should be removed analytically. The proposed ink-
selection algorithm, described in chapter V, was capable of removing the impossible ink
combinations resulting in significant reduction, i.e., from 18,564 down to 32 for a
database of 18 inks. Small numbers of candidate ink sets were much more feasible to
evaluate for spectral reconstruction.
The Yule-Nielsen modified spectral Neugebauer equation was an obvious choice in
building halftone printing models for performance evaluation. Whereas, parameters such
as dot gain factor and reflectance spectra of ink overprints of a printing process were
frequently unknown. The halftone model building effort was not quite feasible at this
phase. Practically, this research suggested a continuous-tone approximation to a halftone
printing process based on the proposed transformation described in Chapter III. The
transformation was revised in describing the colorant mixing for a halftone printing
220
process. The verification of the IT8/7.3 target printed by SWOP standard conditions
showed that the continuous-tone approximation was reasonable.
Since the optimal ink set for minimizing metamerism is image dependent, the
optimal ink set varies from image to image. In order to construct an analytical printing
model using the Yule-Nielsen modified Neugebauer equation, the ink overprints, known as
the Neugebauer primaries, were the basis information required to use this equation. Due to
the number of overprints increasing exponentially with each additional ink, the number of
overprints was enormous when the number of primary inks was large. This would lead to
an unreasonable number of samples to be prepared and measured, in addition to
necessitating resampling and remeasurement upon changes in consumable (e.g., ink and
paper). This research, therefore, included the analytical prediction of ink overprints using
the Kubelka-Munk theory for translucent materials as opposed to exhaustively printing
and measuring. Two optical constants, absorption and scattering coefficients, of each ink
were estimated numerically using the reflectance measurements of an ink printed on a
white and black surfaces and the reflectance factors of the white and black surfaces,
respectively. Hence, the surface spectral reflectance factor of an ink overprint was a
function of the absorption and scattering coefficients, thickness of the top-most ink layer,
and the spectral reflectance factor of the layer underneath the top-most layer.
This research tested the algorithm discussed in Chapter VI using the DuPont
Waterproof® system, a halftone proofing process. Twenty-five overprints, which were at
221
most three-color overprints, were generated from a combination of six primaries. The
prediction results were excellent. Relatively poorer predictions resulted for some of the
three-color overprints. There was a systematic trend in which the samples were too dark.
It was reasoned that the prediction using this algorithm for dark overprints was sensitive
to noise. Noise was greatly amplified when it underwent the transformation using the
translucent equation which is highly non-linear. Nevertheless, the colorimetric and spectral
accuracy was still high.
The development for the synthesis stage of this research was to construct a highly
accurate six-color printing model. DuPont Waterproof® was also utilized to print six-color
ramps and a verification target of 6,250 samples. The six-color printing model was
comprised of ten four-color printing sub-models. The modeling of the six-color printing
process was a collection of modeling ten four-color printing processes. Four different
models were evaluated: the first-order Yule-Nielsen modified spectral Nuegebauer
equation together with the theoretical to effective dot area transfer function, the second-
order improvement published by Iino and Berns, an alternative second-order modification,
and a higher-order matrix transformation. These models were described and evaluated in
Chapter VII. By inspecting two-dimensional vector plots of CIELAB L* versus a*, it was
revealed that the first-order estimation tended to under predict the spectral reflectance
factors, due to an overestimation of the effective dot areas of multiple-ink mixtures for the
dark samples. The error of overestimating effective dot areas was mainly caused by an
222
“optical-trapping” effect. A second-order modification proposed by Iino and Berns,
accounting for optical-trapping, was tested to improve the model accuracy. Although it
yielded approximately the same accuracy as that of the first-order approximation for this
type of printing process, it was capable of correcting the prediction of some of the dark
colors. It was found that the prediction errors were mostly chromatic errors. The third
algorithm, developed during this dissertation as an alternate model of optical trapping, was
more effective in reducing the errors for these dark samples. The difference between the
newly developed algorithm and that described by Iino and Berns was that, conceptually,
Iino and Berns' algorithms suggested that, for example, for cyan and magenta ink
mixtures, whose theoretical dot areas are m% and k%, respectively, the dot gain of the
cyan is normalized to the dot-gain locus of the cyan when the magenta is present at k%.
Similarly, the dot gain of the magenta is normalized to the dot gain locus of the magenta
when the cyan is present at m%. Whereas, the proposed algorithms recommend that the
more optimal dot gain of the cyan is normalized to the dot-gain locus of the magenta
presenting at 50% theoretical dot area whenever the magenta is present at any theoretical
dot area other than zero. It was found that the new approach tended to over predict
spectral reflectance factor (or under estimate the effective dot areas) for the 6,250
mixtures. It is not a surprising nature of the newly developed algorithms since the
approximation by the dot gain of a primary given a secondary presenting at 50%
theoretical will overestimate the effective dot area of the primary when the secondary is
223
actually present at less than 50% theoretical dot area and vice versa. Nevertheless, this
drawback is corrected by Eq. (7-21), as a consequence, yielding the best accuracy.
By inspecting the vector plots of CIELAB values, it was also found that the
reproduction trend was approximately opposite to that of the first-order approximation.
Thus, it was concluded that the real effective dot areas for predicting the 6,250 samples
was linearly located between the effective dot areas estimated by the first-order
approximation and that estimated by the new algorithm. A weighted sum was statistically
estimated and found to yield the highest colorimetric and spectral accuracy. The matrix
method was undesirable for this research since it required a large sample database for
modeling process. In addition, it occasionally produced negative effective dot areas.
This research tested the six-color spectral printing model by estimating the dot
areas required to reproduce a Gretag Macbeth Color Checker. The six color separations,
defined in terms of theoretical dot areas, would lead to color-reproduction accuracy of an
average of 0.57 and a maximum of 5.57 ∆E*94 and an average of 0.78 and a maximum of
2.09 units of metamerism index. Thus, the colorimetric and spectral accuracy was
satisfactory. Since the six primaries were not optimized for the Gretag Macbeth Color
Checker (having been arbitrarily defined by DuPont), the maximum errors were due to
spectral and colorimetric gamut limitations. Comparisons among the spectral gamut
limitation for two three-color printing processes (Fujix Pictrography 3000 and Kodak
Professional 8670 PS thermal printer), one four-color printing process (DuPont
224
Waterproof® system using CMYK), and the proposed six-color technology were made
computationally. The color formulation by the three- and four-color printing processes
were highly metameric relative to the proposed six-color technology. It was concluded
that the spectral-based color separation algorithm was capable of minimizing metamerism
while maintaining high colorimetric accuracy.
The six separation records for the 24 color patches of the Gretag Macbeth Color
Checker were printed using DuPont Waterproof® using the defined six primaries.
Accuracy comparisons among the original, prediction by the six-color model, and
reproduction based on the six-color separations in terms of the theoretical dot areas
predicted by the proposed six-color printing model were shown. It was found that the
colorimetric and spectral error between the prediction and the reproduction was
systematic owing to the print to print variation and the calibration status of the employed
printing process. However, the average colorimetric and spectral error with 1.61 ∆E*94
and 0.42 units of metamerism index, respectively was satisfactory. The stability of the
DuPont Waterproof® is considered high. Hence, the validity of the proposed six-color
printing model is substantiated by this verification.
This research has accomplished a theoretical development and several prototypical
implementations. There is still a need for improvement in order to implement this research
in practice. The colorant-estimation model was implemented by a constrained rotation
process using MATLAB. It takes minutes to hours converging to a set of statistical
225
primaries depending on the initial estimation and the requested numerical precision. It is
possible to statistically guess the initial set of colorants by directly selecting the spectra
located at the vertices of the colorimetric gamut of an input spectral image. The Monte
Carlo simulation suggested by Ohta (1973) may be an alternative to expedite the
converging speed.
The Pantone 14 basic and four process colors were utilized as the ink database for
ink selection. There is only one green in the ink database. From the development of
optimal ink-selection, the performance is partially limited by the lack of green inks to
choose from though it is not the absolute dominant factor. This implies that a wide variety
of candidate inks are desirable, in general.
The development for ink overprint prediction was verified by a proofing process.
Color patches were printed by holding colorant on a laminate, then applying on a
substrate. Unlike a conventional printing process, the phenomenon of colorant penetration
into the substrate was negligible. The penetration of ink causes the estimation of the
scattering coefficient to be difficult. Further research could test the ink overprint
predictions using process-printed samples.
During the development of the six-color separation algorithm, the spectral image
acquisition system was still under development. Hence, it was not possible to test each
component developed during this dissertation within a system. Clearly, the next step is to
226
combine these results with an spectral image acquisition system and develop and test the
complete spectral color reproduction system.
This research has demonstrated the capability of spectral reproduction six-color
spectral printing process. Colorimetric and spectral accuracy for reproducting the Gretag
Macbeth Color Checker was very high even though the six primaries (CMYKGO) were
not defined from the ink selection algorithm. However, the derived six-color process was
not capable of reproducing the blue and purple colors with high accuracy. It is
hypothesized that by adding one degree of freedom to derive a seven-color printing
process, the capability of spectral reproduction can be greatly improved. It is also
hypothesized that by a properly defined ink set of seven colors can approximately bypass
the process of ink selection based on the predicting accuracy of the Gretag Macbeth Color
Checker estimated by the derived six-color printing model.
Since the proposed color separation algorithm is constrained to use one black and
three chromatic inks to reproduce a pixel from an original, the modeling process for a
seven-color printing model is at the expense of establishing ten more four-color sub-
models. There are 20 (C(6, 3)) four-color sub-models for a seven-color printing process
based on the proposed ink-limiting scheme. It is worth mentioning that the file size will be
the same for storing the k-color separation records, where k ≥ 5. To be more specific,
consider the file structure of the separation records diagram as below:
227
Tag of an inkset
Theoretical dotarea of the 1st
ink
Theoretical dotarea of the 2nd
ink
Theoretical dotarea of the 3rd
ink
Theoretical dotarea of theblack ink
1 0.05 0.81 0.21 0.435 0.66 0.32 0.12 0.02.3 . . . .. . . . .. . . . .8 0.83 0.90 0.11 0.11
The tag of an ink set specifies which ink combination should be used for the particular
pixel and the corresponding four theoretical dot areas, shown in the same row, required
for spectral reproduction.
For a six-color printing process the integer tag is ranged from one to ten, from 1 to
20 (C(6, 3)) for a seven-color printing process, and from 1 to 35 (C(7,3)) for a eight-color
printing process. Only 6 bits are required to store the information of the four-ink
combinations up to eight-color printing process. There are only five channels for a
multiple-ink printing process abiding by the proposed ink-limiting scheme. Hence, it is
suggested by this research to speculate the performance of a seven-color printing process,
which is an extension of the proposed six-color technology, without large additional
modeling efforts and storage issues.
228
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X. APPENDICES
This research utilized MATLAB V5.2 as the algorithm developing tool. A
collection of program scripts developed for the calculation of fundamental colorimetry,
such as XYZ tristimulus values, L*a*b*, L*ch, ∆E*ab, ∆E*
94, metamerism index,…, and so
forth, Kubelka-Munk translucent equation, functions for Saunderson correction, and
database for colorimetry, such as color matching functions, ASTM weights for standard
illuminants (D65, D50 , A,…, etc.) is stored as the MATLAB library or toolbox named
under "munsell". When running the attached program scripts in each appendix, the
"munsell" library is indispensable. User should include the "munsell" in the search path
when working with the MATLAB environment. The program scripts developed for each
module of this research are shown in the following appendices.
This research adopted the spectral range from 400 nm to 700 nm at 10 nm interval.
Hence, all the spectral data should be of 31 components. Most sets of spectral data were
stored as column vectors. Whereas, a few sets of spectral data from large measurements
are stored as row vectors. User should be aware of the dimensionality of these matrices
before running the program.
237
APPENDIX A : MATLAB PROGRAMS FOR THE "MUNSELL"LIBRARY
asaunder.m
%This is the fuchtion of inverse Saunderson correction. The argument R_inf%is the corrected reflectance factor by Saunderson correction.
function R_meas=asaunder(R_inf)
k1=0.04;k2=0.6;
R_meas=((1-k1)*(1-k2)*R_inf)./(1-k2*R_inf);
cost_eig.m
%This the objective function of pos_tran.m performing all-positive%vector rotation. the first argument is the initially estimated coefficient%matrix which might rotate the set of input eigenvector (second argument)%matrix to an all positive basis vector matrix close to the third argument%(target-Spectra).
function [f, g]=cost_eig(coefficient, eigenvec, target_spectra)
new_basis = eigenvec*coefficient';
f=rms(target_spectra, new_basis); % rms: function for root-mean-square error
g(1,:)= [-new_basis]'; %the constraints for the positivity of the resultant vector
del_e94.m
%This program calculates the color difference in units of CIE94 color difference equation for two samples with ref1 and ref2, respectively,%where the white_point is a 31x1 reflectance vector of the reference white.%The illum_option is 1x2 vector of specifying the illuminant and observer
function De94=del_e94(ref1, ref2, white_point, illum_option)
if (nargin <= 1) fprintf('The first two argument are matrices of reflectance factors of the standard and trial samples,\n'); fprintf('The third argument is a reflectance vector of a white point.\n'); fprintf('The fourth argument is character string for a viewing illuminant.\n'); fprintf('It could be illumD65 or illumA.\n\n'); return;elseif nargin == 2 fprintf('The default white point is perfect white reflector.\n'); fprintf('The default illuminant is D65 of 2 degree ASTM weight.\n\n'); white_point=ones(31,1); illum_option=[1 1];elseif nargin == 3 fprintf('The default illuminant is D65 of 2 degree ASTM weight.\n\n'); illum_option=[1 1];end
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lch1=lch_ab(ref1, white_point, illum_option);lch2=lch_ab(ref2, white_point, illum_option);
Lab1=lab(ref1, white_point, illum_option);Lab2=lab(ref2, white_point, illum_option);
del_L=lch1(:,1)-lch2(:,1);del_C=lch1(:,2)-lch2(:,2);
[m,n]=size(lch1);
del_H=zeros(m,1);
for i=1:m if (Lab1(i,2)*Lab2(i,3)) == (Lab2(i,2)*Lab1(i,3)) del_H(i,1)=0; elseif (Lab1(i,2)*Lab2(i,3)) > (Lab2(i,2)*Lab1(i,3)) del_H(i,1)= -sqrt(2*(lch1(i,2)*lch2(i,2)-Lab1(i,2)*Lab2(i,2)-Lab1(i,3)*Lab2(i,3))); else del_H(i,1)= sqrt(2*(lch1(i,2)*lch2(i,2)-Lab1(i,2)*Lab2(i,2)-Lab1(i,3)*Lab2(i,3))); endend
Cab=(lch1(:,2).*lch2(:,2)).^(0.5);%Cab=lch1(:,2);SL=1;SC=1+0.045*Cab;SH=1+0.015*Cab;
De94=sqrt((del_L./SL).^2 + (del_C./SC).^2 + (del_H./SH).^2);
etok.m
%This program is the objective function of the constrained optimization%process for the colorant estimation subsystem, where k_ink is the variable%of a set of estimated colorants represented in a linear color mixing space%such as K/S space. The pseudo is the coefficient of a set of eigenvectors,%named as eigenvec. The k_black is a black colorant in linear color mixing%space to be constrained for its pre-existence in the set of mixtures.
function [f,g]=etok(k_ink, pseudo, eigenvec, k_black)
k1=eigenvec*pseudo;k1=(abs(k1)+k1)/2;
%conc=pinv([k_ink k_black])*k1; %if need to constrain the pre-existence of black %k2=[k_ink k_black]*conc; %then use these two line
conc=pinv(k_ink)*k1;k2=k_ink*conc;
f=rms(k1,k2); %calculate the RMS error between two reconstructed set of samples
g=-conc; % concentration is constrained to be all-positive
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k_pca.m
%This program calculates a required numbers (num_of_eig) of eigenvectors%(K_eig) for a set of sample (kovs) in linear color mixing space, where%the prin_fig is the option for displaying the plot of eigenvectors%if that is required.
function [K_eig, report]=k_pca(kovs, num_of_eig, prin_fig)
if nargin == 0 fprintf('The first argument is a matrix of K/S,\n'); fprintf('The second argument is the number of required eigenvectors.\n');elseif nargin == 1 fprintf('The default number of required eigenvectors is six.\n'); num_of_eig=6; prin_fig=0;elseif nargin == 2 prin_fig=0;end
lambda=[400:10:700];
sigma=cov(kovs');[vk,ek]=eig(sigma);ek=flipud(diag(ek));tot_per_var=sum(ek(1:num_of_eig))/sum(ek);per_var= ek/sum(ek);
cum_var=zeros(31,1);cum_var(1)=per_var(1);
for i=2:31 cum_var(i)=cum_var(i-1)+per_var(i);end
if prin_fig==1 figure subplot(2,1,1) plot(ek, 'r+') title('The 31 eigenvalues based on K/S.') xlabel('The ith eigenvectors.') ylabel('Eigenvalue')
subplot(2,1,2) plot(cum_var*100, 'r+') title('The cumulative variance% of i eigenvectors based on K/S.') xlabel('The ith eigenvectors.') ylabel('Variance%')
fprintf('The total percent variance described by %g\n', num_of_eig) fprintf('eigenvectors in K/S space is %f\n', tot_per_var*100);
if num_of_eig <=6 figure subplot(2,3,1) plot(lambda, vk(:,31),'r') title('The 1st eigenvector.') %xlabel('Wavelength') ylabel('K/S') axis([400 700 -1 1]);
subplot(2,3,2)
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plot(lambda, vk(:,30),'r') title('The 2nd eigenvector.') %xlabel('Wavelength') %ylabel('Pseudo-K/S) axis([400 700 -1 1]);
subplot(2,3,3) plot(lambda, vk(:,29),'r') title('The 3rd eigenvector.') %xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,4) plot(lambda, vk(:,28),'r') title('The 4th eigenvector.') xlabel('Wavelength') ylabel('K/S') axis([400 700 -1 1]);
subplot(2,3,5) plot(lambda, vk(:,27),'r') title('The 5th eigenvector.') xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,6) plot(lambda, vk(:,26),'r') title('The 6th eigenvector.') xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);
else figure subplot(2,3,1) plot(lambda, vk(:,31),'r') title('The 1st eigenvector.') %xlabel('Wavelength') ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,2) plot(lambda, vk(:,30),'r') title('The 2nd eigenvector.') %xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,3) plot(lambda, vk(:,29),'r') title('The 3rd eigenvector.') %xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,4) plot(lambda, vk(:,28),'r') title('The 4th eigenvector.') xlabel('Wavelength') ylabel('Pseudo-K/S') axis([400 700 -1 1]);
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subplot(2,3,5) plot(lambda, vk(:,27),'r') title('The 5th eigenvector.') xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,6) plot(lambda, vk(:,26),'r') title('The 6th eigenvector.') xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);
figure subplot(2,3,1) plot(lambda, vk(:,25),'r') title('The 7th eigenvector.') %xlabel('Wavelength') ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,2) plot(lambda, vk(:,24),'r') title('The 8th eigenvector.') %xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,3) plot(lambda, vk(:,23),'r') title('The 9th eigenvector.') %xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,4) plot(lambda, vk(:,22),'r') title('The 10th eigenvector.') xlabel('Wavelength') ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,5) plot(lambda, vk(:,21),'r') title('The 11th eigenvector.') xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);
subplot(2,3,6) plot(lambda, vk(:,20),'r') title('The 12th eigenvector.') xlabel('Wavelength') %ylabel('Pseudo-K/S') axis([400 700 -1 1]);endend % if prin_fig=='on'
report=[ek(1:num_of_eig) per_var(1:num_of_eig) cum_var(1:num_of_eig)];vk=fliplr(vk);K_eig=vk(:,1:num_of_eig);
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km_trans.m
%This program calculated the surface reflectance of translucent material by%the Kubleka-Munk basic equation
function reflectance=km_trans(thickness, ks, Rg)
a=1+(ks(:,1)./ks(:,2));
b=sqrt((a.^2)-1);
reflectance= (1-Rg.*(a-b.*coth(b.*ks(:,2)*thickness)))./(a-Rg+b.*coth(b.*ks(:,2)*thickness));
lab.m
%This function calculates the CIELAB value of a sample with reflectance factor,%stored as reflect1, where the white_point is a 31x1 reflectance vector%of the reference white. The illum_option is 1x2 vector of specifying%the illuminant and observer
function Lab= lab(reflect1, white_point, illum_option)
if nargin == 0 fprintf('The first argument is a matrix of reflectance,\n'); fprintf('The second argument is a reflectance vector of a white point.\n'); fprintf('The third argument is character string for a viewing illuminant.\n'); fprintf('It could be illumD65_2, illumD65_10, illumA_2, or illumA_10.\n\n'); return;elseif nargin == 1 fprintf('The default white point is perfect white reflector.\n'); fprintf('The default illuminant is D65 of 2 degree ASTM weight.\n'); white_point=ones(31,1); illum_option=[1 1];elseif nargin == 2 fprintf('The default illuminant is D65 of 2 degree ASTM weight.\n'); illum_option=[1 1];end
if (illum_option(1)==1)&(illum_option(2)==1) load wd65_2.dat; XYZ_1=[wd65_2'* reflect1]'; XnYnZn =[wd65_2'* white_point]';elseif (illum_option(1)==1)&(illum_option(2)==2) load wd65_10.dat; XYZ_1=[wd65_10'* reflect1]'; XnYnZn =[wd65_10'* white_point]';elseif (illum_option(1)==2)&(illum_option(2)==1) load wa_2.dat; XYZ_1=[wa_2'* reflect1]'; XnYnZn =[wa_2'* white_point]';elseif (illum_option(1)==2)&(illum_option(2)==2) load wa_10.dat; XYZ_1=[wa_10'* reflect1]'; XnYnZn =[wa_10'* white_point]';
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elseif (illum_option(1)==3)&(illum_option(2)==1) load wd50_2.dat; XYZ_1=[wd50_2'* reflect1]'; XnYnZn =[wd50_2'* white_point]';elseif (illum_option(1)==3)&(illum_option(2)==2) load wd50_10.dat; XYZ_1=[wd50_10'* reflect1]'; XnYnZn =[wd50_10'* white_point]';
elseif (illum_option(1)==4)&(illum_option(2)==1) load wf2_2.dat; XYZ_1=[wf2_2'* reflect1]'; XnYnZn =[wf2_2'* white_point]';elseif (illum_option(1)==4)&(illum_option(2)==2) load wf2_10.dat; XYZ_1=[wf2_10'* reflect1]'; XnYnZn =[wf2_10'* white_point]';
elseif (illum_option(1)==5)&(illum_option(2)==1) load wf7_2.dat; XYZ_1=[wf7_2'* reflect1]'; XnYnZn =[wf7_2'* white_point]';end
Lab=xyz_lab(XYZ_1, XnYnZn);
lch_ab.m
%This function calculates the CIELCH value of a sample with reflectance factor,%stored as reflect1, where the white_point is a 31x1 reflectance vector%of the reference white. The illum_option is 1x2 vector of specifying%the illuminant and observer
function LCH=lch_ab(reflectance, white_point, illum_option)
if nargin == 0 fprintf('The first argument is a matrix of reflectance,\n'); fprintf('The second argument is a reflectance vector of a white point.\n'); fprintf('The third argument is character string for a viewing illuminant.\n'); fprintf('It could be illumD65 or illumA.\n\n'); return;elseif nargin == 1 fprintf('The default white point is perfect white reflector.\n'); fprintf('The default illuminant is D65.\n\n'); white_point=ones(31,1); illum_option=[1 1];elseif nargin == 2 fprintf('The default illuminant is D65.\n\n'); illum_option=[1 1];end
LAB=lab(reflectance, white_point, illum_option);
LCH(:,1)=LAB(:,1);LCH(:,2)=sqrt((LAB(:,2).^2)+(LAB(:,3).^2));
[m,n]=size(LAB);
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for i=1:m if (LAB(i,2)==0) & (LAB(i,3)==0) LCH(i,3)=0; elseif (LAB(i,2)==0) & (LAB(i,3)>0) LCH(i,3)=pi/2; elseif (LAB(i,2)==0) & (LAB(i,3)<0) LCH(i,3)=-pi/2; elseif (LAB(i,2)<0) & (LAB(i,3)==0) LCH(i,3)=pi; else LCH(i,3)=atan2(LAB(i,3), LAB(i,2))*180/pi; endend
meta_idx.m
%This function calculates the index of metaterism between the standard and the trial samples.%This function needs two to four arguments.%The 1st argument is a 31xn matrix of standard reflectance,%The 2nd argument is a 31xn matrix of trial reflectance,%The 3rd argument is the 31x1 reflectance vector of a white point, and%The 4th argument is an 1x3 vector of options.%% option(1) = 1: standard illuminant is D65.% = 2: standard illuminant is A.% = 3: standard illuminant is D50.% = 4: standard illuminant is f2.% = 5: standard illuminant is f7.%This illuminant is used for a standard viewing illuminant.%% option(2) = 1: referential illuminant is D65.% = 2: referential illuminant is A.% = 3: referential illuminant = D50.% = 4: standard illuminate is f2.% = 5: standard illuminate is f7.
%This illuminate is used for indicating the degree of metamerism.%% option(3) = 1: 2 degree observer.% = 2: 10 degree observer.%This indicates the color matching function used for calculation.%% option(4) = 1: color difference metric is delta Eab.% = 2: color difference metric is delta Eab94.%This color difference metric is used for calculating the index of metamerism.%%The defult option = [1 2 1 2].
function meta_index=meta_idx(standard, trial, white_point, option)
if nargin < 3 fprintf('The first argument is a matrix of standard reflectance,\n\n'); fprintf('The second argument is a matrix of trial reflectance.\n\n'); fprintf('The third argument is the 31x1 reflectance vector of a white point.\n'); fprintf('The default white point is prd.\n'); fprintf('The 4th argument is an 1x3 vector of options.\n'); fprintf('Type >>help meta_idx for details. \n\n');
elseif nargin == 2
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fprintf('The default white point is prd and.\n'); fprintf('The default defult option = [1 2 1 2], Type >>help meta_idx for details.\n'); white_point=ones(31,1); option = [1 2 1 2];elseif nargin == 3 fprintf('The default defult option = [1 2 1 2], Type >>help meta_idx for details.\n'); option = [1 2 1 2];end
if option(1)==option(2) fprintf('ERROR :The standard and referencial viewing illuminant are the same.\n'); fprintf(' The results of metameric index are not reliable.\n\n');end
corrected_spectra=pcorrect(standard, trial, [option(1) option(3)]);
if option(4)==1 meta_index=del_e(standard, corrected_spectra, white_point, [option(2) option(3)]);elseif option(4)==2 meta_index=del_e94(standard, corrected_spectra, white_point, [option(2) option(3)]);end
off_fn.m
%This is the objective function for the optimization process of estimating%the offset vector of the empirical transformation derived for opaque%colorant represented in the resultant linear color mixing space, where%b is the offset vector, sample is a reflectance matrix of a set of sample%requiring reproduction, and po is the power (default=0.5).
function [f,g]=off_fn(b, sample, po, numofeig)[n,m]=size(sample);
tmp=b;for i=2:m tmp=[tmp b];end
k_paint=tmp-(sample.^po);[eig_k, repk]=k_pca(k_paint,numofeig);psu_con=pinv(eig_k)*k_paint;ks_recon=eig_k*psu_con;ks_recon=(abs(ks_recon)+ks_recon)/2;%R_pred=(tmp-ks_recon).^(1/po);
f=rms(k_paint, ks_recon);%f=rms(R_pred, sample);
g=-k_paint;
pcorrect.m
%This function calculate the corrected spectrum for two sample which are paramour%based on the article of Fairman (1987). Refer to the program meta_idx.m for the%definitions of the input arguments.
function corrected_spectra=pcorrect(standard, trial, illum_option)
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if nargin < 2 fprintf('The first argument is a matrix of standard reflectance,\n'); fprintf('The second argument is a matrix of trial reflectance.\n'); fprintf('The third argument is character string for a viewing illuminant.\n'); fprintf('It could be illumD65, illumA, or illumd50.\n\n');elseif nargin == 2 fprintf('The default illuminant is D65 of 2 degree ASTM weight.\n'); illum_option=[1 1];end
if (illum_option(1)==1)&(illum_option(2)==1) load wd65_2.dat; light_source=wd65_2;elseif (illum_option(1)==1)&(illum_option(2)==2) load wd65_10.dat; light_source=wd65_10;elseif (illum_option(1)==2)&(illum_option(2)==1) load wa_2.dat; light_source=wa_2;elseif (illum_option(1)==2)&(illum_option(2)==2) load wa_10.dat; light_source=wa_10;elseif (illum_option(1)==3)&(illum_option(2)==1) load wd50_2.dat; light_source=wd50_2;elseif (illum_option(1)==3)&(illum_option(2)==2) load wd50_10.dat; light_source=wd50_10;elseif (illum_option(1)==4)&(illum_option(2)==1) load wf2_2.dat; light_source=wf2_2;elseif (illum_option(1)==4)&(illum_option(2)==2) load wf2_10.dat; light_source=wf2_10;elseif (illum_option(1)==5) load wf7_2.dat; light_source=wf7_2;
end
R=light_source*inv(light_source'*light_source)*light_source';
[m,n]=size(R);
identity=diag(ones(m,1));
corrected_spectra=R*standard + (identity-R)*trial;
saunder.m
%This fuction performs the Saunderson's correction to the discontinuity%of the refractive index for two material in optical contact with each other
function R_inf=saunder(R_meas)
k1=0.04;k2=0.6;
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R_inf=R_meas./(1-k1-k2+k1*k2+k2*R_meas);
xyz.m
%This program calculates the XYZ tristimulus. Refer to the lab.m for the definition of arguments.
function [XYZ, xy]=xyz(reflect1, illum_option)
if nargin == 0 fprintf('The first argument is a matrix of reflectance,\n'); fprintf('The second argument is character string for a viewing illuminant.\n'); fprintf('It could be illumD65_2, illumD65_10, illumA_2, or illumA_10.\n\n'); return;elseif nargin == 1 fprintf('The default illuminant is D65 of 2 degree ASTM weight.\n'); illum_option=[1 1];end
if (illum_option(1)==1)&(illum_option(2)==1) load wd65_2.dat; XYZ=[wd65_2'* reflect1]';elseif (illum_option(1)==1)&(illum_option(2)==2) load wd65_10.dat; XYZ=[wd65_10'* reflect1]';elseif (illum_option(1)==2)&(illum_option(2)==1) load wa_2.dat; XYZ=[wa_2'* reflect1]';elseif (illum_option(1)==2)&(illum_option(2)==2) load wa_10.dat; XYZ=[wa_10'* reflect1]';elseif (illum_option(1)==3)&(illum_option(2)==1) load wd50_2.dat; XYZ=[wd50_2'* reflect1]';elseif (illum_option(1)==3)&(illum_option(2)==2) load wd50_10.dat; XYZ=[wd50_10'* reflect1]';
elseif (illum_option(1)==4)&(illum_option(2)==1) load wf2_2.dat; XYZ=[wf2_2'* reflect1]';elseif (illum_option(1)==4)&(illum_option(2)==2) load wf2_10.dat; XYZ=[wf2_10'* reflect1]';
elseif (illum_option(1)==5)&(illum_option(2)==1) load wf7_2.dat; XYZ=[wf7_2'* reflect1]';
end
[m,n]=size(XYZ);
for i=1:m xy(i,1)=XYZ(i,1)/sum(XYZ(i,:)); xy(i,2)=XYZ(i,2)/sum(XYZ(i,:));end
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APPENDIX B : MATLAB PROGRAMS FOR THE COLORANTESTIMATION SUBSYSTEM
The following diagram depicts the program structure of the process for the colorant estimation subsystem.
Main Program
(colorant_est.m)
Optimization for theoffset vector of the
empirical transformationPrincipal component analysis Colorant estimation
colorant_est.m
%This program was implemented based on the algorithm of the colorant estimation research.%It estimates the underlying colorant of a set of 105 opaque mixtures obtained by hand mixing%six opaque Poster Colorants
close all; clear all;
load target105.txt %the 105 patches of the hand mixed poster colorspaint=target105; %now paint is the matrix of the reflectance spectra %of the 105 patches
lambda=[400:10:700]';
load white.txt;R_paper=white; %white is the reflectance spectrum of the reference white %now R_paper is the reference white
%************************************************************************************%This section of scripts performance the optimization to obtain the offset vector of the%empirical transformation for opaque color represented in a linear colorant mixing space%named psi space%************************************************************************************
b=ones(31,1); %initializing the offset vector of the empirical transformation
numeig=6; %number of eigenvector is six due to the 105 samples were mixed with six poster colors
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vlb=zeros(31,1);vub=ones(31,1);options(1)=1;options(2)=1e-3;options(3)=1e-3;options(4)=1e-3;options(14)=10000000;
po=1/2; %power of the empirical transformation
b=constr('off_fn', b, options, vlb, vub, '', paint, po, numeig);
[n,m]=size(paint);
tmp=b;for i=2:m tmp=[tmp b];end
b=tmp; %now b is a 31x105 matrixclear tmp;
k_meas=b-paint.^(po); %the 105 mixtures were transformed to a linear colorant mixing representation
[eig6_k, report]=k_pca(k_meas, numeig, 1); %deriving the six eigenvectors from the psi space
fprintf('The total percent variance described by 6 eigenvectors in psi space is %f\n', report(numeig,3)*100);
%************************************************************************************% Spectral reconstruction by PCA analysis% Model : K_reconstruction = eig6_k * C%************************************************************************************
pseudo_con100=pinv(eig6_k(:,1:6))*k_meas;k_recon=eig6_k(:,1:6)*pseudo_con100;R_predicted=(b-k_recon).^(1/po);
delta_e94=del_e94(R_predicted, paint, R_paper);
fprintf('The statistical results of the delta E94')fprintf( 'Mean \t\t%f\n', mean(delta_e94) );fprintf( 'Standard Deviation \t\t%f\n', std(delta_e94) );fprintf( 'Maximum \t\t%f\n', max(delta_e94) );fprintf( 'Minimum \t\t%f\n\n', min(delta_e94) );
spectral_RMS = RMS( R_predicted, paint );
midx=meta_idx(R_predicted, paint, R_paper);fprintf('The statistical results of the metamerism index')fprintf( 'Mean \t\t%f\n', mean(midx) );fprintf( 'Standard Deviation \t\t%f\n', std(midx) );fprintf( 'Maximum \t\t%f\n', max(midx) );fprintf( 'Minimum \t\t%f\n\n', min(midx) );
fprintf( 'The root mean square error by 6 eigenvectors in reflectance space is %f\n\n', spectral_RMS );
figure[yy,xx]=hist(delta_e94);bar(xx,yy,'r')title('The histogram color difference between measured and predicted by PCA.')xlabel('Delta E94')
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ylabel('Frequency')
%************************************************************************************%This section is performing the colorant estimation for the 105 mixtures based on the%proposed constrained vector rotation theory% pos_est.txt is the stored text file of the best estimation in psi space%************************************************************************************
k_ink=b(:,1:6)-primary.^(1/2); %k_ink is the initial estimated six colorant for the 105 mixtures
vlb=ones(31,6)*0.05;vub=b(:,1:6);options(1)=1;options(2)=1e-3;options(3)=1e-3;options(4)=1e-3;options(14)=100000000000000000;
Black=ones(31,1)*0.95; %if the black is constrained to be the pre-existing colorant
k_ink=constr('etok', k_ink, options, vlb, vub, '',pseudo_con100, eig6_k(:,1:6), black);
conc=pinv([black k_ink])*eig6_k(:,1:6)*pseudo_con100;k_recon=[black k_ink]*conc;
%conc=pinv(k_ink)*eig6_k(:,1:6)*pseudo_con100; %if the black is not constrained, use these two lines%k_recon=k_ink*conc;
R_predicted=(b-k_recon).^(1/po);
delta_e94=del_e94(R_predicted,paint, R_paper);fprintf('The average color difference between measured and predicted by choosed ink-set %f\n', mean(delta_e94));fprintf( 'Mean \t\t%f\n', mean(delta_e94) );fprintf( 'Standard Deviation \t\t%f\n', std(delta_e94) );fprintf( 'Maximum \t\t%f\n', max(delta_e94) );fprintf( 'Minimum \t\t%f\n\n', min(delta_e94) );
spectral_RMS = RMS( R_predicted, paint );
fprintf( 'The root mean square error of the 100 measurements is %f\n\n', spectral_RMS );
figure[yy,xx]=hist(delta_e94);bar(xx,yy,'r')title('The histogram of color difference between measured and predicted by 6C K-M.')xlabel('Delta E94')ylabel('Frequency')
midx=meta_idx(R_predicted, paint, R_paper);fprintf('The average color difference between measured and predicted undek illum A by choosed ink-set %f\n', mean(midx));fprintf( 'Mean \t\t%f\n', mean(midx) );fprintf( 'Standard Deviation \t\t%f\n', std(midx) );fprintf( 'Maximum \t\t%f\n', max(midx) );fprintf( 'Minimum \t\t%f\n\n', min(midx) );
figure[yy,xx]=hist(midx);bar(xx,yy,'r')title('The histogram of color difference between measured and predicted by parametric method.')xlabel('Delta E under illum A for corrected paramers')ylabel('Frequency')
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APPENDIX C : MATLAB PROGRAMS FOR THE INK SELECTIONSUBSYSTEM
The following diagram depicts the program structure of the process for the ink selection subsystem.
Main Program
(inkselection.m)
Highest chroma selection
Performance evaluation for 32 candidate ink sets by
continuous tone approximation
Vector correlation analysis
inkselection.m
%This program performs the analysis based on the ink selection algorithm
close all; clear all;load pan18.txt; %the ink database of 14 Pantone basic color and 4 process CMYKload pos_rest.txt; %the statistical primaries of the 105 mixtures
lambda=[400:10:700]';
figure;for i=1:18 subplot(3,6,i); plot(lambda, pan18(:,i)); axis([400 700 0 1]);end
load panwhite.txtR_paper=panwhite; %the reference white is the coated paper printed with Pantone inks
pan16=[pan18(:,1:13) pan18(:,15:17)]; % the two black inks are the sure candidates % only 16 colors need to be correlated[r,c]=size(pan16);
po=1/3.5; %the power of the empirical transformation is 1/3.5 for transparent/translucent %ink in optical contact with an opaque substrate (R_paper)
tmp=R_paper.^po; %the offest vector in this case is R_paper.^(1/3.5)
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for i=2:c tmp=[tmp R_paper.^po];end
b=tmp;clear tmp;
%************************************************************************************% Vector correlation analysis%************************************************************************************coef_tab=zeros(5,c);
for i=1:5 for j=1:c tmpcorr=corrcoef(b(:,1)-pos_rest(:,i).^po, b(:,1)-pan16(:,j).^po); coef_tab(i,j)=tmpcorr(1,2); endend
coef_tab=coef_tab'; %correlation coefficients are stored in coef_tab
%************************************************************************************% Selecting the two highest chroma values%************************************************************************************
lab_pan16=lab(pan16,R_paper,[3 1]);chroma=sqrt(lab_pan16(:,2).^2+lab_pan16(:,3).^2);
[coef1,idx1]=sortrows([coef_tab(:,1) chroma]);coef1=flipud(coef1);idx1=flipud(idx1);
[coef2,idx2]=sortrows([coef_tab(:,2) chroma]);coef2=flipud(coef2);idx2=flipud(idx2);
[coef3,idx3]=sortrows([coef_tab(:,3) chroma]);coef3=flipud(coef3);idx3=flipud(idx3);
[coef4,idx4]=sortrows([coef_tab(:,4) chroma]);coef4=flipud(coef4);idx4=flipud(idx4);
[coef5,idx5]=sortrows([coef_tab(:,5) chroma]);coef5=flipud(coef5);idx5=flipud(idx5);
tmpcoef=[coef1(:,1) coef2(:,1) coef3(:,1) coef4(:,1) coef5(:,1)];tmpchroma=[coef1(:,2) coef2(:,2) coef3(:,2) coef4(:,2) coef5(:,2)];tmpidx=[idx1 idx2 idx3 idx4 idx5];
%************************************************************************************% Performance analysis for each candidate ink set by Continuous tone approximation%************************************************************************************
load target105.txt %the reflectance spectra of the 105 mixtures[m,n]=size(target105);load inkset.txt %32 sets of ink combinations formed by the 11 selected inks
tmp=R_paper.^po;for i=2:n
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tmp=[tmp R_paper.^po];end
b=tmp;clear tmp;
k_tar=b-target105.^po; %the 105 mixtures represented in psi space
pan18=[pan16 pan18(:,14) pan18(:,18)];
performance=zeros(32,9); %accuracy of the 32 sets are stored in performancefor i=1:32 conc=zeros(6,n); primary=[pan18(:,inkset(i,1)) pan18(:,inkset(i,2)) pan18(:,inkset(i,3))]; primary=[primary pan18(:,inkset(i,4)) pan18(:,inkset(i,5)) pan18(:,inkset(i,6))]; k_prmy=b(:,1:6)-primary.^po; for j=1:n conc(:,j)=nnls(k_prmy, k_tar(:,j)); end k_recon=k_prmy*conc; R_predicted=(b-k_recon).^(1/po); delta_e94=del_e94(target105, R_predicted,R_paper,[3 1]); midx=meta_idx(target105, R_predicted, R_paper, [3 1 2 1]); performance(i,1)=mean(midx); performance(i,2)=std(midx); performance(i,3)=max(midx); performance(i,4)=min(midx); performance(i,5)=mean(delta_e94); performance(i,6)=std(delta_e94); performance(i,7)=max(delta_e94); performance(i,8)=min(delta_e94); performance(i,9)=rms(target105,R_predicted);end
[mdx,imdx]=sortrows(performance);
load sam94set.txttemp=sam94set;
figure;plot(lambda,temp(:,1),'b', lambda,temp(:,2),'m:',lambda,temp(:,3),'g-.', lambda,temp(:,4),'k--')xlabel('Wavelength')ylabel('Reflectance factor')axis([400 700 0 1])legend('Measured', 'Set 23','Set 24','Set 19')
figure;plot(lambda,temp(:,1),'b', lambda,temp(:,5),'m:',lambda,temp(:,6),'g-.', lambda,temp(:,7),'k--')xlabel('Wavelength')ylabel('Reflectance factor')axis([400 700 0 1])legend('Measured', 'Set 26','Set 14','Set 9')
figure;plot(lambda,temp(:,1),'b', lambda,temp(:,2),'m:',lambda,temp(:,5),'k-.')xlabel('Wavelength')ylabel('Reflectance factor')axis([400 700 0 1])legend('Measured', 'Set 23','Set 26')
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APPENDIX D : MATLAB PROGRAMS FOR THE INK OVERPRINTPREDICTION SUBSYSTEM
There are two parts of this subsystem. The first estimates the absorption (K) and scattering (S) coefficientsof translucent ink. The second uses the estimated K and S to predicted an overprint by using the K, S, andthickness data of the topmost layer ink and the reflectance data the layer underneath the topmost layer asthe parameters of the Kubelka-Munk basic equation. The first process is the program named ksfind.m andthe second is named overprint.m
The following diagram is the structure chart of the first component for K and S estimation for translucentinks.
Main Program
(ksfind.m)
K&S estimation for Yellow ink
K&S estimation for magenta ink
K&S estimation for cyan ink
K&S estimation for red ink
K&S estimation for green ink
K&S estimation for blue ink
Objective function(ksest.m)
Objective function(ksest.m)
Objective function(ksest.m)
Objective function(ksest.m)
Objective function(ksest.m)
Objective function(ksest.m)
ksfind.m
%This program estimates the absorption (K) and scattering (S) coefficients for translucent%inks
clear all; close all;load black_b.txt; %reflectance spectrum of the black surface of contrast paperload white_b.txt; %reflectance spectrum of the white surface of contrast paperload r_on_ba.txt; %reflectance spectra of primary inks printed on contrast paper %data order is ink printed on white followed by ink printed on black
r_on_ba2=saunder(r_on_ba); %reflectance spectra of primary inks printed on contrast % paper after saunderson's correctionlambda=[400:10:700]';
%************************************************************************************% Determining the K and S for yellow ink%************************************************************************************
ks_yel(:,1)=white_b-r_on_ba2(:,1); %initialing K for yellow inkks_yel(:,2)=r_on_ba2(:,2)-black_b; %initialing S for yellow ink %K&S data are stored as K followed by Sks_yel=abs(ks_yel);
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vlb=ones(31,2)*0.000000000001;vub=ones(31,2)*100;options(1)=1;options(2)=1e-3;options(3)=1e-3;options(4)=1e-3;options(14)=1000000;
ks_yel=constr('ksest', ks_yel, options, vlb, vub, '',r_on_ba(:,1), r_on_ba(:,2), white_b, black_b);
figureplot(lambda, ks_yel(:,1),'b', lambda, ks_yel(:,2), 'm')title('The estimated spectral absorption and scattering coefficient of yellow ink.')xlabel('Wavelength')ylabel('Absorption and Scattering Coefficient')legend('b', 'Absorbtion', 'm','Scattering')
ay=1+(ks_yel(:,1)./ks_yel(:,2));by= sqrt(ay.^2-1); %a and b parameters of the Kubelka-Munk basic equation
white_b=saunder(white_b);Ry_est= (1-white_b.*(ay-by.*coth(by.*ks_yel(:,2))))./(ay-white_b+by.*coth(by.*ks_yel(:,2))); %Reflectance estimation by the Kubelka-Munk basic equationRy_est=asaunder(Ry_est);deltaE94=del_e94(r_on_ba(:,1), Ry_est, white_b)T=num2str( deltaE94, 4 );figureplot(lambda, r_on_ba(:,1),'b', lambda, Ry_est,'m')title('The measured and predicted reflectance of yellow ink.')xlabel('Wavelength')ylabel('Reflectance factor')legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)
%************************************************************************************% Determining the K and S for magenta ink%************************************************************************************load white_b.txt; %reflectance spectrum of the white surface of contrast paperks_mag(:,1)=white_b-r_on_ba2(:,3); %initialing K for magenta inkks_mag(:,2)=r_on_ba2(:,4)-black_b; %initialing S for magenta inkks_mag=abs(ks_mag);
vlb=ones(31,2)*0.000000000001;vub=ones(31,2)*100;options(1)=1;options(2)=1e-3;options(3)=1e-3;options(4)=1e-3;options(14)=1000000;
ks_mag=constr('ksest', ks_mag, options, vlb, vub, '',r_on_ba(:,3), r_on_ba(:,4), white_b, black_b);
figureplot(lambda, ks_mag(:,1),'b', lambda, ks_mag(:,2), 'm')title('The estimated spectral absorption and scattering coefficient of magenta ink.')xlabel('Wavelength')ylabel('Absorption and Scattering Coefficient')legend('b', 'Absorbtion', 'm','Scattering')
am=1+(ks_mag(:,1)./ks_mag(:,2));bm= sqrt(am.^2-1); %a and b parameters of the Kubelka-Munk basic equation
white_b=saunder(white_b);
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Rm_est= (1-white_b.*(am-bm.*coth(bm.*ks_mag(:,2))))./(am-white_b+bm.*coth(bm.*ks_mag(:,2))); %Reflectance estimation by the Kubelka-Munk basic equationRm_est=asaunder(Rm_est);
deltaE94=del_e94(r_on_ba(:,3), Rm_est, white_b)T=num2str( deltaE94, 4 );figureplot(lambda, r_on_ba(:,3),'b', lambda, Rm_est,'m')title('The measured and predicted reflectance of magenta ink.')xlabel('Wavelength')ylabel('Reflectance factor')legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)
%************************************************************************************% Determining the K and S for cyan ink%************************************************************************************load white_b.txt; %reflectance spectrum of the white surface of contrast paperks_cyan(:,1)=white_b-r_on_ba2(:,5); %initialing K for cyan inkks_cyan(:,2)=r_on_ba2(:,6)-black_b; %initialing S for cyan inkks_cyan=abs(ks_cyan);
vlb=ones(31,2)*0.000000000001;vub=ones(31,2)*10;options(1)=1;options(2)=1e-3;options(3)=1e-3;options(4)=1e-3;options(14)=1000000;
ks_cyan=constr('ksest', ks_cyan, options, vlb, vub, '',r_on_ba(:,5), r_on_ba(:,6), white_b, black_b);
figureplot(lambda, ks_cyan(:,1),'b', lambda, ks_cyan(:,2), 'm')title('The estimated spectral absorption and scattering coefficient of cyan ink.')xlabel('Wavelength')ylabel('Absorption and Scattering Coefficient')legend('b', 'Absorbtion', 'm','Scattering')
ac=1+(ks_cyan(:,1)./ks_cyan(:,2));bc= sqrt(ac.^2-1); %a and b parameters of the Kubelka-Munk basic equation
white_b=saunder(white_b);Rc_est= (1-white_b.*(ac-bc.*coth(bc.*ks_cyan(:,2))))./(ac-white_b+bc.*coth(bc.*ks_cyan(:,2))); %Reflectance estimation by the Kubelka-Munk basic equationRc_est=asaunder(Rc_est);
deltaE94=del_e94(r_on_ba(:,5), Rc_est, white_b)T=num2str( deltaE94, 4 );figureplot(lambda, r_on_ba(:,5),'b', lambda, Rc_est,'m')title('The measured and predicted reflectance of cyan ink.')xlabel('Wavelength')ylabel('Reflectance factor')legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)
%************************************************************************************% Determining the K and S for red ink%************************************************************************************load white_b.txt; %reflectance spectrum of the white surface of contrast paperks_red(:,1)=white_b-r_on_ba2(:,7); %initialing K for red inkks_red(:,2)=r_on_ba2(:,8)-black_b;ks_red=abs(ks_red);
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vlb=ones(31,2)*0.000000000001;vub=ones(31,2)*10;options(1)=1;options(2)=1e-3;options(3)=1e-3;options(4)=1e-3;options(14)=1000000;
ks_red=constr('ksest', ks_red, options, vlb, vub, '',r_on_ba(:,7), r_on_ba(:,8), white_b, black_b);
figureplot(lambda, ks_red(:,1),'b', lambda, ks_red(:,2), 'm')title('The estimated spectral absorption and scattering coefficient of red ink.')xlabel('Wavelength')ylabel('Absorption and Scattering Coefficient')legend('b', 'Absorbtion', 'm','Scattering')
ar=1+(ks_red(:,1)./ks_red(:,2));br= sqrt(ar.^2-1); %a and b parameters of the Kubelka-Munk basic equation
white_b=saunder(white_b);Rr_est= (1-white_b.*(ar-br.*coth(br.*ks_red(:,2))))./(ar-white_b+br.*coth(br.*ks_red(:,2))); %Reflectance estimation by the Kubelka-Munk basic equationRr_est=asaunder(Rr_est);
deltaE94=del_e94(r_on_ba(:,7), Rr_est, white_b)T=num2str( deltaE94, 4 );figureplot(lambda, r_on_ba(:,7),'b', lambda, Rr_est,'m')title('The measured and predicted reflectance of red ink.')xlabel('Wavelength')ylabel('Reflectance factor')legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)
%************************************************************************************% Determining the K and S for green ink%************************************************************************************load white_b.txt; %reflectance spectrum of the white surface of contrast paperks_grn(:,1)=white_b-r_on_ba2(:,9); %initialing K for green inkks_grn(:,2)=r_on_ba2(:,10)-black_b; %initialing S for green inkks_grn=abs(ks_grn);
vlb=ones(31,2)*0.000000000001;vub=ones(31,2)*10;options(1)=1;options(2)=1e-3;options(3)=1e-3;options(4)=1e-3;options(14)=1000000;
ks_grn=constr('ksest', ks_grn, options, vlb, vub, '',r_on_ba(:,9), r_on_ba(:,10), white_b, black_b);
figureplot(lambda, ks_grn(:,1),'b', lambda, ks_grn(:,2), 'm')title('The estimated spectral absorption and scattering coefficient of green ink.')xlabel('Wavelength')ylabel('Absorption and Scattering Coefficient')legend('b', 'Absorbtion', 'm','Scattering')
ag=1+(ks_grn(:,1)./ks_grn(:,2));bg= sqrt(ag.^2-1); %a and b parameters of the Kubelka-Munk basic equation
white_b=saunder(white_b);
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Rg_est= (1-white_b.*(ag-bg.*coth(bg.*ks_grn(:,2))))./(ag-white_b+bg.*coth(bg.*ks_grn(:,2))); %Reflectance estimation by the Kubelka-Munk basic equationRg_est=asaunder(Rg_est);
deltaE94=del_e94(r_on_ba(:,9), Rg_est, white_b)T=num2str( deltaE94, 4 );figureplot(lambda, r_on_ba(:,9),'b', lambda, Rg_est,'m')title('The measured and predicted reflectance of green ink.')xlabel('Wavelength')ylabel('Reflectance factor')legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)
%************************************************************************************% Determining the K and S for blue ink%************************************************************************************load white_b.txt; %reflectance spectrum of the white surface of contrast paperks_blu(:,1)=white_b-r_on_ba2(:,11); %initialing K for blue inkks_blu(:,2)=r_on_ba2(:,12)-black_b; %initialing S for blue inkks_blu=abs(ks_blu);
vlb=ones(31,2)*0.000000000001;vub=ones(31,2)*10;options(1)=1;options(2)=1e-3;options(3)=1e-3;options(4)=1e-3;options(14)=1000000;
ks_blu=constr('ksest', ks_blu, options, vlb, vub, '',r_on_ba(:,11), r_on_ba(:,12), white_b, black_b);
figureplot(lambda, ks_blu(:,1),'b', lambda, ks_blu(:,2), 'm')title('The estimated spectral absorption and scattering coefficient of blue ink.')xlabel('Wavelength')ylabel('Absorption and Scattering Coefficient')legend('b', 'Absorbtion', 'm','Scattering')
ab=1+(ks_blu(:,1)./ks_blu(:,2));bb= sqrt(ab.^2-1); %a and b parameters of the Kubelka-Munk basic equation
white_b=saunder(white_b);Rb_est= (1-white_b.*(ab-bb.*coth(bb.*ks_blu(:,2))))./(ab-white_b+bb.*coth(bb.*ks_blu(:,2))); %Reflectance estimation by the Kubelka-Munk basic equationRb_est=asaunder(Rb_est);
deltaE94=del_e94(r_on_ba(:,11), Rb_est, white_b)T=num2str( deltaE94, 4 );figureplot(lambda, r_on_ba(:,11),'b', lambda, Rb_est,'m')title('The measured and predicted reflectance of blue ink.')xlabel('Wavelength')ylabel('Reflectance factor')legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)
ks=[ks_yel ks_mag ks_cyan ks_red ks_grn ks_blu];
save 'du_ks.txt' ks -ASCII -DOUBLE -TABS
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ksest.m
%This is program of the objective function for estimating the absorption and scattering%coefficients for translucent inks
function [f,g]=ksest(ks, Ronw, Ronbk, white, black)
aw=1+(ks(:,1)./ks(:,2));bw= sqrt((aw.^2)-1); %a and b parameters of the Kubelka-Munk basic equation
white=saunder(white);black=saunder(black);Rw_est= (1-white.*(aw-bw.*coth(bw.*ks(:,2))))./(aw-white+bw.*coth(bw.*ks(:,2))); %Reflectance estimation by the Kubelka-Munk basic equationRw_est=asaunder(Rw_est);ak=1+(ks(:,1)./ks(:,2));bk= sqrt(ak.^2-1); %a and b parameters of the Kubelka-Munk basic equation
Rk_est= (1-black.*(ak-bk.*coth(bk.*ks(:,2))))./(ak-black+bk.*coth(bk.*ks(:,2))); %Reflectance estimation by the Kubelka-Munk basic equationRk_est=asaunder(Rk_est);f=rms([Ronw Ronbk], [Rw_est Rk_est]);
g=-[Rw_est Rk_est];
The following diagram is the structure chart of the ink overprint prediction.
Main Program
(overprint.m)
Overpr1.m
Thicknessestimation
Ink overprintestimation
thickest.m km_trans.m
Overpr2.m
Thicknessestimation
Ink overprintestimation
thickest.m km_trans.m
Overpr3.m
Thicknessestimation
Ink overprintestimation
thickest.m km_trans.m
Overpr4.m
Thicknessestimation
Ink overprintestimation
thickest.m km_trans.m
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overprint.m
%This program performs the spectral reflectance factor prediction of ink overprintload ks_dup.sn1; %the absorption and scattering information of the CMYRGB primaries %are stored in a text file named ks_dup.sn1
figuresubplot(2,3,1)plot(lambda, ks_dup(:,1),'b',lambda, ks_dup(:,2)*10,'k:')title('Cyan')ylabel('K or S')axis([400 700 0 3])
subplot(2,3,2)plot(lambda, ks_dup(:,3),'b',lambda, ks_dup(:,4)*10,'k:')title('Magenta')axis([400 700 0 3])
subplot(2,3,3)plot(lambda, ks_dup(:,5),'b',lambda, ks_dup(:,6)*10,'k:')title('Yellow')axis([400 700 0 3])
subplot(2,3,4)plot(lambda, ks_dup(:,7),'b',lambda, ks_dup(:,8)*10,'k:')title('Red')xlabel('Wavelength')ylabel('K or S')axis([400 700 0 3])
subplot(2,3,5)plot(lambda, ks_dup(:,9),'b',lambda, ks_dup(:,10)*10,'k:')title('Green')xlabel('Wavelength')axis([400 700 0 3])
subplot(2,3,6)plot(lambda, ks_dup(:,11),'b',lambda, ks_dup(:,12)*10,'k:')title('Blue')xlabel('Wavelength')axis([400 700 0 3])
overpr1; %prediction of the 1st set of ink overprints proceduced by Waterproofo_est=over_est;overpr2; %prediction of the 2nd set of ink overprints proceduced by Waterproofo_est=[o_est over_est];overpr3; %prediction of the 3rd set of ink overprints proceduced by Waterproofo_est=[o_est over_est];overpr4; %prediction of the 4th set of ink overprints proceduced by Waterproofo_est=[o_est over_est];
load duover.txt; %25 ink overprints produced by Waterproof
deltaE94=del_e94(duover, o_est, ones(31,1), [3 1]);fprintf('The average color difference between measured and predicted DuPont overprints is \n');fprintf( 'Mean \t\t%f\n', mean(deltaE94) );fprintf( 'Standard Deviation \t\t%f\n', std(deltaE94) );fprintf( 'Maximum \t\t%f\n', max(deltaE94) );fprintf( 'Minimum \t\t%f\n\n', min(deltaE94) );
figure[yy,xx]=hist(deltaE94);bar(xx,yy,'r')
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title('The histogram color difference between measured and predicted DuPont overprints.')xlabel('Delta E94')ylabel('Frequency')
spectral_RMS = RMS(duover, o_est);
midx=meta_idx(duover, o_est, ones(31,1));fprintf('The average color difference between measured and predicted DuPont overprints under illum A is %f\n', mean(midx));fprintf( 'Mean \t\t%f\n', mean(midx) );fprintf( 'Standard Deviation \t\t%f\n', std(midx) );fprintf( 'Maximum \t\t%f\n', max(midx) );fprintf( 'Minimum \t\t%f\n\n', min(midx) );
fprintf( 'The root mean square error of predicted DuPont overprints is %f\n\n', spectral_RMS );
figure[yy,xx]=hist(midx);bar(xx,yy,'r')title('The histogram of the index of metamerism between measured and predicted DuPont overprints.')xlabel('Index of Metamerism')ylabel('Frequency')
deltaE94=[deltaE94(1:11);deltaE94(15:21); deltaE94(28:30); deltaE94(34:37)];midx=[midx(1:11);midx(15:21); midx(28:30); midx(34:37)];
fprintf('The average color difference between measured and predicted DuPont overprints is \n');fprintf( 'Mean \t\t%f\n', mean(deltaE94) );fprintf( 'Standard Deviation \t\t%f\n', std(deltaE94) );fprintf( 'Maximum \t\t%f\n', max(deltaE94) );fprintf( 'Minimum \t\t%f\n\n', min(deltaE94) );
figure[yy,xx]=hist(deltaE94);bar(xx,yy,'r')title('The histogram color difference between measured and predicted DuPont overprints.')xlabel('Delta E94')ylabel('Frequency')
fprintf('The average color difference between measured and predicted DuPont overprints under illum A is %f\n', mean(midx));fprintf( 'Mean \t\t%f\n', mean(midx) );fprintf( 'Standard Deviation \t\t%f\n', std(midx) );fprintf( 'Maximum \t\t%f\n', max(midx) );fprintf( 'Minimum \t\t%f\n\n', min(midx) );
figure[yy,xx]=hist(midx);bar(xx,yy,'r')%title('The histogram of the index of metamerism between measured and predicted DuPont overprints.')xlabel('Metameric Index')ylabel('Frequency')
figuresubplot(2,2,1)plot(lambda,duover(:,5),'b',lambda,o_est(:,5),'m:')%xlabel('Wavelength')ylabel('Reflectance')title('Red on Magenta')%legend('measured','predicted','DE94 0.2', 'MI 0.0')axis([400 700 0 1])
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subplot(2,2,2)plot(lambda,duover(:,21),'b',lambda,o_est(:,21),'m:')title('Yellow on Magenta')%xlabel('Wavelength')%ylabel('Reflectance')%legend('measured','predicted','DE94 0.4', 'MI 0.1')axis([400 700 0 1])
subplot(2,2,3)plot(lambda,duover(:,15),'b',lambda,o_est(:,15),'m:')title('Red on Yellow')xlabel('Wavelength')ylabel('Reflectance')%legend('measured','predicted','DE94 0.3', 'MI 0.1')axis([400 700 0 1])
subplot(2,2,4)plot(lambda,duover(:,16),'b',lambda,o_est(:,16),'m:')title('Green on Yellow')xlabel('Wavelength')%ylabel('Reflectance')%legend('measured','predicted','DE94 0.4', 'MI 0.1')axis([400 700 0 1])
figuresubplot(2,2,1)plot(lambda,duover(:,5)-o_est(:,5),'b')%xlabel('Wavelength')ylabel('Delta R')title('Red on Magenta')%legend('measured','predicted','DE94 0.2', 'MI 0.0')axis([400 700 -0.05 0.05])
subplot(2,2,2)plot(lambda,duover(:,21)-o_est(:,21),'b')title('Yellow on Magenta')%xlabel('Wavelength')%ylabel('Reflectance')%legend('measured','predicted','DE94 0.4', 'MI 0.1')axis([400 700 -0.05 0.05])
subplot(2,2,3)plot(lambda,duover(:,15)-o_est(:,15),'b')title('Red on Yellow')xlabel('Wavelength')ylabel('Delta R')%legend('measured','predicted','DE94 0.3', 'MI 0.1')axis([400 700 -0.05 0.05])
subplot(2,2,4)plot(lambda,duover(:,16)-o_est(:,16),'b')title('Green on Yellow')xlabel('Wavelength')%ylabel('Reflectance')%legend('measured','predicted','DE94 0.4', 'MI 0.1')axis([400 700 -0.05 0.05])
figuresubplot(2,2,1)plot(lambda,duover(:,34),'b',lambda,o_est(:,34),'m:')%xlabel('Wavelength')ylabel('Reflectance')
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title('Green on Red')%legend('measured','predicted','DE94 2.1', 'MI 0.4')axis([400 700 0 1])
subplot(2,2,2)plot(lambda,duover(:,17),'b',lambda,o_est(:,17),'m:')title('Blue on Yellow')%xlabel('Wavelength')%ylabel('Reflectance')%legend('measured','predicted','DE94 0.9', 'MI 0.5')axis([400 700 0 1])
subplot(2,2,3)plot(lambda,duover(:,4),'b',lambda,o_est(:,4),'m:')title('Blue on Cyan')xlabel('Wavelength')ylabel('Reflectance')%legend('measured','predicted','DE94 1.4', 'MI 0.7')axis([400 700 0 1])
subplot(2,2,4)plot(lambda,duover(:,20),'b',lambda,o_est(:,20),'m:')title('Blue on Yellow on Cyan')xlabel('Wavelength')%ylabel('Reflectance')%legend('measured','predicted','DE94 1.2', 'MI 1.1')axis([400 700 0 1])
figuresubplot(2,2,1)plot(lambda,duover(:,34)-o_est(:,34),'b')%xlabel('Wavelength')ylabel('Delta R')title('Green on Red')%legend('measured','predicted','DE94 2.1', 'MI 0.4')axis([400 700 -0.05 0.05])
subplot(2,2,2)plot(lambda,duover(:,17)-o_est(:,17),'b')title('Blue on Yellow')%xlabel('Wavelength')%ylabel('Reflectance')%legend('measured','predicted','DE94 0.9', 'MI 0.5')axis([400 700 -0.05 0.05])
subplot(2,2,3)plot(lambda,duover(:,4)-o_est(:,4),'b')title('Blue on Cyan')xlabel('Wavelength')ylabel('Delta R')%legend('measured','predicted','DE94 1.4', 'MI 0.7')axis([400 700 -0.05 0.05])
subplot(2,2,4)plot(lambda,duover(:,20)-o_est(:,20),'b')title('Blue on Yellow on Cyan')xlabel('Wavelength')%ylabel('Reflectance')%legend('measured','predicted','DE94 1.2', 'MI 1.1')axis([400 700 -0.05 0.05])
load prmy_est.txtload primary.txt
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figuresubplot(1,2,1)plot(lambda,ks_dup(:,1),'b',lambda,ks_dup(:,2),'k:')%grid ontitle('Estimated K & S')xlabel('Wavelength')ylabel('K or S')axis([400 700 -0.1 2])
subplot(1,2,2)plot(lambda,primary(:,3)-prmy_est(:,1),'b')grid ontitle('Difference Spectrum')xlabel('Wavelength')ylabel('Delta R')axis([400 700 -0.02 0.02])
overpr1.m
%This program performs ink overprints of the first set of Waterproof samplesclear all; close all;
lambda=[400:10:700]';
load set1.txt; %measured reflectance of overprint
[r,c]=size(set1);
inkidx=set1(1,3:c);
Rg=set1(2:32,2); %spectral reflectance factor of the coated paper
prints=set1(2:r,3:c);
clear set1;
i=1;while inkidx(i) < 7i=i+1;end
num_of_prmy=i-1;pidx=inkidx(:,num_of_prmy+1:c-2); %index of overprintprimary=prints(:,1:num_of_prmy);overprint=prints(:,num_of_prmy+1:c-2);clear prints;
load ks_dup.sn1; %The determined k and S data are stored in ks_dup.sn1
%************************************************************************************% Thickness estimation%************************************************************************************thickness=ones(1,6)*0.001;prmy_est=zeros(31,6);
for i=1:num_of_prmy ks=ks_dup(:,(inkidx(i)-1)*2+1:(inkidx(i)-1)*2+2); vlb=0; vub=10;
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options(1)=1; options(2)=1e-5; options(3)=1e-5; options(4)=1e-5; options(14)=1000000;
thickness(inkidx(i))=constr('thickest', thickness(inkidx(i)), options, vlb, vub, '',ks, saunder(Rg), saunder(primary(:,i))); prmy_est(:,inkidx(i))=asaunder(km_trans(thickness(inkidx(i)), ks, saunder(Rg))); deltaE94=del_e94(primary(:,i), prmy_est(:,inkidx(i)), Rg, [3 1]) T=num2str( deltaE94, 4 );
figure plot(lambda, primary(:,i),'b', lambda, prmy_est(:,inkidx(i)),'m') %title('The measured and predicted reflectance of yellow ink.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)end
[r,c]=size(pidx);over_est=zeros(31,c);
%thickness=[0.9754 0.9584 0.95605 1.02565 0.961775 0.987025];
%************************************************************************************% Ink overprint prediction%************************************************************************************
for i=1:c %c tmpstr=num2str(pidx(i)); num_of_color=length(tmpstr); if num_of_color==2 substrate=prmy_est(:,str2num(tmpstr(2))); ith_prmy=str2num(tmpstr(1)); else substrate=over_est(:,1); ith_prmy=str2num(tmpstr(1)); end ks=ks_dup(:,(ith_prmy-1)*2+1:(ith_prmy-1)*2+2); over_est(:,i)=asaunder(km_trans(thickness(ith_prmy), ks, saunder(substrate))); deltaE94=del_e94(overprint(:,i), over_est(:,i), Rg, [3 1]); T=num2str( deltaE94, 4 ); figure plot(lambda, overprint(:,i),'b', lambda, over_est(:,i),'m') title('The measured and predicted reflectance of overprint.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)end
overpr2.m
%This program performs ink overprints of the second set of Waterproof samples
lambda=[400:10:700]';load set2.txt; %measured reflectance of overprints[r,c]=size(set2);inkidx=set2(1,3:c);
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Rg=set2(2:32,2); %spectral reflectance factor of the coated paper
prints=set2(2:r,3:c);
clear set2;
i=1;while inkidx(i) < 7i=i+1;end
num_of_prmy=i-1;
pidx=inkidx(:,num_of_prmy+1:c-2); %index of overprint
primary=prints(:,1:num_of_prmy);
overprint=prints(:,num_of_prmy+1:c-2);
clear prints;
load ks_dup.sn1;
%************************************************************************************% Thickness estimation%************************************************************************************thickness=ones(1,6)*0.001;
prmy_est=zeros(31,6);
for i=1:num_of_prmy ks=ks_dup(:,(inkidx(i)-1)*2+1:(inkidx(i)-1)*2+2); vlb=0; vub=10; options(1)=1; options(2)=1e-5; options(3)=1e-5; options(4)=1e-5; options(14)=1000000; thickness(inkidx(i))=constr('thickest', thickness(inkidx(i)), options, vlb, vub, '',ks, saunder(Rg), saunder(primary(:,i))); prmy_est(:,inkidx(i))=asaunder(km_trans(thickness(inkidx(i)), ks, saunder(Rg))); deltaE94=del_e94(primary(:,i), prmy_est(:,inkidx(i)), Rg, [3 1]) T=num2str( deltaE94, 4 );
figure plot(lambda, primary(:,i),'b', lambda, prmy_est(:,inkidx(i)),'m') title('The measured and predicted reflectance of yellow ink.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)end
[r,c]=size(pidx);over_est=zeros(31,c);
%thickness=[0.9754 0.9584 0.95605 1.02565 0.961775 0.987025];
%************************************************************************************% Ink overprint prediction%************************************************************************************
for i=1:c %c
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tmpstr=num2str(pidx(i)); num_of_color=length(tmpstr); if num_of_color==2 substrate=prmy_est(:,str2num(tmpstr(2))); ith_prmy=str2num(tmpstr(1)); else substrate=over_est(:,1); ith_prmy=str2num(tmpstr(1)); end ks=ks_dup(:,(ith_prmy-1)*2+1:(ith_prmy-1)*2+2); over_est(:,i)=asaunder(km_trans(thickness(ith_prmy), ks, saunder(substrate))); deltaE94=del_e94(overprint(:,i), over_est(:,i), Rg, [3 1]); T=num2str( deltaE94, 4 ); figure plot(lambda, overprint(:,i),'b', lambda, over_est(:,i),'m') title('The measured and predicted reflectance of overprint.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)end
overpr3.m
%This program performs ink overprints of the second set of Waterproof samples
lambda=[400:10:700]';load set3.txt; %measured reflectance of overprints[r,c]=size(set3);inkidx=set3(1,3:c);Rg=set3(2:32,2); %spectral reflectance factor of the coated paperprints=set3(2:r,3:c);
clear set3;
i=1;while inkidx(i) < 7 i=i+1;end
num_of_prmy=i-1;
pidx=inkidx(:,num_of_prmy+1:c-2); %index of overprintprimary=prints(:,1:num_of_prmy);overprint=prints(:,num_of_prmy+1:c-2);clear prints;
load ks_dup.sn1;
%************************************************************************************% Thickness estimation%************************************************************************************
thickness=ones(1,6)*0.001;
prmy_est=zeros(31,6);
for i=1:num_of_prmy ks=ks_dup(:,(inkidx(i)-1)*2+1:(inkidx(i)-1)*2+2); vlb=0;
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vub=10; options(1)=1; options(2)=1e-5; options(3)=1e-5; options(4)=1e-5; options(14)=1000000;
thickness(inkidx(i))=constr('thickest', thickness(inkidx(i)), options, vlb, vub, '',ks, saunder(Rg), saunder(primary(:,i))); prmy_est(:,inkidx(i))=asaunder(km_trans(thickness(inkidx(i)), ks, saunder(Rg))); deltaE94=del_e94(primary(:,i), prmy_est(:,inkidx(i)), Rg, [3 1]) T=num2str( deltaE94, 4 ); figure plot(lambda, primary(:,i),'b', lambda, prmy_est(:,inkidx(i)),'m') title('The measured and predicted reflectance of yellow ink.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)end
[r,c]=size(pidx);over_est=zeros(31,c);
%thickness=[0.9754 0.9584 0.95605 1.02565 0.961775 0.987025];
%************************************************************************************% Ink overprint prediction%************************************************************************************for i=1:c %c tmpstr=num2str(pidx(i)); num_of_color=length(tmpstr); if num_of_color==2 substrate=prmy_est(:,str2num(tmpstr(2))); ith_prmy=str2num(tmpstr(1)); else substrate=over_est(:,1); ith_prmy=str2num(tmpstr(1)); end ks=ks_dup(:,(ith_prmy-1)*2+1:(ith_prmy-1)*2+2); over_est(:,i)=asaunder(km_trans(thickness(ith_prmy), ks, saunder(substrate))); deltaE94=del_e94(overprint(:,i), over_est(:,i), Rg, [3 1]); T=num2str( deltaE94, 4 ); figure plot(lambda, overprint(:,i),'b', lambda, over_est(:,i),'m') title('The measured and predicted reflectance of overprint.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)end
overpr4.m
%This program performs ink overprints of the second set of Waterproof samples
lambda=[400:10:700]';load set4.txt; %measured reflectance of overprints[r,c]=size(set4);inkidx=set4(1,3:c);Rg=set4(2:32,2); %spectral reflectance factor of the coated paper
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prints=set4(2:r,3:c);
clear set4;
i=1;while inkidx(i) < 7 i=i+1;end
num_of_prmy=i-1;
pidx=inkidx(:,num_of_prmy+1:c-2); %index of overprint
primary=prints(:,1:num_of_prmy);
overprint=prints(:,num_of_prmy+1:c-2);
clear prints;
load ks_dup.sn1;
%************************************************************************************% Thickness estimation%************************************************************************************thickness=ones(1,6)*0.001;
prmy_est=zeros(31,6);
for i=1:num_of_prmy ks=ks_dup(:,(inkidx(i)-1)*2+1:(inkidx(i)-1)*2+2); vlb=0; vub=10; options(1)=1; options(2)=1e-5; options(3)=1e-5; options(4)=1e-5; options(14)=1000000;
thickness(inkidx(i))=constr('thickest', thickness(inkidx(i)), options, vlb, vub, '',ks, saunder(Rg), saunder(primary(:,i))); prmy_est(:,inkidx(i))=asaunder(km_trans(thickness(inkidx(i)), ks, saunder(Rg))); deltaE94=del_e94(primary(:,i), prmy_est(:,inkidx(i)), Rg, [3 1]) T=num2str( deltaE94, 4 ); figure plot(lambda, primary(:,i),'b', lambda, prmy_est(:,inkidx(i)),'m') title('The measured and predicted reflectance of yellow ink.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)end
[r,c]=size(pidx);over_est=zeros(31,c);
%thickness=[0.9754 0.9584 0.95605 1.02565 0.961775 0.987025];
%************************************************************************************% Ink overprint prediction%************************************************************************************for i=1:c %c tmpstr=num2str(pidx(i)); num_of_color=length(tmpstr);
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if num_of_color==2 substrate=prmy_est(:,str2num(tmpstr(2))); ith_prmy=str2num(tmpstr(1)); else substrate=over_est(:,1); ith_prmy=str2num(tmpstr(1)); end ks=ks_dup(:,(ith_prmy-1)*2+1:(ith_prmy-1)*2+2); over_est(:,i)=asaunder(km_trans(thickness(ith_prmy), ks, saunder(substrate))); deltaE94=del_e94(overprint(:,i), over_est(:,i), Rg, [3 1]); T=num2str( deltaE94, 4 ); figure plot(lambda, overprint(:,i),'b', lambda, over_est(:,i),'m') title('The measured and predicted reflectance of overprint.') xlabel('Wavelength') ylabel('Reflectance factor') axis([400 700 0 1]) legend('b','Measured', 'm','Predicted', 'bk', 'Delta_E94', 'bk', T)end
thickest.m
%This is the objective fuction of estimating the actual ink thickness of an printed on%an opaque substrate
function [f,g]=thickest(thick, ks, Rg, R_sample)
R_est=km_trans(thick, ks, Rg); % Kubelka-Munk equation for translucent materialf=rms(R_est, R_sample);g=-R_est;
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APPENDIX E : MATLAB PROGRAMS FOR THE PROPOSED SIX-COLOR FORWARD PRINTING MODEL
Since a six-color printing model is the union of 10 four-color sub-models, the following structure chartonly shows the program structure for a CMYK modeling process.
Main Program
(secondorder.m)
Finding data for modeling
optical-trapping(optrap1.m)
Determining the dot gain
correction function
(interink.m)
Determining the effective dot
area
The first order estimation
(theo2eff.m)
The second order dot gain
correction(iino_q.m)
Reflectance estimation byNeugebauer
equation(neug4c.m)
...
...
Determiningn-factor
(nfind.m)
CMYKmodeling
YGOKmodeling
Demichel equation
(dmi4c.m)
secondorder.m
%This program is a forward second order six-color printing model proposed by this research%modeling optical trapping effect to modified the accuracy of the first order printing model%which is comprised of theoretical to effective dot area transfer function and the Yule-Nielsen modified%spectral Neugebauer equation
clear all; close all;load ave_ramp.txt %the spectral reflectance factors of 13-step ramps
c_ramp=(ave_ramp(1:13, 7:37))';m_ramp=(ave_ramp(14:26, 7:37))';y_ramp=(ave_ramp(27:39, 7:37))';
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g_ramp=(ave_ramp(40:52, 7:37))';o_ramp=(ave_ramp(53:65, 7:37))';k_ramp=(ave_ramp(66:78, 7:37))';
clear ave_ramp;
lambda=[400:10:700]';
R_paper=c_ramp(:,13);
dc=[100;90;80;70;60;50;40;30;20;15;10;5;0]/100; %the theoretical dot areas of each ramp
n=nfind(c_ramp,m_ramp,y_ramp,g_ramp,o_ramp,k_ramp,R_paper,20,0.1);%determining the Yule-Nielsen n-factor (n=2.2)
%the effective dot area of primary ramps solved by n-modified Murray-Davies equationa_c=inv_murr(c_ramp,R_paper,n);a_m=inv_murr(m_ramp,R_paper,n);a_y=inv_murr(y_ramp,R_paper,n);a_g=inv_murr(g_ramp,R_paper,n);a_o=inv_murr(o_ramp,R_paper,n);a_k=inv_murr(k_ramp,R_paper,n);
%prediction of primary ramps by n-modified Murray-Davies equationreconc=murray(c_ramp(:,1), a_c, R_paper, n);reconm=murray(m_ramp(:,1), a_m, R_paper, n);recony=murray(y_ramp(:,1), a_y, R_paper, n);recong=murray(g_ramp(:,1), a_g, R_paper, n);recono=murray(o_ramp(:,1), a_o, R_paper, n);reconk=murray(k_ramp(:,1), a_k, R_paper, n);
figure;plot(dc, [a_c a_m a_y a_g a_o a_k]);axis([0 1 0 1])xlabel('Theoretical dot area')ylabel('Effective dot area')title('Mechanical dot gain of six-color ramps')
figure;plot(dc, a_c,'k-',dc,a_m,'k:',dc, a_y,'k-.',dc, a_k,'k--');axis([0 1 0 1])xlabel('Theoretical dot area')ylabel('Effective dot area')%title('Mechanical dot gain of six-color ramps')legend('cyan','magenta', 'yellow','black')
figure;plot(dc, a_c-dc,'k-',dc,a_m-dc,'k:',dc, a_y-dc,'k-.',dc, a_k-dc,'k--');axis([0 1 0 0.4])xlabel('Theoretical dot area')ylabel('Dot-gain in units of fractional dot area')%title('Mechanical dot gain of six-color ramps')legend('cyan','magenta', 'yellow','black')
tmp1=a_c-dc;tmp2=(a_c-dc)*0.75;tmp3=(a_c-dc)*0.50;tmp4=(a_c-dc)*0.25;
figure;plot(dc, tmp1,'k-',dc,tmp2,'k:',dc, tmp3,'k-.',dc, tmp4,'k--');
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axis([0 1 0 0.35])xlabel('Theoretical dot area')ylabel('Dot-gain in units of fractional dot area')%title('Mechanical dot gain of six-color ramps')legend('mag=0%','mag=25%', 'mag=50%','mag=75%')hold onplot([0.1; 0.1],[0; tmp1(11)],'k:');plot([0.3; 0.3],[0; tmp1(8)],'k:');plot([0.5; 0.5],[0; tmp1(6)],'k:');plot([0.7; 0.7],[0; tmp1(4)],'k:');
figuresubplot(1,2,1)plot([0;0.25;0.5;0.75], [1; tmp2(6)/tmp1(6); tmp3(6)/tmp1(6); tmp4(6)/tmp1(6)].^(2));xlabel('a_t_h_e_o_,_m')ylabel('f_c_m')axis([0 0.8 0 1])title('The correction scalar by Iino and Berns')subplot(1,2,2)plot([0;0.1;0.3;0.5;0.7], [1; 0.85; 0.87; 0.86; 0.9]);xlabel('a_t_h_e_o_,_c')ylabel('g_m_c')axis([0 0.8 0.5 1])title('The correction scalar by the proposed algorithm')
c=c_ramp(:,1);m=m_ramp(:,1);y=y_ramp(:,1);g=g_ramp(:,1);o=o_ramp(:,1);k=k_ramp(:,1);w=R_paper;
figure;plot(lambda,c,'c',lambda,m,'m',lambda,y,'y',lambda,g,'g',lambda,o,'r',lambda,k,'k',lambda,w,'k:')xlabel('Wavelength')ylabel('Reflectance factor')axis([400 700 0 1])legend('Cyan','Magenta','Yellow','Green','Orange','Black','Substrate')
figure;plot(lambda, c_ramp,'b', lambda, reconc,'m:')xlabel('Wavelength')ylabel('Reflectance factor')title('Measured and predicted c_ramp')axis([400 700 0 1])legend('b','Measured', 'm:', 'Predicted')
figure;plot(lambda, m_ramp,'b', lambda, reconm,'m:')xlabel('Wavelength')ylabel('Reflectance factor')title('Measured and predicted m_ramp')axis([400 700 0 1])legend('b','Measured', 'm:', 'Predicted')
figure;plot(lambda, y_ramp,'b', lambda, recony,'m:')xlabel('Wavelength')ylabel('Reflectance factor')title('Measured and predicted y_ramp')
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axis([400 700 0 1])legend('b','Measured', 'm:', 'Predicted')
figure;plot(lambda, g_ramp,'b', lambda, recong,'m:')xlabel('Wavelength')ylabel('Reflectance factor')title('Measured and predicted g_ramp')axis([400 700 0 1])legend('b','Measured', 'm:', 'Predicted')
figure;plot(lambda, o_ramp,'b', lambda, recono,'m:')xlabel('Wavelength')ylabel('Reflectance factor')title('Measured and predicted o_ramp')axis([400 700 0 1])legend('b','Measured', 'm:', 'Predicted')
figure;plot(lambda, k_ramp,'b', lambda, reconk,'m:')xlabel('Wavelength')ylabel('Reflectance factor')title('Measured and predicted k_ramp')axis([400 700 0 1])legend('b','Measured', 'm:', 'Predicted')
dac=[0 25 50 70 90]/100; %the theoretical dot areas for modeling ink trappingdac=dac';
sc=0.62; %scalar for adjusting the actual effective dot areas
%************************************************************************************% The forward CMYK submodel%************************************************************************************load cmyk.txtdc_1=cmyk(:,2:5)/100; %the theoretical dot areas of the CMYK targetcmyk5555=cmyk(:,6:36)'; %the 625 CMYK mixtures
load neprmy1.txt %the 16 Neugebauer primaries of CMYK printing process
%optrap1=trapfind(cmyk); %the procedure searching for data in the verification target %for modeling optical-trapping effect
%fij=interink(optrap1, neprmy1, n, a_c, a_m, a_y, a_k, dc, dac);%save 'fij_1.txt' fij -ASCII -DOUBLE -TABS %the fij is the dot gain correction scalar function discussed in the thesis %here, fij is the gij described in the thesis
load fij_1.txt; %the dot gain correction scalar function for CMYK printing processfij=fij_1;
qscalar=iino_q(fij, dc_1, dac); %the dot gain correction scalars for the CMYK verification target
eff=theo2eff(dc_1, dc, c_ramp, m_ramp, y_ramp, k_ramp, R_paper, n); %the first order estimated theoretical dot areas for CMYK verification target
eff1=dc_1+qscalar.*(eff-dc_1); %the effective dot areas for CMYK verification target after the second order %improvement by dot gain correction
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eff1=(dc_1+qscalar.*(eff-dc_1))*sc + eff*(1-sc); %the final effective dot areas for CMYK verification target lies between the first %order estimation and the second order estimation
pred1=neug4c(eff1, neprmy1, n); %spectral reconstruction by n-modified Neugebauer equation
del1=del_e94(cmyk5555, pred1, R_paper, [3 1]);
%************************************************************************************% The forward CMGK submodel%************************************************************************************load cmgk.txtdc_2=cmgk(:,2:5)/100;cmgk5555=cmgk(:,6:36)';
load neprmy2.txt
%optrap2=trapfind(cmgk);
%fij=interink(optrap2, neprmy2, n, a_c, a_m, a_g, a_k, dc, dac);%save 'fij_2.txt' fij -ASCII -DOUBLE -TABS
load fij_2.txt;fij=fij_2;
qscalar=iino_q(fij, dc_2, dac);
eff=theo2eff(dc_2, dc, c_ramp, m_ramp, g_ramp, k_ramp, R_paper, n);
eff2=dc_2+qscalar.*(eff-dc_2);eff2=(dc_2+qscalar.*(eff-dc_2))*sc + eff*(1-sc);
pred2=neug4c(eff2, neprmy2, n);
del2=del_e94(cmgk5555, pred2, R_paper, [3 1]);
%************************************************************************************% The forward CMOK submodel%************************************************************************************load cmok.txtdc_3=cmok(:,2:5)/100;cmok5555=cmok(:,6:36)';
load neprmy3.txt
%optrap3=trapfind(cmok);
%fij=interink(optrap3, neprmy3, n, a_c, a_m, a_o, a_k, dc, dac);%save 'fij_3.txt' fij -ASCII -DOUBLE -TABS
load fij_3.txt;fij=fij_3;
qscalar=iino_q(fij, dc_3, dac);
eff=theo2eff(dc_3, dc, c_ramp, m_ramp, o_ramp, k_ramp, R_paper, n);
eff3=dc_3+qscalar.*(eff-dc_3);
eff3=(dc_3+qscalar.*(eff-dc_3))*sc + eff*(1-sc);
pred3=neug4c(eff3, neprmy3, n);
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del3=del_e94(cmok5555, pred3, R_paper, [3 1]);
%************************************************************************************% The forward CYGK submodel%************************************************************************************load cygk.txtdc_4=cygk(:,2:5)/100;cygk5555=cygk(:,6:36)';
load neprmy4.txt
%optrap4=trapfind(cygk);
%fij=interink(optrap4, neprmy4, n, a_c, a_y, a_g, a_k, dc, dac);%save 'fij_4.txt' fij -ASCII -DOUBLE -TABS
load fij_4.txt;fij=fij_4;
qscalar=iino_q(fij, dc_4, dac);
eff=theo2eff(dc_4, dc, c_ramp, y_ramp, g_ramp, k_ramp, R_paper, n);
eff4=dc_4+qscalar.*(eff-dc_4);
eff4=(dc_4+qscalar.*(eff-dc_4))*sc + eff*(1-sc);
pred4=neug4c(eff4, neprmy4, n);
del4=del_e94(cygk5555, pred4, R_paper, [3 1]);
%************************************************************************************% The forward CYOK submodel%************************************************************************************load cyok.txtdc_5=cyok(:,2:5)/100;cyok5555=cyok(:,6:36)';
load neprmy5.txt
%optrap5=trapfind(cyok);
%fij=interink(optrap5, neprmy5, n, a_c, a_y, a_o, a_k, dc, dac);%save 'fij_5.txt' fij -ASCII -DOUBLE -TABS
load fij_5.txt;fij=fij_5;
qscalar=iino_q(fij, dc_5, dac);
eff=theo2eff(dc_5, dc, c_ramp, y_ramp, o_ramp, k_ramp, R_paper, n);
eff5=dc_5+qscalar.*(eff-dc_5);eff5=(dc_5+qscalar.*(eff-dc_5))*sc + eff*(1-sc);
pred5=neug4c(eff5, neprmy5, n);
del5=del_e94(cyok5555, pred5, R_paper, [3 1]);
%************************************************************************************% The forward CGOK submodel%************************************************************************************load cgok.txt
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dc_6=cgok(:,2:5)/100;cgok5555=cgok(:,6:36)';
load neprmy6.txt
%optrap6=trapfind(cgok);
%fij=interink(optrap6, neprmy6, n, a_c, a_g, a_o, a_k, dc, dac);%save 'fij_6.txt' fij -ASCII -DOUBLE -TABS
load fij_6.txt;fij=fij_6;
qscalar=iino_q(fij, dc_6, dac);
eff=theo2eff(dc_6, dc, c_ramp, g_ramp, o_ramp, k_ramp, R_paper, n);
eff6=dc_6+qscalar.*(eff-dc_6);eff6=(dc_6+qscalar.*(eff-dc_6))*sc + eff*(1-sc);
pred6=neug4c(eff6, neprmy6, n);
del6=del_e94(cgok5555, pred6, R_paper, [3 1]);
%************************************************************************************% The forward MYGK submodel%************************************************************************************load mygk.txtdc_7=mygk(:,2:5)/100;mygk5555=mygk(:,6:36)';
load neprmy7.txt
%optrap7=trapfind(mygk);
%fij=interink(optrap7, neprmy7, n, a_m, a_y, a_g, a_k, dc, dac);%save 'fij_7.txt' fij -ASCII -DOUBLE -TABS
load fij_7.txt;fij=fij_7;
qscalar=iino_q(fij, dc_7, dac);
eff=theo2eff(dc_7, dc, m_ramp, y_ramp, g_ramp, k_ramp, R_paper, n);
eff7=dc_7+qscalar.*(eff-dc_7);eff7=(dc_7+qscalar.*(eff-dc_7))*sc + eff*(1-sc);
pred7=neug4c(eff7, neprmy7, n);
del7=del_e94(mygk5555, pred7, R_paper, [3 1]);
%************************************************************************************% The forward MYOK submodel%************************************************************************************load myok.txtdc_8=myok(:,2:5)/100;myok5555=myok(:,6:36)';
load neprmy8.txt
%optrap8=trapfind(myok);
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%fij=interink(optrap8, neprmy8, n, a_m, a_y, a_o, a_k, dc, dac);%save 'fij_8.txt' fij -ASCII -DOUBLE -TABS
load fij_8.txt;fij=fij_8;
qscalar=iino_q(fij, dc_8, dac);
eff=theo2eff(dc_8, dc, m_ramp, y_ramp, o_ramp, k_ramp, R_paper, n);
eff8=dc_8+qscalar.*(eff-dc_8);eff8=(dc_8+qscalar.*(eff-dc_8))*sc + eff*(1-sc);
pred8=neug4c(eff8, neprmy8, n);
del8=del_e94(myok5555, pred8, R_paper, [3 1]);
%************************************************************************************% The forward MGOK submodel%************************************************************************************load mgok.txtdc_9=mgok(:,2:5)/100;mgok5555=mgok(:,6:36)';
load neprmy9.txt
%optrap9=trapfind(mgok);
%fij=interink(optrap9, neprmy9, n, a_m, a_g, a_o, a_k, dc, dac);%save 'fij_9.txt' fij -ASCII -DOUBLE -TABS
load fij_9.txt;fij=fij_9;
qscalar=iino_q(fij, dc_9, dac);
eff=theo2eff(dc_9, dc, m_ramp, g_ramp, o_ramp, k_ramp, R_paper, n);
eff9=dc_9+qscalar.*(eff-dc_9);eff9=(dc_9+qscalar.*(eff-dc_9))*sc + eff*(1-sc);
pred9=neug4c(eff9, neprmy9, n);
del9=del_e94(mgok5555, pred9, R_paper, [3 1]);
%************************************************************************************% The forward YGOK submodel%************************************************************************************load ygok.txtdc_10=ygok(:,2:5)/100;ygok5555=ygok(:,6:36)';
load neprmy10.txt
%optrap10=trapfind(ygok);
%fij=interink(optrap10, neprmy10, n, a_y, a_g, a_o, a_k, dc, dac);%save 'fij_10.txt' fij -ASCII -DOUBLE -TABS
load fij_10.txt;fij=fij_10;
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qscalar=iino_q(fij, dc_10, dac);
eff=theo2eff(dc_10, dc, y_ramp, g_ramp, o_ramp, k_ramp, R_paper, n);
eff10=dc_10+qscalar.*(eff-dc_10);eff10=(dc_10+qscalar.*(eff-dc_10))*sc + eff*(1-sc);
pred10=neug4c(eff10, neprmy10, n);
del10=del_e94(ygok5555, pred10, R_paper, [3 1]);
tmp1=[mean(del1);mean(del2);mean(del3);mean(del4);mean(del4);mean(del6)];meande=[tmp1;mean(del7);mean(del8);mean(del9);mean(del10)]
tmp1=[max(del1);max(del2);max(del3);max(del4);max(del4);max(del6)];maxde=[tmp1;max(del7);max(del8);max(del9);max(del10)]
tmp1=[std(del1);std(del2);std(del3);std(del4);std(del4);std(del6)];stdde=[tmp1;std(del7);std(del8);std(del9);std(del10)]
measured=[cmyk5555 cmgk5555 cmok5555 cygk5555 cyok5555 cgok5555];measured=[measured mygk5555 myok5555 mgok5555 ygok5555];
predicted=[pred1 pred2 pred3 pred4 pred5 pred6 pred7 pred8 pred9 pred10];
delta_e94=del_e94(measured, predicted, R_paper, [3 1]);
fprintf('The statistical color difference between measured and predicted ramps is \n');fprintf( 'Mean \t\t%f\n', mean(delta_e94) );fprintf( 'Standard Deviation \t\t%f\n', std(delta_e94) );fprintf( 'Maximum \t\t%f\n', max(delta_e94) );fprintf( 'Minimum \t\t%f\n\n', min(delta_e94) );
spectral_RMS = RMS( measured, predicted );
midx=meta_idx(measured, predicted, R_paper, [3 2 1 2]);fprintf('The statistical color difference between measured and predicted under illum A %f\n', mean(midx));fprintf( 'Mean \t\t%f\n', mean(midx) );fprintf( 'Standard Deviation \t\t%f\n', std(midx) );fprintf( 'Maximum \t\t%f\n', max(midx) );fprintf( 'Minimum \t\t%f\n\n', min(midx) );
figure[yy,xx]=hist(delta_e94);bar(xx,yy,'r')%title('The histogram color difference between measured and predicted')xlabel('Delta E94')ylabel('Frequency')
nfind.m
%This program searches and optimizes the best n-factor for reconstructing the ramps with%minimum error
function n_factor=nfind(ramp1, ramp2, ramp3, ramp4, ramp5, ramp6, R_paper, num_of_step, step)
n=1;idx_deltaE=zeros(num_of_step,2);for i=1:num_of_step dotarea1=inv_murr(ramp1,R_paper,n);
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dotarea2=inv_murr(ramp2,R_paper,n); dotarea3=inv_murr(ramp3,R_paper,n); dotarea4=inv_murr(ramp4,R_paper,n); dotarea5=inv_murr(ramp5,R_paper,n); dotarea6=inv_murr(ramp6,R_paper,n);
recon1=murray(ramp1(:,1), dotarea1, R_paper, n); recon2=murray(ramp2(:,1), dotarea2, R_paper, n); recon3=murray(ramp3(:,1), dotarea3, R_paper, n); recon4=murray(ramp4(:,1), dotarea4, R_paper, n); recon5=murray(ramp5(:,1), dotarea5, R_paper, n); recon6=murray(ramp6(:,1), dotarea6, R_paper, n);
measured=[ramp1 ramp2 ramp3 ramp4 ramp5 ramp6]; predicted=[recon1 recon2 recon3 recon4 recon5 recon6];
idx_deltaE(i,1)=mean(real(del_e94(measured, predicted, R_paper, [3 1]))); idx_deltaE(i,2)=n;
n=n+step;end
figure;plot(idx_deltaE(:,2), idx_deltaE(:,1));xlabel('n-factor')ylabel('Delta E94 for D50')
[sorted, idx]=sortrows(idx_deltaE);
n_factor=sorted(1,2);
trapfind.m
%This fuction search the ink mixture in the verification target for modeling optical trapping
function trapdata=trapfind(cmyk)
load trapdac.txt % the theoretical dot areas of ink mixtures which are required for modeling optical trapping
dc_1=cmyk(:,2:5)/100;cmyk5555=cmyk(:,6:36)';
[p,q]=size(trapdac);[s,t]=size(dc_1);
trap_idx=zeros(p,1);for i=1:p for j=1:s if trapdac(i,:)-dc_1(j,:)*100==0 trap_idx(i)=j; end endend
optrap1=cmyk(trap_idx(1),:);
for i=2:p optrap1=[optrap1 ;cmyk(trap_idx(i),:)];end
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trapdata=optrap1(:,6:36)';
interink.m
%This program models the optical-trapping and builds dot gain correction functions for each%verification target
function fij=interink(optrap1, neprmy1, n, a_c, a_m, a_y, a_k, dc, dac)
rcm=optrap1(:,1:5); rcy=optrap1(:,6:10); rck=optrap1(:,11:15);rmc=optrap1(:,16:20); rmy=optrap1(:,21:25); rmk=optrap1(:,26:30);ryc=optrap1(:,31:35); rym=optrap1(:,36:40); ryk=optrap1(:,41:45);rkc=optrap1(:,46:50); rkm=optrap1(:,51:55); rky=optrap1(:,56:60);
fcm=r_neug2c(rcm, [neprmy1(:,1:3) neprmy1(:,7)], n);fcy=r_neug2c(rcy, [neprmy1(:,1:2) neprmy1(:,4) neprmy1(:,6)], n);fck=r_neug2c(rck, [neprmy1(:,1:2) neprmy1(:,9:10)], n);fmc=r_neug2c(rmc, [neprmy1(:,1:3) neprmy1(:,7)], n);fmy=r_neug2c(rmy, [neprmy1(:,1) neprmy1(:,3:5)], n);fmk=r_neug2c(rmk, [neprmy1(:,1) neprmy1(:,3) neprmy1(:,9) neprmy1(:,11)], n);fyc=r_neug2c(ryc, [neprmy1(:,1) neprmy1(:,2) neprmy1(:,4) neprmy1(:,6)], n);fym=r_neug2c(rym, [neprmy1(:,1) neprmy1(:,3) neprmy1(:,4) neprmy1(:,5)], n);fyk=r_neug2c(ryk, [neprmy1(:,1) neprmy1(:,4) neprmy1(:,9) neprmy1(:,12)], n);fkc=r_neug2c(rkc, [neprmy1(:,1) neprmy1(:,2) neprmy1(:,9) neprmy1(:,10)], n);fkm=r_neug2c(rkm, [neprmy1(:,1) neprmy1(:,3) neprmy1(:,9) neprmy1(:,11)], n);fky=r_neug2c(rky, [neprmy1(:,1) neprmy1(:,4) neprmy1(:,9) neprmy1(:,12)], n); %r_neug2c() is a inverse Neugebauer equation for a two color case
tmp=[fcm(:,2) fcy(:,2) fck(:,2) fmc(:,1) fmy(:,2) fmk(:,2)];tmp=[tmp fyc(:,1) fym(:,1) fyk(:,2) fkc(:,1) fkm(:,1) fky(:,1)];
c=interp1(dc, a_c, dac, 'cubic');m=interp1(dc, a_m, dac, 'cubic');y=interp1(dc, a_y, dac, 'cubic');k=interp1(dc, a_k, dac, 'cubic');
tmp2=ones(5,12);tmp2(2:5,1)=(tmp(2:5,1)-dac(2:5))./(m(2:5,1)-dac(2:5));tmp2(2:5,2)=(tmp(2:5,2)-dac(2:5))./(y(2:5,1)-dac(2:5));tmp2(2:5,3)=(tmp(2:5,3)-dac(2:5))./(k(2:5,1)-dac(2:5));tmp2(2:5,4)=(tmp(2:5,4)-dac(2:5))./(c(2:5,1)-dac(2:5));tmp2(2:5,5)=(tmp(2:5,5)-dac(2:5))./(y(2:5,1)-dac(2:5));tmp2(2:5,6)=(tmp(2:5,6)-dac(2:5))./(k(2:5,1)-dac(2:5));tmp2(2:5,7)=(tmp(2:5,7)-dac(2:5))./(c(2:5,1)-dac(2:5));tmp2(2:5,8)=(tmp(2:5,8)-dac(2:5))./(m(2:5,1)-dac(2:5));tmp2(2:5,9)=(tmp(2:5,9)-dac(2:5))./(k(2:5,1)-dac(2:5));tmp2(2:5,10)=(tmp(2:5,10)-dac(2:5))./(c(2:5,1)-dac(2:5));tmp2(2:5,11)=(tmp(2:5,11)-dac(2:5))./(m(2:5,1)-dac(2:5));tmp2(2:5,12)=(tmp(2:5,12)-dac(2:5))./(y(2:5,1)-dac(2:5));
fij=tmp2;
%This program determines the global correction scalar for the dot gain of each primary%ramp due to the ink and optical trapping
function qscalar=iino_q(fij, dc_1, dac)
fmc=interp1(dac, fij(:,4), dc_1(:,1), 'cubic');fyc=interp1(dac, fij(:,7), dc_1(:,1), 'cubic');fkc=interp1(dac, fij(:,10), dc_1(:,1), 'cubic');
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fcm=interp1(dac, fij(:,1), dc_1(:,2), 'cubic');fym=interp1(dac, fij(:,8), dc_1(:,2), 'cubic');fkm=interp1(dac, fij(:,11), dc_1(:,2), 'cubic');
fcy=interp1(dac, fij(:,2), dc_1(:,3), 'cubic');fmy=interp1(dac, fij(:,5), dc_1(:,3), 'cubic');fky=interp1(dac, fij(:,12), dc_1(:,3), 'cubic');
fck=interp1(dac, fij(:,3), dc_1(:,4), 'cubic');fmk=interp1(dac, fij(:,6), dc_1(:,4), 'cubic');fyk=interp1(dac, fij(:,9), dc_1(:,4), 'cubic');
%the following check for the existence of secondary%fmc should be corrected by the existence of magenta in as the secondary ink
fmc_c=fmc.*(((1./fmc).*(~dc_1(:,2)))+(~~dc_1(:,2))); %corrected fmcfyc_c=fyc.*(((1./fyc).*(~dc_1(:,3)))+(~~dc_1(:,3)));fkc_c=fkc.*(((1./fkc).*(~dc_1(:,4)))+(~~dc_1(:,4)));
fcm_c=fcm.*(((1./fcm).*(~dc_1(:,1)))+(~~dc_1(:,1)));fym_c=fym.*(((1./fym).*(~dc_1(:,3)))+(~~dc_1(:,3)));fkm_c=fkm.*(((1./fkm).*(~dc_1(:,4)))+(~~dc_1(:,4)));
fcy_c=fcy.*(((1./fcy).*(~dc_1(:,1)))+(~~dc_1(:,1)));fmy_c=fmy.*(((1./fmy).*(~dc_1(:,2)))+(~~dc_1(:,2)));fky_c=fky.*(((1./fky).*(~dc_1(:,4)))+(~~dc_1(:,4)));
fck_c=fck.*(((1./fck).*(~dc_1(:,1)))+(~~dc_1(:,1)));fmk_c=fmk.*(((1./fmk).*(~dc_1(:,2)))+(~~dc_1(:,2)));fyk_c=fyk.*(((1./fyk).*(~dc_1(:,3)))+(~~dc_1(:,3)));
qscalar=[fmc_c.*fyc_c.*fkc_c fcm_c.*fym_c.*fkm_c fcy_c.*fmy_c.*fky_c fck_c.*fmk_c.*fyk_c];
neug2c.m
%This program performs the spectral reflectance estimation for a two-color printing process.%The estimation is based the n-modified spectral Neugebauer equation
function R_predicted=neug2c(cm, neuprimary, nf)
a=demi2c(cm);a1=a(:,1);
%notice a1 is the area of papera2=a(:,2);a3=a(:,3);a4=a(:,4);
clear a;
R1=neuprimary(:,1);%notice R1 is the R_paper
R2=neuprimary(:,2);R3=neuprimary(:,3);R4=neuprimary(:,4);
[n,m]=size(cm);
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[p,q]=size(neuprimary);tmp=zeros(p,n);
for i=1:n tmp(:,i)=a1(i)*(R1.^(1/nf))+a2(i)*(R2.^(1/nf))+a3(i)*(R3.^(1/nf))+a4(i)*(R4.^(1/nf)); tmp(:,i)=tmp(:,i).^nf;end
R_predicted=tmp;
neug4c.m%This program performs the spectral reflectance estimation for a four-color printing process.%The estimation is based the n-modified spectral Neugebauer equation
function R_predicted=neug4c(cmyk, neuprimary, nf)
a=demi4c(cmyk);a1=a(:,1);
%notice a1 is the area of papera2=a(:,2);a3=a(:,3);a4=a(:,4);a5=a(:,5);a6=a(:,6);a7=a(:,7);a8=a(:,8);a9=a(:,9);a10=a(:,10);a11=a(:,11);a12=a(:,12);a13=a(:,13);a14=a(:,14);a15=a(:,15);a16=a(:,16);
clear a;
R1=neuprimary(:,1);%notice R1 is the R_paper
R2=neuprimary(:,2);R3=neuprimary(:,3);R4=neuprimary(:,4);R5=neuprimary(:,5);R6=neuprimary(:,6);R7=neuprimary(:,7);R8=neuprimary(:,8);R9=neuprimary(:,9);R10=neuprimary(:,10);R11=neuprimary(:,11);R12=neuprimary(:,12);R13=neuprimary(:,13);R14=neuprimary(:,14);R15=neuprimary(:,15);R16=neuprimary(:,16);
[n,m]=size(cmyk);[p,q]=size(neuprimary);tmp=zeros(p,n);
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for i=1:n tmp(:,i)=a1(i)*(R1.^(1/nf))+a2(i)*(R2.^(1/nf))+a3(i)*(R3.^(1/nf))+a4(i)*(R4.^(1/nf)); tmp(:,i)=tmp(:,i)+a5(i)*(R5.^(1/nf))+a6(i)*(R6.^(1/nf))+a7(i)*(R7.^(1/nf))+a8(i)*(R8.^(1/nf)); tmp(:,i)=tmp(:,i)+a9(i)*(R9.^(1/nf))+a10(i)*(R10.^(1/nf))+a11(i)*(R11.^(1/nf))+a12(i)*(R12.^(1/nf)); tmp(:,i)=tmp(:,i)+a13(i)*(R13.^(1/nf))+a14(i)*(R14.^(1/nf))+a15(i)*(R15.^(1/nf))+a16(i)*(R16.^(1/nf)); tmp(:,i)=tmp(:,i).^nf;end
R_predicted=tmp;
demi4c.m
%This function calculated the dot area coverage based on the Demichel's probability model
function area=demi4c(cmyk)
f1 =(1-cmyk(:,1)).*(1-cmyk(:,2)).*(1-cmyk(:,3)).*(1-cmyk(:,4)); %white
f2 = cmyk(:,1).*(1-cmyk(:,2)).*(1-cmyk(:,3)).*(1-cmyk(:,4)); %cyanf3 =(1-cmyk(:,1)).* cmyk(:,2).*(1-cmyk(:,3)).*(1-cmyk(:,4)); %magentaf4 =(1-cmyk(:,1)).*(1-cmyk(:,2)).* cmyk(:,3).*(1-cmyk(:,4)); %yellowf5 =(1-cmyk(:,1)).* cmyk(:,2).* cmyk(:,3).*(1-cmyk(:,4)); %redf6 = cmyk(:,1).*(1-cmyk(:,2)).* cmyk(:,3).*(1-cmyk(:,4)); %greenf7 = cmyk(:,1).* cmyk(:,2).*(1-cmyk(:,3)).*(1-cmyk(:,4)); %bluef8 = cmyk(:,1).* cmyk(:,2).* cmyk(:,3).*(1-cmyk(:,4)); %3cblack
f9 =(1-cmyk(:,1)).*(1-cmyk(:,2)).*(1-cmyk(:,3)).* cmyk(:,4); %1cblack
f10 = cmyk(:,1).*(1-cmyk(:,2)).*(1-cmyk(:,3)).* cmyk(:,4); %kcyanf11 =(1-cmyk(:,1)).* cmyk(:,2).*(1-cmyk(:,3)).* cmyk(:,4); %kmagentaf12 =(1-cmyk(:,1)).*(1-cmyk(:,2)).* cmyk(:,3).* cmyk(:,4); %kyellowf13 =(1-cmyk(:,1)).* cmyk(:,2).* cmyk(:,3).* cmyk(:,4); %kredf14 = cmyk(:,1).*(1-cmyk(:,2)).* cmyk(:,3).* cmyk(:,4); %kgreenf15 = cmyk(:,1).* cmyk(:,2).*(1-cmyk(:,3)).* cmyk(:,4); %kbluef16 = cmyk(:,1).* cmyk(:,2).* cmyk(:,3).* cmyk(:,4); %4cblack
area=[f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16];
theo2eff.m
%The module builds an transfer function for theoretical to effective dot area%using one-dimensional look up table and cubic spline for interpolation
function eff=theo2eff(cmyk, dc, c_ramp, m_ramp, y_ramp, k_ramp, R_paper, nf);
efflut_c=inv_murr(c_ramp, R_paper, nf);efflut_m=inv_murr(m_ramp, R_paper, nf);efflut_y=inv_murr(y_ramp, R_paper, nf);efflut_k=inv_murr(k_ramp, R_paper, nf);
[row, col]=size(cmyk);
eff=zeros(row,col);
eff(:,1)=interp1(dc, efflut_c, cmyk(:,1),'cubic');eff(:,2)=interp1(dc, efflut_m, cmyk(:,2),'cubic');eff(:,3)=interp1(dc, efflut_y, cmyk(:,3),'cubic');eff(:,4)=interp1(dc, efflut_k, cmyk(:,4),'cubic');
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APPENDIX F : MATLAB PROGRAMS FOR THE SPECTRAL-BASED SIX-COLOR SEPARATION MINIMIZING METAMERISM
The following diagram depicts the structure of the MLTLAB program which performs the spectral-basedsix-color separation to determine the ink amount corresponding to each printing primary.
Main Program
(checksep.m)
Four-inkselector
(selector.m)
Six color backward printing model
(six_sep.m)
Six color forward printing model
(neuge6.m)
CMYK backward printing model(cmyk_sep.m)
YGOK backward printing model(ygok_sep.m)
... CMYK forward printing model(cmyk_sub.m)
YGOK forward printing model(ygok_sub.m)
...Nonnegative leastsquare fitting
(nnls.m)
Objective functionminimizingmetamerism(sep1_obj.m)
Objective functionminimizingmetamerism(sep1_obj.m)
Spectralreconstruction by
the proposed printingmodel
(forward2nd.m)
Spectralreconstruction by
the proposed printingmodel
(forward2nd.m)
checksep.m
%This program performs spectral-based six-color separation of the Gretag Macbeth Color Checker%minimizing the metamerism between the original and its reproduction
clear all; close all;load ave_ramp.txt %the six-color ramps for the forward modeling
c_ramp=(ave_ramp(1:13, 7:37))';m_ramp=(ave_ramp(14:26, 7:37))';y_ramp=(ave_ramp(27:39, 7:37))';g_ramp=(ave_ramp(40:52, 7:37))';o_ramp=(ave_ramp(53:65, 7:37))';k_ramp=(ave_ramp(66:78, 7:37))';
clear ave_ramp;
lambda=[400:10:700]';
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R_paper=c_ramp(:,13);
dc=[100;90;80;70;60;50;40;30;20;15;10;5;0]/100; %theoretical dot area for ramps
n=2.2; %the determined n-factor is 2.2
prmy_ramp=[c_ramp m_ramp y_ramp g_ramp o_ramp k_ramp];primary=[c_ramp(:,1) m_ramp(:,1) y_ramp(:,1) g_ramp(:,1) o_ramp(:,1) k_ramp(:,1)];
load checker.txt %spectral reflectance of the Gretag Macbeth Color Checker
po=1/3.5; %po can be form 1/4 to 1/3
checktag=selector(checker, R_paper, primary, po); %the selector determines best four ink for a spectrum requiring six-color reproduction
dc_est=six_sep(checktag, checker, prmy_ramp,R_paper, n); %the six-color backward printing model
check_dc =dc_est(:,2:5);
R_predicted=neuge6(check_dc, checktag, prmy_ramp, R_paper, n); %the proposed six-color forward printing model
delta_e94=del_e94(R_predicted, checker, R_paper, [3 1]);
fprintf('The average color difference between measured and predicted is \n');fprintf( 'Mean \t\t%f\n', mean(delta_e94) );fprintf( 'Standard Deviation \t\t%f\n', std(delta_e94) );fprintf( 'Maximum \t\t%f\n', max(delta_e94) );fprintf( 'Minimum \t\t%f\n\n', min(delta_e94) );
spectral_RMS = RMS( R_predicted, checker);
midx=meta_idx(R_predicted, checker, R_paper, [3 2 1 2]);fprintf('The average color difference between measured and predicted under illum A %f\n', mean(midx));fprintf( 'Mean \t\t%f\n', mean(midx) );fprintf( 'Standard Deviation \t\t%f\n', std(midx) );fprintf( 'Maximum \t\t%f\n', max(midx) );fprintf( 'Minimum \t\t%f\n\n', min(midx) );
fprintf( 'The root mean square error of reflectance factor is %f\n\n', spectral_RMS );
figure[yy,xx]=hist(delta_e94);bar(xx,yy,'r')title('The histogram color difference between measured and predicted')xlabel('Delta E94')ylabel('Frequency')
lab_meas=lab(checker,R_paper,[3 1]);lab_pred=lab(R_predicted,R_paper, [3 1]);primary=[c_ramp(:,1) m_ramp(:,1) y_ramp(:,1) g_ramp(:,1) o_ramp(:,1) k_ramp(:,1)];lab_prmy=lab(primary,R_paper,[3 1]);
figure;plot(lab_meas(:,2),lab_meas(:,3),'*')hold onplot(lab_pred(:,2),lab_pred(:,3),'mo')hold onplot(lab_prmy(:,2), lab_prmy(:,3),'g+')
figure;
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plot(lab_meas(:,2),lab_meas(:,1),'*')hold onplot(lab_pred(:,2),lab_pred(:,1),'mo')
figure;plot(lab_meas(:,3),lab_meas(:,1),'*')hold onplot(lab_pred(:,3),lab_pred(:,1),'mo')
delta_e94=delta_e94(1:24);midx=midx(1:24);
a=[midx delta];[y,i]=sort(a);
figure;subplot(2,2,1)plot(lambda,checker(:,1),'k', lambda,R_predicted(:,1),'k:')axis([400 700 0 1])%xlabel('Wavelength')ylabel('Reflectance')legend('Dark Skin')
subplot(2,2,2)plot(lambda,checker(:,2),'k', lambda,R_predicted(:,2),'k:')axis([400 700 0 1])%xlabel('Wavelength')%ylabel('Reflectance')legend('Light Skin')
subplot(2,2,3)plot(lambda,checker(:,14),'k', lambda,R_predicted(:,14),'k:')axis([400 700 0 1])xlabel('Wavelength')ylabel('Reflectance')legend('Green')
subplot(2,2,4)plot(lambda,checker(:,6),'k', lambda,R_predicted(:,6),'k:')axis([400 700 0 1])xlabel('Wavelength')%ylabel('Reflectance')legend('Blue Green')
figure;subplot(2,2,1)plot(lambda,checker(:,8),'k', lambda,R_predicted(:,8),'k:')axis([400 700 0 1])%xlabel('Wavelength')ylabel('Reflectance')legend('Purplish Blue')
subplot(2,2,2)plot(lambda,checker(:,13),'k', lambda,R_predicted(:,13),'k:')axis([400 700 0 1])%xlabel('Wavelength')%ylabel('Reflectance')legend('Blue')
subplot(2,2,3)plot(lambda,checker(:,18),'k', lambda,R_predicted(:,18),'k:')axis([400 700 0 1])
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xlabel('Wavelength')ylabel('Reflectance')legend('Cyan')
subplot(2,2,4)plot(lambda,checker(:,12),'k', lambda,R_predicted(:,12),'k:')axis([400 700 0 1])xlabel('Wavelength')%ylabel('Reflectance')legend('Orange Yellow')
load check_dc.txt;checktag=check_dc(:,1);check_dc=check_dc(:,2:5);
predicted=neuge6(check_dc, checktag, prmy_ramp, R_paper, n);
load check_6c_print.txt
printed=check_6c_print;
figure;subplot(2,3,1)plot(lambda, checker(:,5),'b', lambda, predicted(:,5),'m',lambda, printed(:,5),'g')ylabel('Reflectance')axis([400 700 0 1])title('Blue flower')
subplot(2,3,2)plot(lambda, checker(:,7),'b', lambda, predicted(:,7),'m',lambda, printed(:,7),'g')%ylabel('Reflectance')axis([400 700 0 1])title('Orange')
subplot(2,3,3)plot(lambda, checker(:,8),'b', lambda, predicted(:,8),'m',lambda, printed(:,8),'g')%ylabel('Reflectance')legend('Original','Predicted','Reproduction')axis([400 700 0 1])title('Purplish blue')
subplot(2,3,4)plot(lambda, checker(:,11),'b', lambda, predicted(:,11),'m',lambda, printed(:,11),'g')ylabel('Reflectance')xlabel('Wavelength')axis([400 700 0 1])title('Yellow green')
subplot(2,3,5)plot(lambda, checker(:,20),'b', lambda, predicted(:,20),'m',lambda, printed(:,20),'g')xlabel('Wavelength')axis([400 700 0 1])title('Neutral 8')
subplot(2,3,6)plot(lambda, checker(:,21),'b', lambda, predicted(:,21),'m',lambda, printed(:,21),'g')xlabel('Wavelength')axis([400 700 0 1])title('Neutral 6.5')
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selector.m
%This program performs the proper four inks for a given input spectrum requiring%six color reproduction
function inktag=selector(sample, R_paper, primary, po)
if nargin==3 po=1/3.5;end
[n,m]=size(sample);
tmp=R_paper.^po;for i=2:m tmp=[tmp R_paper.^po];end
k_prmy=tmp(:,1:6)-primary.^po;
k_c=k_prmy(:,1);k_m=k_prmy(:,2);k_y=k_prmy(:,3);k_g=k_prmy(:,4);k_o=k_prmy(:,5);k_k=k_prmy(:,6);
k_cmyk=[k_prmy(:,1) k_prmy(:,2) k_prmy(:,3) k_prmy(:,6)];k_cmgk=[k_prmy(:,1) k_prmy(:,2) k_prmy(:,4) k_prmy(:,6)];k_cmok=[k_prmy(:,1) k_prmy(:,2) k_prmy(:,5) k_prmy(:,6)];k_cygk=[k_prmy(:,1) k_prmy(:,3) k_prmy(:,4) k_prmy(:,6)];k_cyok=[k_prmy(:,1) k_prmy(:,3) k_prmy(:,5) k_prmy(:,6)];k_cgok=[k_prmy(:,1) k_prmy(:,4) k_prmy(:,5) k_prmy(:,6)];k_mygk=[k_prmy(:,2) k_prmy(:,3) k_prmy(:,4) k_prmy(:,6)];k_myok=[k_prmy(:,2) k_prmy(:,3) k_prmy(:,5) k_prmy(:,6)];k_mgok=[k_prmy(:,2) k_prmy(:,4) k_prmy(:,5) k_prmy(:,6)];k_ygok=[k_prmy(:,3) k_prmy(:,4) k_prmy(:,5) k_prmy(:,6)];
k_sample=tmp-(sample.^po); %tranform ink samples to psi space
tmp=tmp(:,1:10);
tmp_tag=zeros(m,1);
for i=1:m R_spectra=sample(:,i); for j=1:9 R_spectra=[R_spectra sample(:,i)]; end
k_spectra=zeros(31,10); conc1=nnls(k_cmyk, k_sample(:,i)); %nonnegative least square fittting by CMYK inks conc2=nnls(k_cmgk, k_sample(:,i)); conc3=nnls(k_cmok, k_sample(:,i)); conc4=nnls(k_cygk, k_sample(:,i)); conc5=nnls(k_cyok, k_sample(:,i)); conc6=nnls(k_cgok, k_sample(:,i)); conc7=nnls(k_mygk, k_sample(:,i)); conc8=nnls(k_myok, k_sample(:,i)); conc9=nnls(k_mgok, k_sample(:,i)); conc10=nnls(k_ygok, k_sample(:,i));
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k_spectra(:,1)=k_cmyk*conc1; k_spectra(:,2)=k_cmgk*conc2; k_spectra(:,3)=k_cmok*conc3; k_spectra(:,4)=k_cygk*conc4; k_spectra(:,5)=k_cyok*conc5; k_spectra(:,6)=k_cgok*conc6; k_spectra(:,7)=k_mygk*conc7; k_spectra(:,8)=k_myok*conc8; k_spectra(:,9)=k_mgok*conc9; k_spectra(:,10)=k_ygok*conc10; R_pred10=(tmp-k_spectra).^(1/po); criteria = sqrt(mean((R_pred10-R_spectra).^2)); %the root-mean-squrae error
%xyz_sample=[xyz(R_spectra, [1 1]) xyz(R_spectra, [2 1]) xyz(R_spectra, [3 1])]'; %xyz_pred=[xyz(R_pred10, [1 1]) xyz(R_pred10, [2 1]) xyz(R_pred10, [3 1])]'; %criteria=sqrt(mean((xyz_pred-xyz_sample).^2)); [y,ink_idx]=sort(criteria);
tmp_tag(i)=ink_idx(1);end
inktag=tmp_tag;
six_sep.m
%This program performs the proposed spectral-based six-color separation algorithm where%the parameter "inktag" is the vector of enumerated ink combination formed by CMYKGO
function dc_est=six_sep(inktag, sample, prmy_ramp,R_paper, n, dc)
[p,q]=size(sample);
prmy_cmyk=[prmy_ramp(:,1:39) prmy_ramp(:,66:78)];prmy_cmgk=[prmy_ramp(:,1:26) prmy_ramp(:,40:52) prmy_ramp(:,66:78)];prmy_cmok=[prmy_ramp(:,1:26) prmy_ramp(:,53:65) prmy_ramp(:,66:78)];prmy_cygk=[prmy_ramp(:,1:13) prmy_ramp(:,27:39) prmy_ramp(:,40:52) prmy_ramp(:,66:78)];prmy_cyok=[prmy_ramp(:,1:13) prmy_ramp(:,27:39) prmy_ramp(:,53:65) prmy_ramp(:,66:78)];prmy_cgok=[prmy_ramp(:,1:13) prmy_ramp(:,40:78)];prmy_mygk=[prmy_ramp(:,14:52) prmy_ramp(:,66:78)];prmy_myok=[prmy_ramp(:,14:39) prmy_ramp(:,53:65) prmy_ramp(:,66:78)];prmy_mgok=[prmy_ramp(:,14:26) prmy_ramp(:,40:78)];prmy_ygok=[prmy_ramp(:,27:78)];
dc_tmp=zeros(q,5);for i=1:q i switch inktag(i) case 1, dc_tmp(i,:)= cmyk_sep(sample(:,i), prmy_cmyk, R_paper, n); case 2, dc_tmp(i,:)= cmgk_sep(sample(:,i), prmy_cmgk, R_paper, n); case 3, dc_tmp(i,:)= cmok_sep(sample(:,i), prmy_cmok, R_paper, n); case 4, dc_tmp(i,:)= cygk_sep(sample(:,i), prmy_cygk, R_paper, n); case 5, dc_tmp(i,:)= cyok_sep(sample(:,i), prmy_cyok, R_paper, n); case 6, dc_tmp(i,:)= cgok_sep(sample(:,i), prmy_cgok, R_paper, n); case 7, dc_tmp(i,:)= mygk_sep(sample(:,i), prmy_mygk, R_paper, n); case 8, dc_tmp(i,:)= myok_sep(sample(:,i), prmy_myok, R_paper, n); case 9, dc_tmp(i,:)= mgok_sep(sample(:,i), prmy_mgok, R_paper, n); case 10, dc_tmp(i,:)= ygok_sep(sample(:,i), prmy_ygok, R_paper, n); otherwise fprintf('error'); end
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end
dc_est=dc_tmp;
cmyk_sep.m
%This program is the backward CMYK printing model
function dot_est=cmyk_sep(sample, ramp, R_paper, n);load fij_1.txt;
fij_1=[fij_1; ones(1,12)];load neprmy1.txt
cmyk_est=ones(1,4)*0.5;vlb=ones(1,4)*0.0001;vub=ones(1,4)*0.99;options(1)=1;options(2)=1e-3;options(3)=1e-3;options(4)=1e-3;options(14)=10000;
cmyk_est=constr('sep1_obj',cmyk_est,options,vlb,vub,'',sample,ramp,fij_1,neprmy1, R_paper, n);
dot_est=[1 cmyk_est];
sep1_obj.m
%This is the objective function for a four-color forward printing model
function [f,g]=sep1_obj(dot_est, sample, ramp, fij, neprmy, R_paper, n);
sc=0.62;dac=[0 25 50 70 90 100]/100;dac=dac';dc=[100;90;80;70;60;50;40;30;20;15;10;5;0]/100;
qscalar=iino_q(fij, dot_est, dac);eff=theo2eff(dot_est, dc, ramp(:,1:13), ramp(:,14:26), ramp(:,27:39), ramp(:,40:52), R_paper, n);eff1=(dot_est+qscalar.*(eff-dot_est))*sc + eff*(1-sc);
pred=neug4c(eff1, neprmy, n);
%xyz_sample=[xyz(sample, [3 1]) xyz(sample, [2 1])]';%xyz_pred=[xyz(pred, [3 1]) xyz(pred, [2 1])]';%f=rms(xyz_sample(1:4), xyz_pred(1:4));
f=del_e94(sample, pred, R_paper, [3 1]);
tmpcmyk=sum(dot_est);g(1)=-tmpcmyk;g(2)=tmpcmyk-3;g(3)=dot_est(4)-0.001;g(4)=-dot_est(4);
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neuge6.m
%This program performs the proposed spectral-based six-color synthesis based on the union%of 10 four-color forward printing models where the parameter "inktag" is the vector%of enumerated ink combination formed by CMYKGO
function R_predicted=neuge6(check_dc, inktag,prmy_ramp,R_paper, n);
[q,p]=size(check_dc);
prmy_cmyk=[prmy_ramp(:,1:39) prmy_ramp(:,66:78)];prmy_cmgk=[prmy_ramp(:,1:26) prmy_ramp(:,40:52) prmy_ramp(:,66:78)];prmy_cmok=[prmy_ramp(:,1:26) prmy_ramp(:,53:65) prmy_ramp(:,66:78)];prmy_cygk=[prmy_ramp(:,1:13) prmy_ramp(:,27:39) prmy_ramp(:,40:52) prmy_ramp(:,66:78)];prmy_cyok=[prmy_ramp(:,1:13) prmy_ramp(:,27:39) prmy_ramp(:,53:65) prmy_ramp(:,66:78)];prmy_cgok=[prmy_ramp(:,1:13) prmy_ramp(:,40:78)];prmy_mygk=[prmy_ramp(:,14:52) prmy_ramp(:,66:78)];prmy_myok=[prmy_ramp(:,14:39) prmy_ramp(:,53:65) prmy_ramp(:,66:78)];prmy_mgok=[prmy_ramp(:,14:26) prmy_ramp(:,40:78)];prmy_ygok=[prmy_ramp(:,27:78)];
tmp=zeros(31,q);for i=1:q switch inktag(i) case 1, tmp(:,i)= cmyk_sub(check_dc(i,:), prmy_cmyk, R_paper, n); case 2, tmp(:,i)= cmgk_sub(check_dc(i,:), prmy_cmgk, R_paper, n); case 3, tmp(:,i)= cmok_sub(check_dc(i,:), prmy_cmok, R_paper, n); case 4, tmp(:,i)= cygk_sub(check_dc(i,:), prmy_cygk, R_paper, n); case 5, tmp(:,i)= cyok_sub(check_dc(i,:), prmy_cyok, R_paper, n); case 6, tmp(:,i)= cgok_sub(check_dc(i,:), prmy_cgok, R_paper, n); case 7, tmp(:,i)= mygk_sub(check_dc(i,:), prmy_mygk, R_paper, n); case 8, tmp(:,i)= myok_sub(check_dc(i,:), prmy_myok, R_paper, n); case 9, tmp(:,i)= mgok_sub(check_dc(i,:), prmy_mgok, R_paper, n); case 10, tmp(:,i)= ygok_sub(check_dc(i,:), prmy_ygok, R_paper, n); otherwise fprintf('error'); endend
R_predicted=tmp;
cmyk_sub.m
%This program performs the spectral reconstruction using CMYK inks
function pred= cmyk_sub(dot_est, ramp, R_paper, n);load fij_1.txt;
fij_1=[fij_1; ones(1,12)];load neprmy1.txt
pred=forward2nd(dot_est, ramp, fij_1, neprmy1, R_paper, n);
forward2nd.m
%This program is the proposed four-color forward printing model
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function pred=forward2nd(dottiest, ramp, fij, neprmy, R_paper, n);
sc=0.62;dac=[0 25 50 70 90 100]/100;dac=dac';dc=[100;90;80;70;60;50;40;30;20;15;10;5;0]/100;
qscalar=iino_q(fij, dot_est, dac);
eff=theo2eff(dot_est, dc, ramp(:,1:13), ramp(:,14:26), ramp(:,27:39), ramp(:,40:52), R_paper, n);
eff1=(dot_est+qscalar.*(eff-dot_est))*sc + eff*(1-sc);
pred=neug4c(eff1, neprmy, n);
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APPENDIX G : THE SIX ESTIMATED COLORANTS FOR THE 105MIXTURES OF POSTER COLORS
Wavelength Primary 1 Primary 2 Primary 3 Primary 4 Primary 5 Primary 6400 0.278 0.457 0.004 0.591 0.556 0.003410 0.319 0.480 0.001 0.559 0.651 0.003420 0.349 0.500 0.000 0.537 0.728 0.002430 0.387 0.513 0.000 0.487 0.789 0.001440 0.454 0.496 0.000 0.359 0.791 0.001450 0.538 0.443 0.002 0.204 0.690 0.001460 0.631 0.363 0.010 0.075 0.522 0.001470 0.708 0.281 0.027 0.012 0.355 0.001480 0.738 0.208 0.052 0.000 0.223 0.001490 0.720 0.149 0.104 0.000 0.125 0.003500 0.704 0.090 0.260 0.020 0.039 0.007510 0.641 0.033 0.542 0.224 0.004 0.008520 0.512 0.004 0.757 0.619 0.002 0.007530 0.373 0.000 0.835 0.887 0.010 0.006540 0.262 0.000 0.841 0.881 0.034 0.005550 0.173 0.005 0.832 0.747 0.080 0.005560 0.107 0.025 0.830 0.552 0.143 0.006570 0.062 0.072 0.845 0.371 0.211 0.008580 0.036 0.157 0.875 0.214 0.251 0.010590 0.027 0.270 0.892 0.108 0.232 0.010600 0.027 0.406 0.891 0.041 0.160 0.009610 0.032 0.531 0.883 0.010 0.094 0.008620 0.037 0.610 0.872 0.002 0.060 0.008630 0.042 0.656 0.865 0.000 0.047 0.009640 0.045 0.685 0.867 0.000 0.043 0.010650 0.049 0.708 0.871 0.001 0.038 0.010660 0.053 0.738 0.877 0.005 0.028 0.009670 0.061 0.776 0.877 0.013 0.013 0.007680 0.075 0.823 0.866 0.034 0.003 0.005690 0.096 0.876 0.849 0.074 0.000 0.004700 0.123 0.901 0.836 0.143 0.000 0.006
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APPENDIX H : THE ESTIMATED SPECTRAL ABSORPTION ANDSCATTERING COEFFICIENTS FOR THE WATERPROOF ®
CMYRGB PRIMARIES
Wavelength Kcyan Scyan Kmagenta Smagenta Kyellow Syellow
400 0.282 0.027 0.514 0.022 0.959 0.036410 0.158 0.021 0.460 0.016 1.140 0.043420 0.118 0.015 0.455 0.016 1.310 0.050430 0.087 0.012 0.449 0.015 1.370 0.051440 0.055 0.009 0.452 0.015 1.350 0.054450 0.034 0.007 0.485 0.017 1.200 0.051460 0.030 0.006 0.544 0.019 1.000 0.044470 0.031 0.005 0.621 0.021 0.877 0.038480 0.036 0.004 0.734 0.023 0.749 0.041490 0.046 0.003 0.872 0.025 0.516 0.041500 0.062 0.002 0.990 0.028 0.279 0.035510 0.086 0.002 1.160 0.034 0.121 0.026520 0.125 0.002 1.450 0.044 0.044 0.020530 0.181 0.003 1.650 0.048 0.015 0.015540 0.258 0.004 1.530 0.041 0.006 0.012550 0.369 0.007 1.530 0.041 0.003 0.010560 0.540 0.011 1.790 0.054 0.002 0.008570 0.755 0.015 2.100 0.087 0.002 0.008580 0.956 0.019 1.150 0.088 0.002 0.006590 1.100 0.022 0.417 0.066 0.001 0.006600 1.240 0.025 0.169 0.046 0.001 0.005610 1.360 0.028 0.086 0.033 0.000 0.005620 1.380 0.029 0.057 0.025 0.000 0.005630 1.350 0.028 0.048 0.020 0.000 0.004640 1.280 0.026 0.044 0.016 0.000 0.003650 1.160 0.023 0.042 0.013 0.000 0.003660 1.020 0.020 0.042 0.012 0.000 0.003670 0.963 0.018 0.041 0.010 0.000 0.003680 0.991 0.018 0.040 0.009 0.000 0.003690 1.060 0.020 0.039 0.009 0.000 0.002700 1.180 0.023 0.038 0.007 0.000 0.002
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Wavelength Kred Sred Kgreen Sgreen Kblue Sblue
400 1.200 0.100 3.130 0.219 0.800 0.100
410 1.100 0.065 2.270 0.156 0.614 0.050
420 1.140 0.075 1.960 0.136 0.489 0.041
430 1.130 0.069 1.670 0.113 0.393 0.035
440 1.150 0.070 1.460 0.103 0.297 0.029
450 1.220 0.075 1.220 0.093 0.221 0.025
460 1.380 0.086 0.961 0.081 0.200 0.022
470 1.740 0.113 0.763 0.074 0.212 0.020
480 2.020 0.121 0.604 0.070 0.256 0.019
490 2.170 0.114 0.416 0.067 0.342 0.019
500 2.250 0.111 0.245 0.062 0.464 0.021
510 2.290 0.111 0.165 0.054 0.629 0.022
520 2.310 0.109 0.159 0.045 0.877 0.027
530 2.320 0.111 0.191 0.039 1.220 0.029
540 2.320 0.127 0.252 0.036 2.240 0.102
550 2.300 0.160 0.346 0.035 2.560 0.060
560 2.250 0.204 0.483 0.036 2.650 0.041
570 2.210 0.232 0.671 0.040 2.660 0.042
580 1.910 0.247 0.941 0.045 2.660 0.043
590 0.690 0.132 1.350 0.058 2.650 0.046
600 0.240 0.101 1.940 0.076 2.640 0.047
610 0.084 0.080 2.460 0.094 2.630 0.049
620 0.032 0.066 2.560 0.095 2.610 0.051
630 0.014 0.056 2.580 0.100 2.590 0.052
640 0.007 0.049 2.600 0.103 2.570 0.054
650 0.004 0.044 2.600 0.107 2.560 0.054
660 0.003 0.039 2.560 0.116 2.540 0.053
670 0.002 0.036 2.410 0.119 2.540 0.051
680 0.001 0.033 2.080 0.111 2.550 0.047
690 0.001 0.031 1.740 0.087 2.560 0.049
700 0.001 0.029 1.620 0.087 2.550 0.052
297
APPENDIX I : THE REFLECTANCE SPECTRA OF THEORIGINAL GRETAG MACBETH COLOR CHECKER
WavelengthDark skin Light skin Blue sky Foliage Blue flower Blue green
400 0.059 0.195 0.249 0.057 0.301 0.262410 0.059 0.214 0.294 0.058 0.377 0.311420 0.059 0.216 0.305 0.060 0.405 0.329430 0.059 0.216 0.311 0.062 0.417 0.343440 0.059 0.217 0.321 0.065 0.423 0.359450 0.059 0.220 0.326 0.067 0.425 0.383460 0.059 0.225 0.323 0.068 0.417 0.421470 0.059 0.233 0.313 0.068 0.401 0.467480 0.059 0.243 0.302 0.070 0.376 0.516490 0.060 0.250 0.290 0.073 0.341 0.563500 0.062 0.256 0.273 0.083 0.304 0.592510 0.067 0.267 0.250 0.110 0.275 0.598520 0.073 0.282 0.227 0.155 0.249 0.584530 0.075 0.288 0.211 0.187 0.225 0.557540 0.077 0.293 0.196 0.188 0.211 0.522550 0.080 0.304 0.175 0.167 0.206 0.479560 0.087 0.311 0.154 0.142 0.200 0.433570 0.100 0.321 0.143 0.123 0.194 0.388580 0.118 0.353 0.151 0.114 0.191 0.344590 0.134 0.396 0.158 0.109 0.198 0.300600 0.144 0.438 0.152 0.104 0.214 0.259610 0.148 0.478 0.145 0.100 0.229 0.232620 0.152 0.515 0.141 0.100 0.239 0.219630 0.158 0.540 0.140 0.103 0.251 0.211640 0.166 0.555 0.141 0.108 0.276 0.206650 0.174 0.563 0.144 0.112 0.308 0.202660 0.179 0.568 0.149 0.115 0.338 0.204670 0.179 0.571 0.151 0.113 0.353 0.210680 0.175 0.574 0.147 0.108 0.350 0.220690 0.171 0.577 0.140 0.104 0.343 0.230700 0.171 0.579 0.133 0.104 0.343 0.238
298
Wavelength Orange Purplishblue
Moderatered
Purple Yellowgreen
Orangeyellow
400 0.055 0.232 0.134 0.173 0.060 0.066410 0.055 0.283 0.139 0.206 0.061 0.067420 0.055 0.312 0.138 0.214 0.062 0.066430 0.055 0.338 0.137 0.197 0.064 0.066440 0.055 0.365 0.136 0.168 0.067 0.066450 0.056 0.377 0.135 0.139 0.073 0.067460 0.056 0.364 0.132 0.114 0.083 0.069470 0.057 0.332 0.127 0.094 0.100 0.073480 0.058 0.286 0.121 0.081 0.128 0.080490 0.061 0.235 0.116 0.072 0.174 0.090500 0.069 0.190 0.111 0.064 0.249 0.103510 0.094 0.157 0.105 0.060 0.356 0.126520 0.134 0.131 0.100 0.058 0.457 0.186530 0.165 0.114 0.097 0.055 0.514 0.289540 0.182 0.103 0.097 0.054 0.530 0.379550 0.204 0.096 0.097 0.054 0.521 0.432560 0.250 0.091 0.099 0.055 0.500 0.479570 0.328 0.087 0.109 0.054 0.476 0.530580 0.423 0.086 0.159 0.053 0.450 0.573590 0.502 0.085 0.271 0.054 0.411 0.602600 0.543 0.085 0.410 0.059 0.361 0.614610 0.556 0.084 0.503 0.071 0.324 0.615620 0.563 0.084 0.550 0.092 0.302 0.620630 0.572 0.084 0.568 0.119 0.289 0.638640 0.582 0.087 0.572 0.145 0.280 0.667650 0.589 0.091 0.573 0.170 0.272 0.695660 0.592 0.098 0.580 0.198 0.273 0.715670 0.588 0.107 0.596 0.232 0.282 0.713680 0.585 0.120 0.617 0.275 0.296 0.695690 0.587 0.139 0.636 0.332 0.313 0.691700 0.595 0.166 0.649 0.399 0.327 0.703
299
Wavelength Blue Green Red YellowMagenta Cyan400 0.114 0.053 0.059 0.057 0.280 0.185410 0.151 0.054 0.058 0.058 0.339 0.224420 0.185 0.055 0.057 0.058 0.351 0.245430 0.223 0.056 0.057 0.060 0.341 0.269440 0.274 0.059 0.057 0.063 0.320 0.302450 0.325 0.064 0.057 0.068 0.293 0.338460 0.332 0.074 0.056 0.078 0.263 0.378470 0.294 0.091 0.055 0.093 0.236 0.415480 0.225 0.118 0.053 0.118 0.207 0.432490 0.155 0.162 0.052 0.159 0.180 0.428500 0.104 0.226 0.051 0.225 0.160 0.403510 0.074 0.299 0.050 0.328 0.143 0.364520 0.059 0.333 0.049 0.456 0.123 0.314530 0.051 0.325 0.049 0.556 0.105 0.262540 0.047 0.307 0.049 0.611 0.100 0.214550 0.045 0.284 0.050 0.641 0.104 0.171560 0.044 0.252 0.052 0.662 0.104 0.135570 0.043 0.221 0.057 0.681 0.109 0.111580 0.043 0.195 0.072 0.696 0.138 0.097590 0.043 0.167 0.116 0.708 0.200 0.088600 0.042 0.133 0.230 0.717 0.291 0.080610 0.043 0.108 0.374 0.724 0.391 0.076620 0.043 0.093 0.487 0.731 0.491 0.074630 0.043 0.084 0.551 0.737 0.575 0.073640 0.043 0.078 0.583 0.743 0.647 0.073650 0.044 0.074 0.600 0.749 0.705 0.074660 0.044 0.071 0.618 0.753 0.748 0.076670 0.044 0.071 0.636 0.755 0.778 0.077680 0.044 0.072 0.657 0.760 0.799 0.076690 0.045 0.075 0.678 0.767 0.814 0.075700 0.046 0.078 0.695 0.771 0.826 0.073
300
Wavelength White Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5Black400 0.437 0.372 0.283 0.178 0.087 0.033410 0.684 0.509 0.340 0.196 0.089 0.034420 0.838 0.557 0.353 0.199 0.090 0.034430 0.883 0.566 0.356 0.200 0.091 0.034440 0.891 0.569 0.357 0.201 0.092 0.034450 0.894 0.570 0.358 0.201 0.092 0.034460 0.895 0.569 0.357 0.202 0.091 0.034470 0.895 0.567 0.356 0.202 0.090 0.034480 0.895 0.565 0.354 0.202 0.090 0.033490 0.894 0.564 0.353 0.201 0.089 0.033500 0.892 0.564 0.352 0.200 0.089 0.033510 0.891 0.564 0.351 0.200 0.089 0.033520 0.888 0.564 0.350 0.198 0.089 0.033530 0.885 0.564 0.349 0.197 0.089 0.033540 0.885 0.564 0.349 0.197 0.089 0.033550 0.885 0.564 0.349 0.197 0.089 0.033560 0.883 0.564 0.352 0.197 0.089 0.033570 0.883 0.565 0.356 0.197 0.089 0.032580 0.885 0.567 0.360 0.199 0.089 0.032590 0.888 0.567 0.360 0.199 0.089 0.032600 0.890 0.566 0.355 0.198 0.089 0.032610 0.889 0.564 0.349 0.196 0.088 0.032620 0.888 0.563 0.344 0.194 0.088 0.032630 0.888 0.561 0.340 0.192 0.087 0.032640 0.890 0.559 0.337 0.190 0.086 0.032650 0.892 0.557 0.335 0.188 0.086 0.032660 0.894 0.555 0.334 0.187 0.085 0.032670 0.894 0.553 0.335 0.186 0.085 0.032680 0.895 0.551 0.337 0.186 0.084 0.032690 0.896 0.549 0.338 0.185 0.084 0.032700 0.896 0.547 0.338 0.185 0.083 0.032
301
APPENDIX J : THE REFLECTANCE SPECTRA OF THEPREDICTED GRETAG MACBETH COLOR CHECKER BY THE
PROPOSED SIX-COLOR SEPARATION ALGORITHM
WavelengthDark skin Light skin Blue sky Foliage Blue flower Blue green
400 0.028 0.079 0.086 0.027 0.113 0.101410 0.041 0.138 0.171 0.039 0.230 0.199420 0.055 0.200 0.263 0.051 0.354 0.311430 0.058 0.221 0.305 0.055 0.408 0.360440 0.059 0.226 0.327 0.058 0.434 0.386450 0.060 0.228 0.333 0.061 0.436 0.406460 0.059 0.228 0.326 0.066 0.419 0.426470 0.059 0.228 0.319 0.073 0.401 0.463480 0.060 0.229 0.308 0.080 0.376 0.492490 0.062 0.240 0.294 0.092 0.350 0.506500 0.065 0.262 0.280 0.112 0.329 0.527510 0.068 0.283 0.262 0.138 0.302 0.564520 0.070 0.292 0.239 0.159 0.267 0.590530 0.073 0.295 0.217 0.167 0.239 0.579540 0.078 0.302 0.200 0.163 0.225 0.539550 0.085 0.303 0.178 0.155 0.202 0.484560 0.089 0.295 0.151 0.144 0.169 0.421570 0.096 0.293 0.132 0.134 0.148 0.366580 0.111 0.329 0.129 0.123 0.163 0.319590 0.132 0.409 0.141 0.113 0.210 0.281600 0.146 0.472 0.147 0.106 0.241 0.252610 0.154 0.506 0.148 0.102 0.253 0.237620 0.158 0.522 0.150 0.101 0.260 0.233630 0.160 0.528 0.152 0.101 0.265 0.233640 0.163 0.532 0.156 0.102 0.272 0.234650 0.166 0.536 0.165 0.103 0.286 0.239660 0.169 0.540 0.176 0.105 0.302 0.246670 0.172 0.543 0.181 0.107 0.310 0.252680 0.173 0.546 0.179 0.110 0.307 0.256690 0.174 0.547 0.174 0.112 0.298 0.259700 0.175 0.548 0.166 0.113 0.285 0.259
302
Wavelength Orange Purplishblue
Moderatered
Purple Yellowgreen
Orangeyellow
400 0.028 0.089 0.055 0.047 0.032 0.031410 0.038 0.191 0.092 0.087 0.044 0.041420 0.046 0.291 0.125 0.122 0.056 0.049430 0.048 0.342 0.136 0.138 0.060 0.051440 0.050 0.371 0.139 0.147 0.063 0.054450 0.052 0.374 0.137 0.146 0.071 0.059460 0.056 0.352 0.133 0.138 0.085 0.067470 0.059 0.328 0.127 0.126 0.103 0.075480 0.065 0.294 0.119 0.111 0.127 0.088490 0.079 0.259 0.116 0.100 0.174 0.117500 0.101 0.230 0.120 0.093 0.260 0.168510 0.124 0.197 0.118 0.081 0.381 0.230520 0.142 0.157 0.105 0.062 0.490 0.284530 0.159 0.129 0.098 0.051 0.541 0.320540 0.189 0.115 0.104 0.049 0.542 0.355550 0.227 0.094 0.103 0.043 0.520 0.394560 0.252 0.068 0.090 0.032 0.484 0.418570 0.295 0.053 0.084 0.027 0.450 0.457580 0.381 0.060 0.138 0.041 0.414 0.528590 0.477 0.082 0.288 0.076 0.376 0.599600 0.540 0.091 0.431 0.099 0.344 0.641610 0.570 0.091 0.513 0.107 0.328 0.661620 0.586 0.093 0.548 0.111 0.323 0.671630 0.593 0.095 0.562 0.114 0.321 0.675640 0.599 0.100 0.568 0.118 0.320 0.676650 0.608 0.113 0.573 0.129 0.320 0.678660 0.618 0.130 0.577 0.143 0.323 0.680670 0.623 0.137 0.582 0.150 0.328 0.684680 0.623 0.133 0.586 0.146 0.337 0.688690 0.620 0.123 0.590 0.139 0.346 0.693700 0.615 0.110 0.593 0.128 0.353 0.696
303
Wavelength Blue Green Red YellowMagenta Cyan400 0.072 0.025 0.029 0.034 0.099 0.082410 0.156 0.037 0.042 0.044 0.191 0.170420 0.235 0.050 0.051 0.052 0.280 0.264430 0.277 0.056 0.054 0.053 0.314 0.313440 0.302 0.060 0.055 0.056 0.321 0.347450 0.302 0.067 0.057 0.065 0.306 0.367460 0.280 0.080 0.058 0.080 0.278 0.371470 0.254 0.099 0.056 0.093 0.250 0.378480 0.220 0.121 0.053 0.114 0.216 0.376490 0.185 0.146 0.054 0.167 0.184 0.366500 0.159 0.190 0.058 0.266 0.162 0.355510 0.129 0.258 0.055 0.395 0.138 0.340520 0.095 0.321 0.043 0.504 0.109 0.314530 0.074 0.345 0.036 0.564 0.095 0.277540 0.066 0.332 0.040 0.596 0.099 0.238550 0.052 0.301 0.039 0.619 0.095 0.192560 0.034 0.261 0.030 0.629 0.080 0.144570 0.026 0.222 0.027 0.648 0.073 0.108580 0.030 0.182 0.064 0.681 0.125 0.089590 0.042 0.144 0.196 0.713 0.268 0.081600 0.043 0.115 0.341 0.731 0.402 0.074610 0.039 0.101 0.427 0.741 0.477 0.069620 0.039 0.096 0.465 0.746 0.510 0.069630 0.040 0.095 0.481 0.748 0.525 0.070640 0.043 0.094 0.490 0.749 0.533 0.073650 0.053 0.094 0.499 0.751 0.545 0.081660 0.066 0.096 0.508 0.753 0.556 0.091670 0.072 0.100 0.514 0.756 0.563 0.095680 0.068 0.106 0.516 0.760 0.563 0.093690 0.060 0.113 0.515 0.764 0.559 0.088700 0.049 0.118 0.512 0.766 0.553 0.081
304
Wavelength White Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5Black400 0.235 0.162 0.108 0.066 0.035 0.017410 0.471 0.316 0.203 0.118 0.057 0.024420 0.756 0.501 0.315 0.178 0.080 0.030430 0.858 0.567 0.355 0.199 0.089 0.032440 0.879 0.582 0.365 0.205 0.092 0.033450 0.873 0.580 0.364 0.206 0.093 0.034460 0.860 0.572 0.360 0.204 0.092 0.034470 0.858 0.572 0.360 0.204 0.093 0.034480 0.855 0.571 0.360 0.204 0.093 0.035490 0.853 0.570 0.359 0.204 0.093 0.035500 0.851 0.569 0.359 0.204 0.093 0.036510 0.850 0.568 0.358 0.203 0.092 0.037520 0.850 0.568 0.357 0.202 0.092 0.037530 0.849 0.566 0.355 0.201 0.091 0.035540 0.848 0.565 0.353 0.199 0.090 0.034550 0.850 0.564 0.352 0.198 0.089 0.032560 0.848 0.561 0.349 0.195 0.087 0.030570 0.853 0.562 0.348 0.194 0.086 0.028580 0.854 0.560 0.347 0.193 0.086 0.028590 0.856 0.561 0.347 0.193 0.087 0.030600 0.857 0.561 0.347 0.193 0.087 0.032610 0.859 0.562 0.348 0.194 0.087 0.032620 0.862 0.564 0.349 0.194 0.088 0.033630 0.863 0.565 0.350 0.196 0.089 0.033640 0.863 0.567 0.353 0.198 0.091 0.035650 0.866 0.570 0.357 0.202 0.093 0.036660 0.868 0.574 0.360 0.205 0.096 0.039670 0.869 0.576 0.363 0.208 0.098 0.040680 0.871 0.578 0.365 0.209 0.099 0.040690 0.873 0.579 0.365 0.209 0.099 0.040700 0.873 0.579 0.365 0.209 0.099 0.039
305
APPENDIX K : THE REFLECTANCE SPECTRA OF THEREPRODUCED GRETAG MACBETH COLOR CHECKER USING
DUPONT WATERPROOF® SYSTEM
WavelengthDark skin Light skin Blue sky Foliage Blue flower Blue green
400 0.029 0.084 0.094 0.028 0.121 0.114410 0.040 0.136 0.177 0.038 0.234 0.210420 0.050 0.186 0.261 0.047 0.347 0.311430 0.052 0.200 0.297 0.049 0.396 0.354440 0.053 0.204 0.316 0.051 0.418 0.379450 0.053 0.206 0.321 0.055 0.418 0.398460 0.053 0.208 0.315 0.059 0.401 0.420470 0.053 0.209 0.308 0.066 0.382 0.458480 0.053 0.210 0.298 0.073 0.358 0.488490 0.055 0.224 0.285 0.085 0.333 0.503500 0.058 0.250 0.273 0.105 0.313 0.527510 0.060 0.273 0.257 0.128 0.288 0.566520 0.062 0.279 0.236 0.146 0.254 0.592530 0.065 0.281 0.216 0.152 0.229 0.578540 0.071 0.287 0.199 0.149 0.215 0.532550 0.078 0.288 0.176 0.143 0.192 0.473560 0.083 0.279 0.147 0.133 0.159 0.407570 0.090 0.275 0.126 0.124 0.136 0.352580 0.105 0.313 0.122 0.114 0.150 0.306590 0.123 0.397 0.131 0.105 0.194 0.269600 0.135 0.460 0.135 0.098 0.219 0.240610 0.141 0.490 0.134 0.094 0.227 0.224620 0.144 0.503 0.136 0.092 0.232 0.219630 0.146 0.509 0.138 0.092 0.237 0.217640 0.148 0.512 0.142 0.093 0.243 0.218650 0.151 0.516 0.151 0.094 0.256 0.223660 0.155 0.519 0.161 0.096 0.272 0.230670 0.157 0.522 0.166 0.099 0.280 0.238680 0.159 0.525 0.164 0.102 0.276 0.243690 0.159 0.526 0.159 0.104 0.267 0.246700 0.160 0.527 0.151 0.106 0.255 0.246
306
Wavelength Orange Purplishblue
Moderatered
Purple Yellowgreen
Orangeyellow
400 0.029 0.095 0.057 0.051 0.032 0.030410 0.036 0.191 0.090 0.091 0.041 0.036420 0.041 0.284 0.117 0.123 0.048 0.040430 0.041 0.329 0.125 0.137 0.050 0.041440 0.041 0.355 0.126 0.145 0.053 0.042450 0.043 0.355 0.125 0.143 0.060 0.046460 0.046 0.333 0.121 0.136 0.074 0.053470 0.049 0.309 0.116 0.124 0.090 0.060480 0.054 0.276 0.107 0.110 0.112 0.070490 0.066 0.243 0.104 0.099 0.160 0.096500 0.086 0.217 0.108 0.093 0.251 0.141510 0.105 0.186 0.105 0.081 0.378 0.193520 0.119 0.150 0.091 0.063 0.490 0.234530 0.135 0.125 0.084 0.052 0.541 0.265540 0.167 0.113 0.090 0.051 0.541 0.303550 0.207 0.093 0.089 0.044 0.517 0.347560 0.235 0.065 0.077 0.032 0.479 0.376570 0.281 0.049 0.071 0.026 0.442 0.423580 0.371 0.055 0.125 0.040 0.404 0.508590 0.465 0.075 0.281 0.072 0.363 0.586600 0.521 0.082 0.427 0.091 0.327 0.626610 0.546 0.080 0.506 0.096 0.306 0.641620 0.559 0.081 0.539 0.098 0.298 0.648630 0.565 0.083 0.551 0.101 0.294 0.650640 0.571 0.088 0.557 0.106 0.292 0.650650 0.580 0.101 0.561 0.116 0.292 0.652660 0.589 0.116 0.565 0.130 0.296 0.654670 0.595 0.124 0.570 0.136 0.304 0.659680 0.594 0.119 0.575 0.133 0.315 0.666690 0.590 0.109 0.579 0.125 0.327 0.672700 0.584 0.097 0.582 0.114 0.335 0.676
307
Wavelength Blue Green Red YellowMagenta Cyan400 0.076 0.026 0.030 0.035 0.105 0.088410 0.156 0.036 0.041 0.041 0.192 0.174420 0.230 0.045 0.048 0.045 0.273 0.260430 0.268 0.049 0.049 0.044 0.303 0.303440 0.291 0.053 0.050 0.046 0.308 0.334450 0.288 0.060 0.051 0.053 0.293 0.352460 0.266 0.072 0.053 0.067 0.266 0.356470 0.240 0.091 0.051 0.080 0.238 0.363480 0.207 0.112 0.048 0.098 0.205 0.362490 0.175 0.138 0.049 0.150 0.174 0.354500 0.151 0.185 0.054 0.249 0.154 0.345510 0.123 0.254 0.051 0.372 0.130 0.332520 0.092 0.317 0.039 0.471 0.103 0.307530 0.073 0.340 0.032 0.526 0.091 0.271540 0.066 0.327 0.036 0.560 0.095 0.231550 0.052 0.296 0.035 0.587 0.092 0.184560 0.034 0.256 0.026 0.601 0.076 0.133570 0.024 0.216 0.023 0.626 0.069 0.097580 0.029 0.176 0.062 0.666 0.120 0.078590 0.041 0.137 0.199 0.701 0.267 0.070600 0.041 0.107 0.341 0.717 0.398 0.062610 0.037 0.091 0.421 0.724 0.465 0.057620 0.036 0.085 0.455 0.727 0.494 0.056630 0.037 0.083 0.469 0.728 0.507 0.058640 0.040 0.082 0.477 0.728 0.515 0.061650 0.050 0.082 0.486 0.730 0.526 0.068660 0.063 0.084 0.495 0.732 0.537 0.078670 0.068 0.089 0.500 0.736 0.544 0.082680 0.065 0.096 0.501 0.740 0.543 0.080690 0.056 0.104 0.500 0.744 0.539 0.075700 0.046 0.110 0.497 0.747 0.533 0.068
308
Wavelength White Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5Black400 0.247 0.173 0.112 0.067 0.036 0.017410 0.481 0.323 0.198 0.112 0.054 0.022420 0.759 0.489 0.289 0.158 0.073 0.026430 0.857 0.547 0.320 0.174 0.079 0.028440 0.879 0.560 0.328 0.178 0.081 0.029450 0.874 0.558 0.328 0.179 0.082 0.030460 0.862 0.551 0.324 0.177 0.082 0.030470 0.860 0.550 0.324 0.177 0.082 0.030480 0.858 0.549 0.324 0.178 0.082 0.031490 0.856 0.548 0.324 0.178 0.083 0.031500 0.854 0.547 0.323 0.178 0.082 0.033510 0.853 0.546 0.322 0.177 0.082 0.033520 0.854 0.546 0.322 0.177 0.082 0.033530 0.853 0.544 0.320 0.175 0.081 0.032540 0.852 0.542 0.319 0.174 0.080 0.031550 0.854 0.541 0.318 0.174 0.080 0.029560 0.852 0.537 0.315 0.171 0.078 0.027570 0.858 0.538 0.314 0.170 0.077 0.025580 0.858 0.537 0.313 0.169 0.077 0.025590 0.860 0.537 0.314 0.170 0.078 0.027600 0.861 0.537 0.314 0.170 0.078 0.028610 0.863 0.538 0.314 0.170 0.078 0.029620 0.866 0.540 0.315 0.171 0.079 0.029630 0.867 0.541 0.317 0.172 0.080 0.030640 0.867 0.543 0.319 0.174 0.082 0.031650 0.869 0.546 0.323 0.178 0.084 0.033660 0.870 0.550 0.327 0.181 0.087 0.035670 0.872 0.552 0.329 0.184 0.089 0.036680 0.874 0.554 0.331 0.185 0.090 0.036690 0.875 0.554 0.331 0.185 0.090 0.036700 0.876 0.554 0.331 0.185 0.090 0.035
309
APPENDIX L : THE REFLECTANCE SPECTRA OF THEPREDICTED GRETAG MACBETH COLOR CHECKER USING
FUJIX PICTROGRAPH 3000
WavelengthDark skin Light skin Blue sky Foliage Blue flower Blue green
400 0.037 0.093 0.110 0.036 0.136 0.127410 0.053 0.160 0.207 0.052 0.265 0.238420 0.066 0.230 0.318 0.064 0.418 0.374430 0.063 0.237 0.341 0.063 0.451 0.410440 0.059 0.232 0.338 0.061 0.447 0.417450 0.054 0.218 0.321 0.058 0.421 0.407460 0.052 0.213 0.311 0.058 0.403 0.409470 0.055 0.219 0.306 0.065 0.389 0.425480 0.062 0.233 0.299 0.078 0.368 0.450490 0.073 0.256 0.290 0.101 0.343 0.489500 0.084 0.281 0.275 0.131 0.313 0.531510 0.089 0.296 0.252 0.156 0.278 0.562520 0.085 0.296 0.224 0.165 0.244 0.571530 0.078 0.287 0.199 0.160 0.216 0.559540 0.072 0.279 0.179 0.150 0.198 0.533550 0.070 0.280 0.167 0.141 0.189 0.499560 0.073 0.289 0.161 0.134 0.189 0.457570 0.086 0.318 0.163 0.134 0.200 0.415580 0.109 0.364 0.168 0.136 0.221 0.366590 0.138 0.421 0.171 0.135 0.243 0.315600 0.159 0.470 0.161 0.123 0.250 0.261610 0.162 0.497 0.142 0.104 0.241 0.215620 0.156 0.505 0.124 0.088 0.225 0.184630 0.150 0.507 0.114 0.080 0.215 0.167640 0.154 0.515 0.114 0.080 0.218 0.167650 0.163 0.529 0.121 0.086 0.228 0.174660 0.173 0.541 0.128 0.092 0.238 0.182670 0.195 0.563 0.147 0.108 0.262 0.204680 0.245 0.605 0.194 0.149 0.316 0.256690 0.327 0.661 0.274 0.222 0.400 0.339700 0.428 0.717 0.374 0.322 0.495 0.438
310
Wavelength Orange Purplishblue
Moderatered
Purple Yellowgreen
Orangeyellow
400 0.035 0.117 0.070 0.066 0.043 0.039410 0.048 0.228 0.115 0.114 0.061 0.053420 0.056 0.360 0.156 0.159 0.077 0.063430 0.053 0.388 0.153 0.160 0.076 0.060440 0.050 0.382 0.144 0.151 0.075 0.058450 0.046 0.358 0.130 0.137 0.072 0.054460 0.046 0.338 0.122 0.127 0.075 0.055470 0.052 0.318 0.120 0.120 0.089 0.064480 0.065 0.287 0.120 0.111 0.118 0.085490 0.090 0.249 0.122 0.101 0.174 0.124500 0.127 0.208 0.121 0.087 0.265 0.186510 0.164 0.167 0.113 0.071 0.375 0.259520 0.188 0.134 0.101 0.056 0.467 0.318530 0.196 0.109 0.090 0.044 0.516 0.349540 0.199 0.094 0.084 0.038 0.527 0.363550 0.206 0.086 0.086 0.036 0.517 0.375560 0.224 0.083 0.098 0.038 0.495 0.393570 0.263 0.087 0.127 0.047 0.471 0.430580 0.328 0.096 0.185 0.063 0.440 0.486590 0.420 0.103 0.279 0.086 0.406 0.557600 0.516 0.100 0.390 0.102 0.365 0.625610 0.589 0.087 0.481 0.104 0.326 0.677620 0.633 0.074 0.535 0.098 0.297 0.708630 0.654 0.067 0.560 0.093 0.281 0.724640 0.670 0.068 0.580 0.096 0.282 0.736650 0.688 0.073 0.600 0.103 0.292 0.751660 0.702 0.079 0.616 0.111 0.302 0.762670 0.718 0.093 0.636 0.128 0.327 0.775680 0.741 0.132 0.670 0.172 0.381 0.792690 0.772 0.202 0.714 0.249 0.461 0.813700 0.805 0.300 0.759 0.349 0.551 0.836
311
Wavelength Blue Green Red YellowMagenta Cyan400 0.101 0.039 0.041 0.043 0.117 0.108410 0.197 0.056 0.060 0.059 0.218 0.207420 0.306 0.071 0.074 0.072 0.329 0.327430 0.326 0.071 0.068 0.070 0.338 0.361440 0.318 0.070 0.062 0.068 0.322 0.367450 0.293 0.068 0.054 0.065 0.291 0.358460 0.271 0.071 0.050 0.067 0.267 0.355470 0.246 0.082 0.049 0.080 0.246 0.357480 0.210 0.106 0.051 0.109 0.221 0.356490 0.169 0.148 0.052 0.165 0.192 0.353500 0.128 0.210 0.053 0.262 0.162 0.343510 0.093 0.275 0.049 0.386 0.132 0.320520 0.067 0.319 0.042 0.499 0.106 0.289530 0.051 0.329 0.036 0.570 0.089 0.256540 0.041 0.315 0.033 0.605 0.081 0.225550 0.037 0.292 0.035 0.623 0.082 0.197560 0.036 0.262 0.041 0.632 0.092 0.171570 0.039 0.233 0.060 0.649 0.120 0.149580 0.046 0.202 0.102 0.669 0.174 0.126590 0.052 0.167 0.181 0.695 0.261 0.102600 0.052 0.130 0.290 0.719 0.363 0.075610 0.044 0.098 0.389 0.740 0.444 0.052620 0.036 0.078 0.451 0.754 0.489 0.039630 0.032 0.068 0.480 0.761 0.509 0.033640 0.032 0.068 0.504 0.769 0.526 0.032650 0.035 0.072 0.527 0.780 0.545 0.035660 0.039 0.078 0.546 0.789 0.561 0.038670 0.048 0.092 0.568 0.800 0.582 0.047680 0.076 0.130 0.607 0.814 0.622 0.074690 0.131 0.200 0.662 0.831 0.675 0.130700 0.218 0.298 0.717 0.850 0.727 0.216
312
Wavelength White Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5Black400 0.213 0.169 0.122 0.082 0.048 0.024410 0.437 0.332 0.227 0.142 0.074 0.033420 0.734 0.537 0.348 0.205 0.098 0.039430 0.819 0.588 0.372 0.212 0.097 0.036440 0.833 0.592 0.370 0.208 0.093 0.034450 0.805 0.569 0.353 0.196 0.086 0.031460 0.791 0.558 0.345 0.191 0.084 0.030470 0.789 0.560 0.349 0.195 0.087 0.031480 0.785 0.563 0.357 0.204 0.093 0.035490 0.790 0.575 0.371 0.217 0.102 0.040500 0.798 0.587 0.383 0.228 0.110 0.044510 0.804 0.590 0.385 0.228 0.110 0.044520 0.805 0.584 0.373 0.217 0.101 0.039530 0.801 0.570 0.356 0.200 0.090 0.033540 0.795 0.557 0.340 0.186 0.080 0.028550 0.791 0.550 0.331 0.178 0.075 0.025560 0.784 0.545 0.328 0.177 0.075 0.025570 0.783 0.551 0.337 0.185 0.080 0.028580 0.779 0.561 0.353 0.200 0.092 0.034590 0.780 0.572 0.369 0.215 0.103 0.040600 0.784 0.576 0.371 0.216 0.103 0.040610 0.789 0.570 0.358 0.203 0.093 0.034620 0.792 0.561 0.342 0.186 0.081 0.028630 0.793 0.554 0.331 0.176 0.074 0.024640 0.796 0.557 0.334 0.178 0.076 0.025650 0.802 0.568 0.345 0.187 0.081 0.028660 0.809 0.578 0.356 0.196 0.087 0.031670 0.819 0.599 0.381 0.219 0.103 0.039680 0.834 0.639 0.435 0.271 0.143 0.062690 0.848 0.689 0.511 0.354 0.215 0.113700 0.861 0.739 0.593 0.453 0.314 0.195
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APPENDIX M : THE REFLECTANCE SPECTRA OF THEPREDICTED GRETAG MACBETH COLOR CHECKER USING
KODAK PROFESSIONAL 8670 PS THERMAL PRINTER
WavelengthDark skin Light skin Blue sky Foliage Blue flower Blue green
400 0.174 0.340 0.254 0.150 0.321 0.299410 0.180 0.399 0.348 0.163 0.439 0.410420 0.139 0.363 0.382 0.132 0.483 0.450430 0.092 0.295 0.373 0.091 0.478 0.443440 0.058 0.231 0.343 0.060 0.448 0.414450 0.040 0.189 0.311 0.043 0.413 0.388460 0.032 0.170 0.286 0.036 0.383 0.374470 0.036 0.180 0.279 0.043 0.364 0.393480 0.057 0.230 0.296 0.074 0.362 0.460490 0.085 0.286 0.304 0.125 0.349 0.534500 0.098 0.311 0.284 0.161 0.314 0.576510 0.093 0.307 0.251 0.170 0.273 0.583520 0.082 0.293 0.219 0.162 0.237 0.570530 0.074 0.280 0.193 0.151 0.212 0.543540 0.068 0.272 0.175 0.140 0.195 0.511550 0.067 0.274 0.164 0.133 0.188 0.481560 0.072 0.286 0.160 0.130 0.190 0.449570 0.087 0.319 0.167 0.134 0.206 0.417580 0.113 0.369 0.177 0.141 0.231 0.378590 0.141 0.424 0.180 0.143 0.252 0.335600 0.163 0.471 0.172 0.134 0.260 0.285610 0.174 0.505 0.155 0.118 0.254 0.237620 0.171 0.520 0.133 0.099 0.238 0.196630 0.159 0.519 0.112 0.081 0.216 0.162640 0.146 0.512 0.095 0.067 0.196 0.138650 0.134 0.505 0.083 0.057 0.181 0.121660 0.125 0.498 0.075 0.051 0.170 0.110670 0.121 0.494 0.070 0.047 0.164 0.104680 0.121 0.495 0.070 0.047 0.164 0.104690 0.125 0.500 0.074 0.050 0.169 0.108700 0.134 0.509 0.081 0.056 0.180 0.117
314
Wavelength Orange Purplishblue
Moderatered
Purple Yellowgreen
Orangeyellow
400 0.237 0.226 0.302 0.195 0.219 0.255410 0.215 0.327 0.331 0.242 0.222 0.234420 0.141 0.383 0.275 0.234 0.167 0.156430 0.079 0.401 0.202 0.198 0.108 0.089440 0.044 0.388 0.143 0.156 0.069 0.051450 0.028 0.361 0.107 0.124 0.049 0.033460 0.023 0.329 0.089 0.104 0.043 0.028470 0.029 0.299 0.090 0.097 0.056 0.036480 0.058 0.279 0.113 0.104 0.112 0.076490 0.112 0.248 0.133 0.104 0.223 0.156500 0.162 0.204 0.131 0.089 0.345 0.242510 0.185 0.161 0.115 0.069 0.432 0.298520 0.190 0.129 0.099 0.053 0.478 0.326530 0.191 0.108 0.089 0.043 0.495 0.340540 0.194 0.094 0.086 0.038 0.496 0.350550 0.202 0.086 0.088 0.036 0.490 0.364560 0.220 0.085 0.100 0.039 0.476 0.384570 0.261 0.092 0.131 0.049 0.463 0.423580 0.328 0.103 0.190 0.067 0.446 0.483590 0.412 0.109 0.272 0.089 0.424 0.553600 0.500 0.106 0.368 0.105 0.393 0.619610 0.581 0.095 0.463 0.112 0.357 0.678620 0.641 0.080 0.538 0.108 0.320 0.720630 0.678 0.065 0.583 0.098 0.287 0.744640 0.697 0.053 0.608 0.087 0.260 0.757650 0.710 0.045 0.623 0.077 0.240 0.766660 0.716 0.040 0.630 0.071 0.226 0.769670 0.718 0.037 0.634 0.068 0.219 0.771680 0.720 0.037 0.637 0.067 0.218 0.772690 0.720 0.039 0.640 0.071 0.223 0.771700 0.720 0.044 0.642 0.078 0.235 0.770
315
Wavelength Blue Green Red YellowMagenta Cyan400 0.175 0.151 0.229 0.271 0.362 0.186410 0.261 0.168 0.222 0.253 0.449 0.280420 0.314 0.140 0.159 0.172 0.439 0.342430 0.335 0.100 0.098 0.101 0.386 0.371440 0.326 0.068 0.059 0.059 0.324 0.371450 0.299 0.051 0.038 0.040 0.272 0.358460 0.265 0.044 0.030 0.035 0.236 0.341470 0.229 0.055 0.031 0.046 0.217 0.337480 0.201 0.103 0.044 0.099 0.210 0.360490 0.164 0.188 0.058 0.212 0.193 0.374500 0.123 0.268 0.060 0.346 0.161 0.358510 0.088 0.311 0.051 0.451 0.128 0.325520 0.065 0.322 0.042 0.517 0.104 0.286530 0.051 0.312 0.037 0.556 0.090 0.249540 0.042 0.293 0.035 0.580 0.084 0.216550 0.038 0.273 0.036 0.599 0.085 0.190560 0.038 0.252 0.043 0.613 0.096 0.169570 0.042 0.232 0.064 0.635 0.126 0.152580 0.050 0.209 0.106 0.664 0.181 0.133590 0.056 0.181 0.175 0.698 0.259 0.110600 0.055 0.147 0.266 0.728 0.346 0.083610 0.048 0.115 0.366 0.754 0.430 0.059620 0.039 0.088 0.452 0.770 0.493 0.041630 0.030 0.067 0.508 0.777 0.528 0.029640 0.023 0.053 0.540 0.779 0.544 0.021650 0.019 0.044 0.558 0.782 0.552 0.016660 0.016 0.038 0.567 0.782 0.555 0.013670 0.015 0.035 0.572 0.781 0.557 0.012680 0.015 0.035 0.576 0.781 0.560 0.012690 0.016 0.037 0.579 0.780 0.565 0.013700 0.018 0.042 0.581 0.781 0.570 0.015
316
Wavelength White Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5Black400 0.559 0.443 0.345 0.253 0.164 0.094410 0.770 0.591 0.445 0.313 0.192 0.102420 0.859 0.632 0.456 0.304 0.171 0.082430 0.888 0.618 0.420 0.260 0.131 0.055440 0.886 0.581 0.370 0.211 0.095 0.034450 0.879 0.547 0.330 0.176 0.072 0.023460 0.874 0.528 0.308 0.157 0.061 0.018470 0.876 0.534 0.314 0.163 0.064 0.019480 0.881 0.570 0.357 0.201 0.089 0.031490 0.888 0.603 0.398 0.239 0.117 0.047500 0.891 0.610 0.407 0.248 0.124 0.052510 0.890 0.598 0.391 0.233 0.113 0.045520 0.888 0.580 0.368 0.212 0.098 0.037530 0.881 0.561 0.347 0.193 0.085 0.030540 0.874 0.547 0.331 0.180 0.077 0.026550 0.874 0.540 0.323 0.173 0.072 0.024560 0.870 0.540 0.324 0.174 0.073 0.025570 0.872 0.553 0.340 0.187 0.082 0.029580 0.873 0.571 0.362 0.208 0.096 0.036590 0.878 0.586 0.380 0.224 0.108 0.042600 0.881 0.591 0.385 0.227 0.110 0.043610 0.885 0.587 0.376 0.218 0.103 0.039620 0.886 0.572 0.357 0.200 0.091 0.032630 0.885 0.553 0.333 0.179 0.076 0.025640 0.885 0.535 0.311 0.159 0.064 0.019650 0.888 0.522 0.294 0.145 0.055 0.016660 0.890 0.511 0.281 0.135 0.049 0.013670 0.892 0.506 0.274 0.129 0.046 0.012680 0.893 0.506 0.274 0.129 0.046 0.012690 0.892 0.511 0.280 0.133 0.049 0.013700 0.891 0.521 0.292 0.143 0.054 0.015
317
APPENDIX N : THE REFLECTANCE SPECTRA OF THEPREDICTED GRETAG MACBETH COLOR CHECKER USING
DUPONT WATERPROOF® WITH CMYK PRIMARIES
WavelengthDark skin Light skin Blue sky Foliage Blue flower Blue green
400 0.025 0.076 0.085 0.025 0.112 0.102410 0.036 0.133 0.171 0.035 0.230 0.199420 0.043 0.191 0.257 0.041 0.353 0.301430 0.045 0.211 0.298 0.044 0.407 0.348440 0.049 0.220 0.323 0.048 0.433 0.379450 0.054 0.226 0.333 0.056 0.436 0.405460 0.061 0.230 0.328 0.067 0.419 0.424470 0.064 0.231 0.320 0.074 0.401 0.439480 0.067 0.233 0.306 0.084 0.376 0.456490 0.079 0.249 0.297 0.110 0.351 0.497500 0.100 0.282 0.295 0.156 0.331 0.558510 0.110 0.309 0.284 0.193 0.305 0.610520 0.101 0.313 0.253 0.197 0.268 0.620530 0.090 0.308 0.223 0.182 0.240 0.587540 0.086 0.309 0.202 0.165 0.224 0.532550 0.076 0.299 0.172 0.139 0.201 0.466560 0.059 0.275 0.134 0.107 0.166 0.392570 0.051 0.263 0.110 0.086 0.145 0.333580 0.070 0.302 0.114 0.088 0.161 0.295590 0.120 0.400 0.141 0.109 0.210 0.274600 0.157 0.478 0.154 0.119 0.243 0.258610 0.172 0.518 0.157 0.120 0.256 0.248620 0.179 0.536 0.160 0.121 0.263 0.247630 0.184 0.545 0.163 0.124 0.269 0.250640 0.190 0.551 0.170 0.129 0.276 0.256650 0.203 0.561 0.184 0.142 0.290 0.272660 0.220 0.571 0.202 0.158 0.308 0.291670 0.228 0.577 0.210 0.165 0.316 0.300680 0.224 0.577 0.206 0.161 0.313 0.296690 0.215 0.573 0.195 0.152 0.303 0.285700 0.202 0.567 0.181 0.139 0.289 0.269
318
Wavelength Orange Purplishblue
Moderatered
Purple Yellowgreen
Orangeyellow
400 0.017 0.088 0.054 0.046 0.031 0.030410 0.019 0.188 0.091 0.087 0.041 0.037420 0.019 0.286 0.123 0.121 0.047 0.042430 0.018 0.335 0.133 0.137 0.049 0.042440 0.019 0.363 0.137 0.146 0.053 0.045450 0.023 0.364 0.137 0.146 0.065 0.052460 0.027 0.342 0.134 0.138 0.085 0.065470 0.029 0.317 0.128 0.127 0.101 0.075480 0.031 0.283 0.120 0.112 0.126 0.089490 0.037 0.248 0.118 0.101 0.191 0.129500 0.045 0.220 0.125 0.096 0.313 0.201510 0.044 0.187 0.123 0.083 0.456 0.285520 0.031 0.149 0.108 0.063 0.546 0.341530 0.025 0.122 0.099 0.050 0.562 0.366540 0.028 0.110 0.104 0.049 0.537 0.383550 0.027 0.091 0.101 0.042 0.493 0.386560 0.020 0.065 0.087 0.030 0.436 0.373570 0.017 0.051 0.081 0.025 0.391 0.369580 0.058 0.060 0.134 0.040 0.368 0.423590 0.240 0.084 0.285 0.077 0.365 0.551600 0.464 0.096 0.430 0.100 0.360 0.655610 0.607 0.096 0.513 0.107 0.355 0.713620 0.672 0.098 0.550 0.111 0.356 0.738630 0.696 0.101 0.565 0.114 0.359 0.747640 0.708 0.106 0.574 0.119 0.365 0.751650 0.716 0.119 0.584 0.131 0.379 0.756660 0.721 0.136 0.593 0.147 0.397 0.759670 0.726 0.143 0.599 0.155 0.406 0.763680 0.731 0.139 0.600 0.151 0.402 0.766690 0.734 0.129 0.598 0.141 0.393 0.768700 0.736 0.116 0.595 0.128 0.380 0.769
319
Wavelength Blue Green Red YellowMagenta Cyan400 0.068 0.027 0.029 0.034 0.099 0.083410 0.144 0.038 0.042 0.042 0.190 0.173420 0.215 0.043 0.051 0.048 0.280 0.264430 0.252 0.046 0.053 0.048 0.313 0.312440 0.272 0.051 0.055 0.052 0.321 0.348450 0.268 0.062 0.057 0.062 0.306 0.369460 0.244 0.079 0.058 0.079 0.279 0.374470 0.217 0.093 0.057 0.093 0.251 0.372480 0.183 0.113 0.054 0.115 0.218 0.366490 0.151 0.165 0.056 0.173 0.186 0.365500 0.127 0.258 0.061 0.285 0.165 0.370510 0.100 0.351 0.058 0.425 0.140 0.362520 0.071 0.387 0.045 0.535 0.111 0.327530 0.055 0.367 0.038 0.588 0.098 0.282540 0.050 0.325 0.042 0.609 0.101 0.239550 0.041 0.269 0.040 0.613 0.098 0.188560 0.027 0.206 0.031 0.603 0.082 0.134570 0.022 0.160 0.028 0.601 0.076 0.097580 0.030 0.140 0.066 0.629 0.127 0.082590 0.047 0.137 0.197 0.692 0.267 0.081600 0.052 0.131 0.338 0.741 0.397 0.076610 0.049 0.126 0.421 0.767 0.470 0.070620 0.049 0.126 0.458 0.779 0.502 0.070630 0.050 0.128 0.474 0.784 0.517 0.072640 0.054 0.134 0.483 0.786 0.525 0.077650 0.064 0.148 0.493 0.790 0.536 0.089660 0.078 0.166 0.503 0.794 0.547 0.104670 0.085 0.174 0.509 0.797 0.554 0.112680 0.081 0.170 0.510 0.800 0.554 0.107690 0.072 0.159 0.508 0.801 0.550 0.098700 0.061 0.145 0.504 0.801 0.545 0.085
320
Wavelength White Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5Black400 0.235 0.157 0.102 0.061 0.032 0.015410 0.471 0.309 0.195 0.112 0.053 0.021420 0.756 0.483 0.292 0.159 0.068 0.023430 0.858 0.549 0.332 0.179 0.075 0.024440 0.879 0.573 0.352 0.193 0.082 0.027450 0.873 0.580 0.362 0.203 0.089 0.031460 0.860 0.577 0.365 0.208 0.095 0.036470 0.858 0.576 0.365 0.209 0.096 0.038480 0.855 0.573 0.363 0.208 0.097 0.039490 0.853 0.580 0.374 0.220 0.107 0.046500 0.851 0.596 0.398 0.245 0.126 0.056510 0.850 0.608 0.414 0.259 0.133 0.056520 0.850 0.605 0.406 0.247 0.119 0.043530 0.849 0.591 0.386 0.226 0.102 0.033540 0.848 0.578 0.369 0.211 0.094 0.032550 0.850 0.558 0.341 0.186 0.079 0.026560 0.848 0.526 0.301 0.151 0.058 0.018570 0.853 0.505 0.274 0.129 0.047 0.016580 0.854 0.513 0.286 0.141 0.057 0.023590 0.856 0.549 0.332 0.183 0.085 0.037600 0.857 0.573 0.364 0.209 0.099 0.039610 0.859 0.585 0.377 0.218 0.102 0.035620 0.862 0.592 0.385 0.224 0.104 0.034630 0.863 0.596 0.390 0.229 0.106 0.035640 0.863 0.601 0.397 0.236 0.111 0.037650 0.866 0.611 0.411 0.250 0.124 0.046660 0.868 0.622 0.427 0.268 0.141 0.058670 0.869 0.628 0.435 0.277 0.148 0.064680 0.871 0.627 0.432 0.273 0.144 0.060690 0.873 0.622 0.424 0.263 0.134 0.052700 0.873 0.615 0.412 0.249 0.121 0.042
321
APPENDIX O : THE ACCURACY OF THE GRETAGSPECTROLINO SPECTROPHOTOMETER
The accuracy of spectral reproduction is sensitive to the accuracy of measuringinstruments. Since a Gretag Spectrolino with an automatic station was used to measurethe 6,250 printed samples, an evaluation of the accuracy of this spectrophotometer isrequired. It is hypothesized that a Gretag SPM 60 is highly accurate, hence, designated asa standard spectrophotometer. Accuracy of the Gretag Spectrolino was evaluated by thecolorimetric and spectral accuracy of the reflectance spectra of the Gretag Macbeth ColorChecker measured by the two instruments. The colorimectric and spectral accuracy isshown below.
∆E*94 Metamerism Index
Mean 0.44 0.04Stdev 0.17 0.03Max 0.72 0.14Min 0.12 0.00RMS 0.004
The following figure is the histogram of the colorimetric error.
0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .80
1
2
3
4
5
6
D e lta E 9 4
Fre
qu
en
cy
According to this accuracy comparison, the agreement between the two instruments werevery high revealed by the very low ∆E*
94, metamerism index, and the RMS error. Hence,
322
this ensures the spectral reproduction quality and prediction accuracy of the proposed six-color separation mechanism such that the resultant spectral reflectance spectra of theactual reproduction is consistent with the theoretical prediction.
Spectral reflectance factors of the Gretag Macbeth Color Checker measured by theGretag SPM 60 spectrophotometer
WavelengthDark skin Light skin Blue sky Foliage Blue flower Blue green
400 0.060 0.206 0.260 0.053 0.301 0.254410 0.061 0.231 0.319 0.054 0.400 0.314420 0.060 0.230 0.325 0.055 0.425 0.331430 0.060 0.228 0.332 0.057 0.434 0.346440 0.060 0.228 0.343 0.060 0.441 0.362450 0.060 0.229 0.349 0.062 0.444 0.386460 0.060 0.230 0.347 0.063 0.441 0.421470 0.060 0.233 0.335 0.064 0.424 0.469480 0.061 0.244 0.322 0.065 0.401 0.514490 0.061 0.268 0.309 0.068 0.363 0.553500 0.063 0.295 0.292 0.074 0.321 0.579510 0.068 0.306 0.266 0.099 0.289 0.590520 0.074 0.302 0.239 0.148 0.259 0.584530 0.077 0.297 0.221 0.184 0.230 0.564540 0.078 0.296 0.207 0.188 0.217 0.537550 0.080 0.291 0.183 0.170 0.214 0.498560 0.086 0.289 0.158 0.144 0.205 0.453570 0.100 0.297 0.144 0.125 0.198 0.406580 0.119 0.363 0.156 0.115 0.199 0.358590 0.138 0.455 0.166 0.110 0.208 0.307600 0.148 0.495 0.159 0.105 0.225 0.259610 0.151 0.503 0.150 0.101 0.238 0.228620 0.155 0.513 0.146 0.100 0.244 0.212630 0.163 0.530 0.144 0.103 0.251 0.202640 0.174 0.555 0.144 0.108 0.275 0.196650 0.185 0.578 0.148 0.112 0.309 0.192660 0.193 0.593 0.155 0.115 0.343 0.192670 0.195 0.590 0.158 0.113 0.360 0.200680 0.189 0.572 0.153 0.109 0.357 0.212690 0.183 0.555 0.146 0.105 0.347 0.224700 0.183 0.552 0.136 0.104 0.346 0.232
323
Wavelength Orange Purplishblue
Moderatered
Purple Yellowgreen
Orangeyellow
400 0.048 0.246 0.136 0.164 0.058 0.068410 0.048 0.309 0.139 0.201 0.059 0.068420 0.048 0.340 0.137 0.216 0.059 0.067430 0.049 0.365 0.136 0.204 0.061 0.068440 0.050 0.391 0.135 0.179 0.064 0.068450 0.051 0.403 0.134 0.150 0.069 0.069460 0.051 0.393 0.132 0.122 0.079 0.071470 0.052 0.361 0.127 0.098 0.095 0.072480 0.053 0.312 0.121 0.081 0.123 0.075490 0.056 0.255 0.116 0.068 0.167 0.079500 0.061 0.204 0.111 0.060 0.245 0.089510 0.084 0.165 0.105 0.055 0.371 0.128520 0.130 0.135 0.100 0.052 0.493 0.223530 0.162 0.114 0.097 0.049 0.553 0.323540 0.176 0.102 0.098 0.048 0.570 0.379550 0.195 0.094 0.098 0.049 0.564 0.415560 0.240 0.087 0.100 0.050 0.549 0.464570 0.327 0.083 0.107 0.051 0.527 0.532580 0.438 0.081 0.151 0.051 0.493 0.586590 0.542 0.080 0.276 0.051 0.445 0.615600 0.594 0.080 0.423 0.056 0.388 0.622610 0.599 0.079 0.516 0.069 0.347 0.620620 0.595 0.079 0.557 0.093 0.326 0.630630 0.588 0.079 0.567 0.120 0.315 0.649640 0.582 0.082 0.568 0.144 0.308 0.677650 0.576 0.087 0.567 0.167 0.304 0.701660 0.568 0.094 0.570 0.193 0.309 0.714670 0.559 0.104 0.583 0.225 0.325 0.705680 0.562 0.118 0.603 0.263 0.349 0.691690 0.571 0.139 0.623 0.312 0.373 0.683700 0.586 0.167 0.637 0.360 0.393 0.690
324
Wavelength Blue Green Red YellowMagenta Cyan400 0.115 0.052 0.049 0.056 0.291 0.180410 0.156 0.052 0.048 0.056 0.363 0.221420 0.193 0.053 0.048 0.057 0.372 0.236430 0.232 0.055 0.048 0.058 0.362 0.260440 0.283 0.057 0.048 0.061 0.340 0.293450 0.349 0.061 0.047 0.065 0.311 0.329460 0.359 0.069 0.047 0.074 0.281 0.371470 0.312 0.085 0.046 0.088 0.250 0.422480 0.235 0.111 0.045 0.111 0.219 0.448490 0.160 0.153 0.044 0.148 0.188 0.445500 0.106 0.222 0.044 0.210 0.165 0.417510 0.075 0.314 0.043 0.318 0.148 0.374520 0.058 0.358 0.043 0.466 0.125 0.321530 0.050 0.343 0.043 0.573 0.104 0.268540 0.046 0.322 0.044 0.627 0.098 0.218550 0.043 0.297 0.045 0.656 0.103 0.173560 0.040 0.257 0.048 0.677 0.103 0.134570 0.040 0.221 0.053 0.698 0.105 0.108580 0.039 0.192 0.065 0.714 0.132 0.094590 0.039 0.162 0.099 0.725 0.197 0.084600 0.039 0.129 0.180 0.730 0.289 0.077610 0.039 0.104 0.315 0.734 0.394 0.072620 0.039 0.089 0.474 0.738 0.509 0.070630 0.039 0.081 0.595 0.738 0.611 0.069640 0.040 0.076 0.665 0.741 0.686 0.069650 0.040 0.073 0.701 0.744 0.736 0.070660 0.041 0.070 0.721 0.747 0.768 0.072670 0.041 0.069 0.733 0.750 0.786 0.073680 0.042 0.070 0.741 0.757 0.798 0.073690 0.043 0.072 0.749 0.763 0.806 0.072700 0.044 0.075 0.755 0.769 0.811 0.069
325
Wavelength White Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5Black400 0.371 0.332 0.265 0.167 0.079 0.033410 0.657 0.509 0.335 0.184 0.082 0.033420 0.823 0.558 0.347 0.187 0.083 0.032430 0.860 0.566 0.349 0.191 0.085 0.032440 0.870 0.571 0.353 0.195 0.087 0.033450 0.880 0.575 0.354 0.197 0.087 0.032460 0.888 0.577 0.353 0.197 0.087 0.032470 0.889 0.573 0.350 0.195 0.086 0.032480 0.890 0.571 0.348 0.194 0.086 0.032490 0.894 0.572 0.347 0.194 0.085 0.032500 0.893 0.573 0.346 0.194 0.086 0.032510 0.894 0.574 0.346 0.195 0.086 0.032520 0.894 0.575 0.346 0.196 0.086 0.032530 0.891 0.575 0.346 0.197 0.087 0.032540 0.891 0.576 0.346 0.197 0.087 0.032550 0.889 0.575 0.345 0.196 0.086 0.032560 0.884 0.574 0.345 0.196 0.086 0.032570 0.882 0.575 0.348 0.196 0.086 0.032580 0.882 0.574 0.350 0.195 0.085 0.032590 0.884 0.574 0.351 0.195 0.085 0.032600 0.884 0.572 0.350 0.194 0.084 0.032610 0.883 0.570 0.349 0.192 0.084 0.032620 0.887 0.570 0.347 0.191 0.083 0.032630 0.886 0.566 0.344 0.189 0.082 0.032640 0.888 0.565 0.342 0.187 0.082 0.032650 0.894 0.564 0.340 0.186 0.081 0.032660 0.897 0.562 0.338 0.184 0.080 0.032670 0.897 0.560 0.336 0.183 0.080 0.032680 0.899 0.559 0.335 0.182 0.079 0.033690 0.899 0.557 0.332 0.180 0.079 0.033700 0.897 0.555 0.330 0.178 0.078 0.033
326
Spectral reflectance factors of the Gretag Macbeth Color Checker measured by theGretag Spectrolino spectrophotometer
WavelengthDark skin Light skin Blue sky Foliage Blue flower Blue green
400 0.059 0.205 0.254 0.051 0.297 0.249410 0.060 0.231 0.313 0.053 0.392 0.309420 0.059 0.232 0.322 0.054 0.421 0.328430 0.059 0.230 0.329 0.056 0.433 0.343440 0.059 0.229 0.338 0.058 0.438 0.360450 0.059 0.229 0.345 0.060 0.441 0.384460 0.059 0.230 0.341 0.061 0.435 0.420470 0.059 0.234 0.331 0.062 0.420 0.466480 0.059 0.246 0.317 0.063 0.395 0.510490 0.059 0.271 0.304 0.066 0.358 0.549500 0.061 0.296 0.286 0.074 0.317 0.575510 0.067 0.306 0.261 0.100 0.285 0.585520 0.073 0.303 0.235 0.145 0.255 0.580530 0.075 0.299 0.218 0.179 0.228 0.561540 0.076 0.297 0.202 0.182 0.215 0.532550 0.079 0.294 0.180 0.165 0.212 0.495560 0.086 0.291 0.156 0.140 0.203 0.449570 0.100 0.307 0.145 0.122 0.197 0.404580 0.119 0.375 0.155 0.112 0.199 0.354590 0.137 0.461 0.163 0.107 0.209 0.304600 0.147 0.500 0.157 0.102 0.225 0.258610 0.151 0.511 0.149 0.098 0.238 0.228620 0.156 0.522 0.145 0.098 0.244 0.212630 0.164 0.542 0.143 0.101 0.254 0.203640 0.175 0.566 0.144 0.105 0.278 0.197650 0.187 0.590 0.148 0.109 0.312 0.192660 0.195 0.604 0.154 0.112 0.344 0.194670 0.196 0.602 0.156 0.110 0.360 0.202680 0.191 0.586 0.152 0.106 0.359 0.214690 0.187 0.572 0.145 0.103 0.352 0.225700 0.188 0.571 0.136 0.103 0.352 0.234
327
Wavelength Orange Purplishblue
Moderatered
Purple Yellowgreen
Orangeyellow
400 0.047 0.242 0.131 0.161 0.058 0.066410 0.047 0.306 0.137 0.199 0.058 0.066420 0.047 0.338 0.135 0.213 0.059 0.065430 0.047 0.364 0.134 0.202 0.060 0.065440 0.048 0.387 0.133 0.178 0.063 0.065450 0.048 0.398 0.132 0.149 0.069 0.067460 0.049 0.387 0.129 0.120 0.078 0.068470 0.050 0.356 0.124 0.096 0.095 0.069480 0.051 0.307 0.119 0.080 0.123 0.072490 0.053 0.251 0.113 0.067 0.169 0.077500 0.060 0.201 0.108 0.058 0.249 0.088510 0.085 0.163 0.103 0.053 0.369 0.130520 0.128 0.134 0.098 0.050 0.485 0.221530 0.159 0.113 0.096 0.048 0.546 0.316540 0.175 0.101 0.096 0.047 0.563 0.372550 0.197 0.093 0.096 0.048 0.559 0.411560 0.246 0.087 0.098 0.049 0.543 0.461570 0.335 0.083 0.109 0.050 0.522 0.527580 0.447 0.081 0.162 0.050 0.487 0.579590 0.547 0.080 0.285 0.050 0.438 0.608600 0.597 0.080 0.425 0.056 0.385 0.616610 0.605 0.079 0.518 0.070 0.346 0.618620 0.601 0.079 0.558 0.095 0.325 0.628630 0.595 0.080 0.571 0.122 0.315 0.649640 0.589 0.083 0.572 0.146 0.308 0.674650 0.583 0.088 0.571 0.169 0.304 0.697660 0.576 0.095 0.576 0.196 0.309 0.708670 0.571 0.106 0.591 0.229 0.326 0.702680 0.574 0.120 0.612 0.269 0.350 0.690690 0.586 0.141 0.633 0.316 0.374 0.684700 0.603 0.169 0.648 0.365 0.393 0.690
328
Wavelength Blue Green Red YellowMagenta Cyan400 0.113 0.051 0.048 0.056 0.281 0.177410 0.152 0.051 0.048 0.056 0.353 0.219420 0.189 0.052 0.047 0.056 0.364 0.238430 0.228 0.053 0.046 0.057 0.357 0.262440 0.279 0.056 0.046 0.060 0.335 0.293450 0.336 0.060 0.046 0.065 0.307 0.331460 0.345 0.068 0.045 0.073 0.275 0.374470 0.301 0.084 0.044 0.087 0.245 0.422480 0.226 0.111 0.043 0.111 0.214 0.446490 0.154 0.154 0.042 0.149 0.184 0.444500 0.102 0.223 0.042 0.213 0.161 0.416510 0.072 0.307 0.041 0.322 0.143 0.372520 0.056 0.349 0.041 0.463 0.121 0.321530 0.048 0.338 0.041 0.568 0.101 0.267540 0.044 0.317 0.042 0.623 0.096 0.218550 0.041 0.291 0.044 0.655 0.101 0.172560 0.039 0.253 0.046 0.677 0.101 0.134570 0.038 0.218 0.052 0.700 0.106 0.109580 0.037 0.188 0.066 0.714 0.136 0.094590 0.037 0.158 0.104 0.725 0.202 0.084600 0.037 0.126 0.189 0.732 0.293 0.077610 0.037 0.102 0.327 0.737 0.402 0.072620 0.038 0.088 0.482 0.740 0.516 0.070630 0.038 0.080 0.600 0.742 0.616 0.070640 0.038 0.075 0.669 0.744 0.688 0.070650 0.039 0.072 0.706 0.747 0.737 0.071660 0.040 0.069 0.726 0.750 0.769 0.072670 0.040 0.068 0.739 0.755 0.789 0.074680 0.041 0.069 0.749 0.762 0.802 0.074690 0.041 0.071 0.758 0.770 0.811 0.073700 0.043 0.074 0.765 0.778 0.819 0.071
329
Wavelength White Neutral 8 Neutral 6.5 Neutral 5 Neutral 3.5Black400 0.374 0.329 0.256 0.159 0.076 0.032410 0.634 0.489 0.323 0.179 0.081 0.032420 0.803 0.544 0.338 0.184 0.082 0.031430 0.856 0.558 0.343 0.188 0.084 0.031440 0.869 0.563 0.345 0.192 0.085 0.031450 0.878 0.567 0.347 0.194 0.086 0.031460 0.883 0.567 0.345 0.194 0.085 0.031470 0.887 0.566 0.343 0.192 0.085 0.031480 0.888 0.564 0.341 0.191 0.084 0.031490 0.892 0.565 0.341 0.191 0.084 0.031500 0.893 0.566 0.340 0.192 0.084 0.031510 0.894 0.567 0.340 0.193 0.085 0.030520 0.896 0.570 0.341 0.194 0.085 0.030530 0.895 0.571 0.341 0.195 0.085 0.030540 0.895 0.571 0.341 0.195 0.085 0.030550 0.895 0.573 0.342 0.195 0.085 0.030560 0.890 0.572 0.342 0.195 0.085 0.030570 0.891 0.574 0.346 0.196 0.085 0.030580 0.889 0.573 0.348 0.195 0.085 0.030590 0.891 0.573 0.349 0.194 0.084 0.030600 0.892 0.572 0.348 0.193 0.084 0.031610 0.893 0.572 0.348 0.192 0.083 0.031620 0.895 0.571 0.346 0.191 0.082 0.031630 0.897 0.569 0.344 0.189 0.082 0.031640 0.898 0.567 0.342 0.188 0.081 0.031650 0.903 0.566 0.340 0.186 0.081 0.031660 0.907 0.564 0.338 0.185 0.080 0.031670 0.909 0.564 0.337 0.184 0.079 0.031680 0.911 0.563 0.336 0.183 0.079 0.031690 0.912 0.562 0.334 0.182 0.078 0.032700 0.913 0.562 0.332 0.180 0.078 0.032