Applications of the Applications of the Kubelka-Munk Color ModelKubelka-Munk Color Model

Kristen Hoffman

Dr. Edul N. Dalal

RIT Center for Imaging Science

Xerox Corporation, Wilson Center for

Research and Technology

Introduction - Goals and AccomplishmentsIntroduction - Goals and Accomplishments

Goal: Ability to model the reflectance of a color xerographic sample

Developed: Predictive color model based on Kubelka-Munk theory

Model extended to– Bidirectional Measurement Geometry– MultiLayer Images– Xerographic Print Samples

Background: Kubelka-Munk Background: Kubelka-Munk TheoryTheoryColor reflection depends on

– Material properties - the absorption and scattering spectra, K() and S()

– Sample thickness, X– Substrate reflectance spectrum, Rp ()

Model applies to– Uniform thickness samples with complete

substrate coverage– Single color images

Background: Saunderson Background: Saunderson Correction ParametersCorrection ParametersTwo parameters

– k1 and k2 - corrections are made for reflections at the sample surface

– Derived for integrating sphere measurement geometry

– Applied to reflectance spectrum before the Kubelka-Munk model

Developed Color ModelDeveloped Color Model

Correction EquationsCorrection Equations

Introduction of kIntroduction of k00 Correction Correction

ParameterParameterk0

– Describes front surface reflection reaching detector of measurement device

– Correlation exists for 45/0 measurement geometry as a function of 75 image gloss

– Depends on refractive index ratio at the air-image boundary

Derived Correction Equations for Derived Correction Equations for Bidirectional Geometry SystemsBidirectional Geometry Systems

measured

measuredcorrected Rkkkkkkk

kRR

2202121

0

1

)(1

)1)(1(

2

210

corrected

correctedmeasured Rk

RkkkR

Link to Derivation:

http://www.cis.rit.edu/~kmh7483/index.html

Multi-Layer ImagesMulti-Layer Images

Examples of Image Layer Examples of Image Layer StructureStructure

Substrate Substrate

(a) Single colorant layer considered in the original Kubelka-Munk model

(b) Multiple colorant layers generally encountered in process color xerographic prints

Rp()

Rp()corr

R1()corr

R2()corr

Rn()corr

R()

k0, k1, k2 for substrate

k0, k1, k2 for toner

K, S for layer n

K, S for layer 2

K, S for layer 1

SaundersonCorrection

Kubelka-Munk

Kubelka-Munk

Kubelka-Munk

Inverse Saunderson

Substrate

Bottom-most toner layer

Second toner layer

Top-most toner layer

Calculated Sample Reflectance

Non Planar Toner LayersNon Planar Toner Layers

Image PhotomicrographsImage Photomicrographs

Toner Layer Thickness Toner Layer Thickness MeasurementsMeasurementsLayer structure digitized

electronically– Measurements made at every 0.5m – Small interval divides print into planar

sectionsK/M applied to each small planar

interval

ResultsResults

Fitted Absorption Spectra for Xerox 5760 CMY Toners

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

350 400 450 500 550 600 650 700

Wavelength (nm)

Ab

so

rpti

on

(m

g/c

m2

)

Cyan

Magenta

Yellow

Fitted Scattering Spectra for Xerox 5760 CMY Toners

0

0.05

0.1

0.15

0.2

0.25

350 400 450 500 550 600 650 700

Wavelength (nm)

Sc

att

eri

ng

(c

m2

/mg

)

Cyan

Magenta

Yellow

Results - Toner Layer Thickness Results - Toner Layer Thickness Probability Distribution ExampleProbability Distribution Example

0

00.0375

0.03750.075

0.0750.1125

0.11250.15

0.150.1875

0.18750.225

0.2250.2625

0.26250.3

0.3

Pro

b Pro

b

Example of Green Probability DistributionRank 1 Eqn 2501 z=3inc-gaw()

Results - Single Layer, 45/0Results - Single Layer, 45/0

dE*CIELAB Average C = 1.83 M = 1.77, Y = 1.26

0

1

2

3

4

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.8 1

Thickness

dE

* (C

IEL

AB

)

dE* Cyan

dE* Magenta

dE* Yellow

Results - Multi-Layer ImageResults - Multi-Layer Image

0

0.5

1

1.5

2

2.5

3

3.5

1

dE

* (C

IEL

AB

)

R G B

Results - Multilayer Non Planar Results - Multilayer Non Planar PrintPrint

0

2

4

6

8

10

1

dE

* (C

IEL

AB

)

Cyan

Magenta

Yellow

Red

Green

Blue

dE*CIELAB Average 5.1, RMS = 5.5

ConclusionsConclusions

Benefits of K/M Color Model– Based on physical parameters of toner

set– No print samples needed– Good predictions (low color error)