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https://ntrs.nasa.gov/search.jsp?R=19730022682 2020-05-25T14:42:50+00:00Z

Computer enhancementof radiographsFinal report MFR-73-3July 15, 1973Contract No. NAS8-28521DCN 1-2-70-00122 (1 F)

byA. DekaneyJ. KeaneJ. Desautels

prepared byMead Technology Laboratories3481 Dayton-Xenia RoadDayton, Ohio 45432

prepared forGeorge C. Marshall Space Flight CenterMarshall Space Flight Center, Alabama

This report was prepared by Mead TechnologyLaboratories under contract NAS8-28521 entitled"Computer Enhancement of Radiographs" for theGeorge C. Marshall Space Flight Center of theNational Aeronautics and Space Administration.

TABLE OF CONTENTS

Section Page

I INTRODUCTION ................... 1A. Setting ................... .. 1B. Problem Addressed . ............. . 1C. Implications . . . ........... . .. . . 2D. Format . . . . .. . . . . . . . . . ... . . 2

II DATAACQUISITION ................. 3A. Film Graininess ................. 3

1. Radiograph Generation. . . . . . . . ..... 32. Film Processing .. ..... ...... . 33. Data Conversion ............... 3

B. Quantum Mottle ........ ......... 51. Parameter Selection . . . . . . . . . . . . . 52. Exposure Procedures . . . . . . ... . . 63. Film Processing ... ........ ... . 84. Data Conversion ............. .. 8

C. Scanner Noise .................. 81. Neutral-Density Filter Specification . . . . . . 82. Data Collection .... . . ........ 8

D. Image Degradation ................ 91. Target Development . . . . . . . . . . . . . 92. Radiograph Generation. . .... . . . . 93. Data Reduction . .............. 12

III DATA ANALYSIS. . . . .............. 14A. Film Granularity ................ 14B. Quantum Mottle . ..... . . . . . ..... 16

1. RMS Granularity . ........... . . 162. Power Spectral Density Computations . . . . . 25

C. Scanner Noise*. .......... ........ 381. RMS Granularity Computation. . . . . . 382. PSD Computation . . . . . . . . . . . . . 39

D. Image Degradation . . . .. . . . . . . . . . 391. Edge Selection ............... 392. MTF Computation . . ............ 45

IV CONCLUSIONS & RECOMMENDATIONS . . . . . . . . . 50A. Conclusions ............. ...... 50B. Recommendations ................ 51

V BIBLIOGRAPHY ................... 52

APPENDICES

Appendix I Mead Micro-Analyzer Lens and Focus Selection . . . 53A. Focus Criterion ................. 53B. Lens Combination Selection ........... 53

TABLE OF CONTENTS

(Con't)

Section Page

Appendix II RMS Film Granularity . . . . . . . . . . . . . . . . 54A. Mean and Variance of a Random Variable . . . . . 54B. Data Sampling ................. 57

1. Independence of Sample Points . . . . . . 572. Sampling Distributions . . . . . . . . . . . 573. Sampling for the Mean ......... . . . 584. Sampling for the Standard Deviation . . . . 60

C. Practical Considerations .. . . . . . . . . 621. Density Wedging . . . . ....... . .. . . 622. Random Flaws ............... 64

Appendix III Power Spectral Density . . ......... . . . 65A. Definitions . . . .............. 65B. Sample Spectrum Statistics . .... .. . . . 66C. Smoothed Spectral Estimates . . . . . . . . . . 68D. Statistics of Smoothed Spectral Estimator . . . . . 69E. Bandwidth ................ . . 71F. Spectral Analysis Design . . . . . . . . . 72

Appendix IV Edge Gradient Analysis ............... 74A. System Performance Measures . . . . . . . . . 74B. Application of Edges as Targets . ........ 75

ii

TABLE OF FIGURES

Number Description Page

1 8keV X-ray System ................... 42 MG-150 Experimental Configuration . ........ . . 73 Wedge Target ...................... 104 Edge Target ... .. .. ..... . .. .. . . .. . 115 Effective Calibration of MSFC's Scanner. . ........ . 226 Comparison of RMS Granularity by Aperture for R (SC),

100 keV, Al (.50 inch) .................. 237 Comparison of RMS Granularity by Aperture for R (DC),

200 keV, Al (.50 inch) .................. 238 Comparison of RMS Granularity by Aperture for M, 200 keV,

Fe (.25 inch) ...................... 249 Comparison of Film Types at 100 keV, 13 .um aperture . . * 24

10 Comparison of Film Types at 100 keV, 27 jum apertures. . . 2611 Comparison of Film Types at 100 keV, 57 um apertures * *. 2612 Comparison of RMS Granularity as a Function of keV for

Film Type R (DC), for aperture 13,pm, 27jpm, 57)m . . . 2713 Comparison of RMS Granularity for Type R (DC) 100 keV

as a Function of Specimen Material and Scanning Aperture * 2714 Effect of Intensifying Screens on RMS Granularity . . . .. 2815 PSD R (SC), 100 keV ..... ............. 3016 PSD R (DC), 100 keV . .. .............. 3117 PSD M, 100 keV.. ................... 3218 PSD R (SC), 200 keV .................. 3319 PSD R (DC), 200 keV .................. 3420 PSD M, 200 keV ................... . . 3521 PSD R (DC), 300keV ................. . 3622 PSD M, 300keV .. ................. 3723 Graphical Display of MSFC Scanner Noise Magnitude . . . 4124 PSD MSFC's Scanner Density Levels, .18, .45, .82, 1.12,

1.50 (12.5pm Aperture) ................. 4225 PSD MSFC's Scanner Density Levels 1.76, 2.47, 2.33

(12.5 um Aperture) ................... 4326 PSD of Type M Film, 100 keV Scanned at MSFC (12.5 ,m

Aperture) ... ................. . . 4427 Envelope Curve of MTF's ........... . . . 49

iii

TABLES

Number Description Page

1 Quantum Mottle Exposures . . . . . . . . . . . . . . . . . 62 Exposure Conditions for Edge Analysis Radiographs . ..... 123 Density Levels of Edges . ................. 134 RMS Granularity R (SC), 8 keV .............. 155 RMS Granularity R (DC), 8 keV . ............. 156 RMS Granularity M, 8 keV . ............... 157 RMS Granularity M, 100 keV, Fe (.25 inch) . . . . . ... 178 RMS Granularity M, 200 keV, Fe (.25 inch) . ....... 179 RMS Granularity M, 300 keV, Fe (.125 inch) . ....... 17

10 RMS Granularity M (with screens), 200 keV, Fe (.25 inch) . 1811 RMS Granularity R (SC) 100 keV, Al (.25 inch) . . . ... 1812 RMS Granularity R (SC) 100 keV, Al (.50 inch) . ..... 1813 RMS Granularity R (SC) 100 keV, Al (.73 inch) . ..... 1914 RMS Granularity R (SC) 200 keV, Al (.25 inch) . ..... 1915 RMS Granularity R (DC) 100 keV, Al (.25 inch) . ..... 1916 RMS Granularity R (DC) 100 keV, Al (.50 inch) . ..... 2017 RMS Granularity R (DC) 100 keV, Al (.75 inch) . ..... 2018 RMS Granularity R (DC) 100 keV, Fe (.25 inch) . . . . . . 2019 RMS Granularity R (DC) 200 keV, Fe (.25 inch) . ..... 2120 RMS Granularity R (DC) 300 keV, Fe (.125 inch) ...... 2121 Radiographs Selected for PSD Analysis . . . . . . . .... 2922 MSFC Optronics Noise Process RMS Granularity. . ...... 4023 Edge Identifications .................... 4524 Edge Responses Suitable for Analysis . . . . . . . . . . . 4525 MTF's for Twenty-Six Edges. . . . . . . . . . . . . ... 46

iv

ACKNOWLEDGEMENTS

Mead Technology Laboratories acknowledges the cooperation of Mr. JimHolloway, Metals and Ceramics Division, NDT and Mechanics Branch (LLN),Air Force Material Laboratory, Wright-Patterson Air Force Base, Dayton,Ohio for his assistance in generating low energy x-ray radiographs and for theuse of AFML's x-ray equipment.

V

ABSTRACT

Examination of three relevant noise processes and the image degrada-tion associated with Marshall Space Flight Center's (MSFC) x-ray/scanningsystem was conducted for application to computer enhancement of radiographsusing MSFC's digital filtering techniques. Graininess of type M, R single coatand R double coat x-ray films was quantified as a function of density levelusing root-mean-square (RMS) granularity. Quantum mottle (including filmgrain) was quantified as a function of the above film types, exposure level,specimen material and thickness, and film density using RMS granularityand power spectral density (PSD). For various neutral-density levels thescanning device used in digital conversion of radiographs was examined fornoise characteristics which were quantified by RMS granularity and PSD.Image degradation of the entire pre-enhancement system (MG-150 x-ray device;film; and Optronics scanner) was measured using edge targets to generateModulation Transfer Functions (MTF). The four parameters were examinedas a function of scanning aperture sizes of approximately 12.5 pm, 25 pm and50 pm.

vi

SECTION I

INTRODUCTION

A. SETTING

X-ray generated radiography provides a valuable nondestructive methodfor determining the soundness of welds. X-ray radiography is widely usedfor the detection of discontinuities such as cracks, porosity, slag inclusions,incomplete fusion, and incomplete penetration, mainly because it offers arapid and sensitive approach.

The mere presence of defects or flaws in a weld does not generallymean that the specimen cannot serve its intended purpose. An individualwho interprets the radiograph must exercise some judgement as to the degreeof imperfection that exists and must determine whether the flaw would pre-vent the specimen from being used. Accurate quantification of the degree ofimperfection or discontinuity, therefore, demands optimum information extrac-tion from x-ray imagery. To this end, the National Aeronautical and SpaceAdministration (NASA) at Marshall Space Flight Center (MSFC) is engaged indigital image enhancement of weld flaws through the application of digitalfiltering techniques to radiographic imagery scanned on a high-speed micro-densitometer.

B. PROBLEM ADDRESSED

MSFC's mode of processing the imagery is to take an x-ray generatedradiograph of the weld where the flaw occurred; scan the radiograph on a highspeed microdensitometer and digitize the image; enhance the digital image bythe computer; then recreate the enhanced image on film. MSFC has recognizedthat this approach to image enhancement is constrained by the sources in thex-ray/scanning system where noise processes and other degradations can beintroduced.

The tasks undertaken in this program were the specification and quanti-fication of several of those source parameters which should be considered inthe development of digital filters applicable to radiographic images processedthrough MSFC's system. The investigation was limited to the x-ray/scanningphases of the processing system, since the film writing phase is dependentupon the particular filtering techniques developed by MSFC. The success ofthe filtering process is dependent on the nature of the system preceding it andshould not be constrained by the device displaying the enhanced imagery.

This study examined the relevant noise processes and image degrada-tion inherent in the system. The noise sources considered in the program werefilm grain, quantum mottle and scanner noise. Film grain was examined with. afixed exposure(keV)level but with varying film type and exposure time. Quantummottle was examined with varying exposure levels, varying film types, varyingspecimen material and thickness, and varying film density levels. Scanner noisewas examined under varying neutral-density filter levels. These sources were

1

quantified through the computation of the root-mean-square (RMS) granularity,which equals the standard deviation of the noise process multiplied by 1000, andthe power spectral density (PSD) function, which shows how the statistical var-iance of the noise process is distributed in frequency space.

The degradation considered in the program was the composite degradationintroduced by the x-ray equipment, the film and the scanning device. The degrada-tion was examined as a function of film type, exposure level, exposure time,film density and format position. Image degradation was measured by computa-tion of the Modulation Transfer Function (MTF), which measures the degradationto a sine wave introduced by the system. The MTF can also be thought of asthe Fourier transform of the point spread function of the system.

C. IMPLICATIONS

The design of digital enhancement filters depends on the system's MTF andnoise characteristics (PSD) in the following manner. The MTF's of the systemfor various values of the physical aspects of the system indicate the resolutionlimits of the system. These resolution limits are important in the establishmentof cut off frequencies for noise suppression filters. For example, in attemptingto suppress noise frequencies beyond the highest image spatial frequencies, theresolution limits can be used as upper cutoff frequencies on low-pass filters.

The design of effective bandpass spatial filters requires that the spectralcontent of the noise be known. The computed noise process PSD's allow for thedetermination of whether or not significant noise bands can be removed withoutseriously affecting image spatial frequencies.

D. FORMAT

The remainder of this report discusses the data acquisition for examina-tion of the relevant system parameters mentioned above, the analysis of the dataand the impact of each parameter on digital filtering. All pertinent tables andfigures are found in the body of the report. The details of the data analysistechniques used in this study are presented in the Appendix.

2

SECTION II

DATA ACQUISITION

In order to evaluate the significance of film graininess, quantum mottle,scanner noise and image degradation, test radiographs and neutral-densityfilters must be generated and converted to digital form. This section discussesthe procedures followed to generate the radiographs and select the neutral-density filters for each of the areas listed above. It also discusses the techniquesemployed to convert these data to digital form.

A. FILM GRAININESS

The purpose of the film graininess portion of the program was to deter-mine the granularity of x-ray film types R single (SC), R double (DC), and M.Proper exposure of film for this investigation required extremely lowenergy x-rays. Low energy x-ray exposure makes it impossible for one x-rayphoton to expose more than one film grain. Insuring that each photon exposedno more than a single grain permitted the generation of radiographs free of quan-tum mottle. In addition, exposure with low energy x-rays permitted grain ex-posure throughout the depth of the emulsion as opposed to exposure with lightwhich tends to expose grains at the surface of the emulsion.

1. Radiograph Generation: In total fourteen exposures were generatedusing low energy x-rays. For film type R (SC) five exposures were made at densitylevels of 1.04, 1.55, 1.80, 1.90, and 2.75D. The type R (DC) film was exposedto the five density levels of 1.18, 1.46, 1.60, 2.06, 2.60D. Four density levelswere generated on type M film at density levels of .99, 1.46, 2.06, and 2.18D.

Radiographs used for the granularity investigation were produced on filmsupplied by MSFC. This film was taken from the same packages as the film usedin the quantum mottle and MTF portions of this investigation which will be dis-cussed later. The film was exposed with an 8 keV x-ray system similar to thatdepicted by Figure 1. As discussed above, the 8 keV x-ray source resulted inradiographs relatively free of quantum mottle. The vacuum path from the x-raysource to the film reduced the attenuation of the 8 keV x-rays. The paper cassettein which the film was packed was removed to further reduce the attenuation of the8 keV x-rays and to eliminate noise caused by the fiberous material in the cassettepaper. All exposures were made with an x-ray tube current of 10 ma. Variationin exposure levels and ultimately film density levels was obtained by varying theexposure time from .5 to 3.0 minutes.

2. Film Processing: The exposed films were processed manually usingKodak x-ray developer and Kodak fixer. Five minute development at 700 F was usedas described in the "Film Processing Supplement" to Radiography In ModernIndustry, by Eastman Kodak Company, Rochester, New York. All processingwas performed within a twenty hour period after the film was exposed, in orderto minimize latent image failure.

3. Data Conversion: The developed fourteen radiographs were scannedon Mead Technology Laboratories' Mann Micro-Analyzer. Set-up procedures for

3

X-Ray Source

Aluminum0 \ Vacuum Tank

OVacuum

Film Door Gauge

VacuumPump

Figure 1. 8 keV X-Ray System

4

Micro-Analyzer scanning are discussed in Appendix I. Three circular scanning

apertures (13, 27, and 57 micrometers) approximating the apertures availableon MSFC's high speed scanner were used to scan the radiographs. To provide

for comparison of results among the aperture settings the three radiograph scans

were taken over the same area. A minimum of 6144 samples were taken in eachscan. A sampling interval equal to or greater than the diameter of the aperturewas used in each scan to assure that the resulting samples would be uncorrelatedas required by the analysis procedures.

B. QUANTUM MOTTLE

The purpose in examining quantum mottle was to characterize it as afunction of x-ray energy, specimen filter, film type, density of the radiograph,and presence or absence of intensifying screen through the computation of RMS

granularity and PSD. Since the scope of each of these parameters is large, theexamination of mottle was limited to specific parameter values pertinent toMSFC's inspection of weld flaws.

1. Parameter Selection: The investigation of quantum mottle was limited

to five material specimens (three aluminum and two steel), three film types,lead screen and no screens, five film density levels, and three exposure levels.The following paragraphs provide a brief statement of the rationale for selectingthe specific parameter values for the quantum mottle experiments.

The choice of film types M, R (DC) and R (SC) was made by mutual agree-ment of MSFC and Mead Technology Laboratories personnel, in order to examinefilms currently being used in MSFC applications and films considered for future

applications. In order to determine the effects of screens on mottle and yet re-strict the number of experiments to a reasonable number, the decision was made

to generate a complete set of radiographs without screens, while running a fewselected experiments for replication with screens.

The specimen materials were selected to be representative of those whichnormally are candidates for ultimate digital filtering enhancement. Therefore,aluminum and steel were chosen as the experimental specimen materials. Mater-ial thicknesses were also chosen to represent normal operations and to limitthe number of experiments to a manageable number. Thus 1/4, 1/2 and 3/4inch aluminum specimens and 1/8 and 1/4 inch steel specimens were specifiedfor the experiments.

The majority of radiographs considered for enhancement by MSFC are

exposed to an approximate density level of 1.7D. To provide for a measure of

completeness in this investigation, the density range under consideration wasmade to vary from this nominal by approximately + .8D. Also, to maintain a

manageable number of experiments, the specific number of densities investi-

gated was limited to five levels. These levels were approximately 1.0, 1.3,1.7, 2.0, and 2.5D.

The energy level normally used by MSFC for weld flaw inspection liesin a neighborhood of 100 keV. Thus, to make the experiments meaningful andalso relatively complete, five energy levels (30, 50, 100, 200, and 300 keV)were initially selected for the investigation. However, performance of the

5

experiment indicated that the two lowest x-ray energies (30, 50) were imprac-tical due to the extended exposure period required and subsequently were deleted fromconsideration. 1/4 inch steel was selected for replication with screen at the200 keV energy level, simulating operational procedures. The complete setof radiographs generated for the quantum mottle examination are described inTable 1 below.

KEV MATERIAL DENSITY

Type in. 1.0 1.3 1.7 2.0 2.5

1/4 1,2 1,2 1,2 1,2 1,2Alum. 1/2 1, 2 1, 2 1, 2 1, 2 1, 2

3/4 1,2 1, 2 1, 2 1, 2 1, 2100

1/8Steel 1/4 2, 3 2, 3 2, 3 2,3 2,3

1/8200 Steel 1/4 1,2,3 1,2,3 1,2,3 1,2,3 1,2,3

1/8 2,3 2, 3 2, 3 2,3 2,3300 Steel 1/4

Table 1. Quantum Mottle Exposures

2. Exposure Procedures: The radiographs for the quantum mottleexamination were exposed at the MSFC nondestructive testing laboratory. Thefocal spot of MSFC's MG-150 x-ray unit was positioned 80 inches above the floorwith the x-ray beam pointing toward the floor. The holder for the film cassettewas mounted to support the film 40 inches from the focal spot. The floor wascovered with a lead sheet at least 1/8 inch thick over the x-ray beam area.

The cassette support was made in the form of a frame to provide supportat the edges of the cassette and to avoid backscatter. The cassette was coveredwith a lead mask 1/4 inch thick which was cut 1/4 inch smaller in the insidedimensions than the support frame. The sizing prevented backscatter from theframe at the edges. The experimental configuration of the x-ray equipment isshown in Figure 2.

For this particular task, the specimen was to act as a filter in absorbingthe soft radiation without creating an image. Hence, the specimen was locatedas close to the x-ray target as possible but was not placed directly on thecassette as in normal radiographs. Placement of the material close to the focalspot caused hidden manufacturing flaws present in the material to be spreadacross the film format in a manner similar to effect of geometric unsharpness.Therefore, contributions of such anomalies to the mottle and grain noise wereminimized.

6

Support

Lead Mask

Film Cassette

1/8" Lead

Figure 2. MG-150 Experimental Configuration

7

3. Film Processing: To maintain maximum processing control and tominimize latent image failure, all radiographs were manually processed withintwenty-four hours of exposure at MSFC. The processing conditions employedwere 4.5 minutes in 700 developer, followed by a 30 second stopbath and 15minutes in a fixing solution. The radiographs were washed and then treatedwith photo flow to prevent water spotting. The finished radiographs were for-warded to Mead Technology Laboratories for analysis.

4. Data Conversion: The techniques employed to reduce the radiographsfor the quantum mottle examination to digital form were identical to those techni-ques employed for the film granularity experiment. (See Section II, A.3) Thesame Micro-Analyzer parameters were employed to analyze these radiographs.Several of the radiographs were not scanned with all three Micro-Analyzerspots. The reason for their omission was that analysis of initial scan data indi-cated new information would not be obtained from these additional scans.

To more meaningfully relate the quantum mottle investigation to MSFC'simage processing facility, several radiographs were selected for scanning withMSFC's Optronics scanner. These film specimens were scanned with 12.5um,25 um, and 50 1um apertures using equivalent sampling intervals. The densitymode selector was set to 0-3D range.

C. SCANNER NOISE

MSFC's Optronics scanner represents a potential noise source in thedigital filtering system. The purpose of examining this element of the systemis to identify the variance and spectrum of the noise process associated with thescanner and to examine them for their influence on the filtering process.

1. Neutral-Density Filter Specification: The noise characteristics ofthe drum type scanner were determined by scanning, digitizing, and analyzingseveral Kodak neutral-density filters. Neutral-density filters were selected astargets since they represent uniform density patches with minimum inherentnoise. Ten filters were used to represent the density range .18 to 2.39D. Thespecific density values of the filters were .18, .48, .82, .98, 1.16, 1.50, 1.76,1.94, 2.19, and 2.39D. It is not necessary to scan the entire specimen to obtaina sufficient sample size, hence a 1 cm. wide rectangular area served as the noisetarget.

2. Data Collection: The filters were prepunched and mounted in theupper left hand corner of the drum. The top of the filter was supported by andclamped to the drum, whereas the left edge was supported by and taped to thedrum. Support was believed sufficient to allow the filters to assume the curvedcontour of the drum.

Scanning was accomplished under the following parameterization:

0-3D Density RangeStart of record set at 2End of record set at 6Left X selector set at 3Right X selector set at 4

8

Each of the ten filters was scanned under the above conditions repeated withapertures and sampling intervals of 12.5pum, 25,um, and 5 0pm.

D. IMAGE DEGRADATION

The purpose of investigating image degradation was to provide MTF datawhich could be utilized in selecting digital filters for application to enhancementof radiographs. The method selected to obtain this information was the analysisof edge targets which consist of two distinct uniformly thick specimens whoseboundary is a plane. By producing radiographs containing edges, scanning theradiographs with the Optronics scanner and analyzing the resulting digital data,the MTF for the entire x-ray scanning system (including x-ray source, film, pro-cessing and scanner) was computed.

In the design of the experiment to compute the MTF, several parameterswhich could affect the MTF of the system were considered for investigation. Theparameters selected were film type, density difference per edge, average densityper edge target, x-ray energy level, and distance from the center of format.These parameters were selected because they were likely to significantly influ-ence the system MTF and because they would normally be variables in actualapplications.

1. Target Development: Generation of radiographs for edge analysisrequired the design and manufacturing of two targets. The wedge target shownin Figure 3 was designed to provide twelve density levels which formed elevenedges of approximately equal density difference. The purposes of the wedgetarget were to provide an overall check on the x-ray system linearity and, ifpossible, to become a source of low modulation edges. The edge target shownin Figure 4 was designed to provide seven edges of varying density differenceswith approximately the same mean density. Both the wedge and edge targetswere machined out of 2419 aluminum.

Since lower modulation flaws would be of primary interest in practice,the targets had to be exposed in such a way to assure that edges of minimumanalyzable density amplitude would be available. The thickness differencesexisting in the targets were designed so that the density difference recorded on theradiographs would range from 0.04 to 0.50D. This range of density differences wouldprovide an input signal for the scanner ranging from less than the scanner'sminimum detectable density difference to ten times the minimum detectabledensity difference.

2. Radiograph Generation: Identical geometry was used to expose allnine radiographs for the MTF investigation. The two targets described earlierwere placed in direct contact with the paper film cassettes at the center of thefilm format. The film and targets were placed on a lead covered table with thefocal spot of the x-ray tube 40 inches above the center of the film. This particu-lar geometric configuration was selected because it simulated actual radiographicconditions encountered at the MSFC nondestructive test facility. Variation inimage density was achieved through a combination of varied exposure time, variedx-ray tube current and insertion of filter material under the target. The variousexposure conditions are listed in Table 2.

9

4.8 in.

0.4 0.4 .4 0.4 0.4 .+ 0 .4. 0.4.4+0 4+0.4+0. 4

NOTE: MATERIAL - 2219 AlView "A" All Dimensions + 0.05 1.5 in.View "B" All Dimensions + 0.002

VIEW "A"

.301

450".271

.254_ .392" .500"

.226.188

.169.147 1

.127.107

VIEW "B"

Figure 3. Wedge Target

.015". 025"

.063 .007"

.5 . 5 5-- .5 1.0 1.0 1.0 .5-

5.5 in.

Al 2219

Figure 4. Edge Target

Radiograph keV Level Film Type Exposure TimeIdentification

100-2-2. 5 100 R (DC) 2. 5 min.70-3-2 70 M 2.0 min.200-1-1 200 R (SC) 1.0 min.100-1-4.2 100 R (SC) 4.2 min.100-3-1 100 M 1.0 min.200-1-1. 5 200 R (SC) 1.5 min.80-3-2 80 M 2 min.100-1-4 100 R (SC) 4 min.100-3-1. 5 100 M 1.5 min.

Table 2. Exposure Conditions for Edge Analysis Radiographs

The radiographs were processed at MSFC immediately following exposureunder the same conditions as those employed for the quantum mottle radiographs.Each step on the radiographs was subsequently read by a MacBeth densitometer.The values of the readings are found in Table 3.

3. Data Reduction: The nine radiographs were digitized on MSFC'sOptronics scanning device in such a manner that the direction of scan was nearlyperpendicular to the edges on the film. At least one hundred scan lines weretaken across each edge in three locations (at approximately the center and atthe two extremities). The scanning was performed in a 0-3D density mode usingthe 12.5 um sampling and aperture settings, with the data recorded on magnetictape for subsequent computer analysis. The length of each scan was constrainedso that a sufficient number of samples would be taken to include the entire edgeresponse.

12

Exposure Conditions Step NumberIdentification 1 2 3 4 5 6 7 8 9 10 11 12

.95 1.04 1.19 1.42 1.51 1.61 1.72 1.84 1.97 2.07 2.16 2.28100-2-2.5 ain. 1.85 1.94 2.08 1.55 1.63 1.88 1.93 2.22

.74 .87 1.08 1.51 1.70 1.95 2.22 2.54 3.02 3.43 3.86 4.401.01 1.04 1.16 .76 .82 1.01 1.07 1.36

.88 .94 1.02 1.15 1.21 1.26 1.32 1.38 1.45 1.50 1.54 1.57200-1-1 .1.54 1.42 1.50 1.22 1.27 1.40 1.43 1.57

1.06 1.17 1.32 1.56 1.66 1.81 1.94 2.08 2.25 2.39 2.52 2.67100-1-4.2 min.1.53 1.57 1.67 1.29 1.36 1.55 1.60 1.84

.78 .89 1.00 1.21 1.31 1.41 1.53 1.65 1.80 1.93 2.05 2.171.27 1.32 1.42 1.05 1.11 1.30 1.36 1.58

1.24 1.35 1.46 1.65 1.73 1.83 1.93 1.99 2.08 2.17 2.22 2.27200-1-1. 5 min.1.95 2.07 2.18 1.75 1.82 2.01 2.05 2.25

1.26 1.50 1.81 2.41 2.71 3.10 3.52 3.98 4.4080-3-2 min.1.70 1.74 1.92 1.31 1.42 1.73 1.82 2.23

.99 1.08 1.22 1.44 1.54 1.64 1.77 1.88 -2.04 2.14 2.24 .2361.90 1.99 2.17 1.57 1.65 1.92 1.98 2.28

.93 1.04 1.18 1.40 1.50 1.62 1.75 1.87 2.04 2.16 2.26 2.361.85 1.93 2.08 1.54 1.63 1..90 1.99 2.31

Table Entries are MacBeth Density Reading of Eadch Step in the Wedge (First entry in box)and Edge (Second entry in the box)

x-y-z, where x = keV Levely = Film Type (1 = R (SC), 2=R (DC), 3=M)z = Exposure Time

Table 3. Density Levels of Edges

SECTION III

DATA ANALYSIS

A. FILM GRANULARITY

The digital data obtained from the Micro-Analyzer scans of the 8 keVgenerated radiographs was analyzed by computing the RMS granularity forthe fourteen film specimens. A complete description of RMS granularity theoryand computational technique is presented in Appendix II. The examination ofthe film granularity as a function of film type, density level and scanning aper-ture is summarized by the RMS granularity numbers presented in Tables 4, 5,and 6.

Careful examination of the granularity data contained in these tables in-dicates that the resulting granularity does not show smooth or uniform trends.The erratic nature of the data, in large measure, can be attributed to difficulitiesencountered in the course of the experiment. These difficulties included thefact that the x-ray system used to expose the radiographs was not ideally suitedfor the performance of low noise experiments, since it was operating at an energylevel below its normal operating range and some scattering resulted from themetal walls of the vacuum chamber. Also, the experiments were conducted overan extended period of time resulting in difficulty maintaining control of the pro-cessing chemistry.

One would normally expect to see two trends in the data presented inTables 4, 5, and 6. As the density level increases for a fixed film and aperturesize, the granularity level should increase 1, and for a fixed film type and densitylevel the granularity should decrease as the aperture size increases 2. Thelatter of these trends occurs in the granularity data while the former does notstrictly occur (for the reasons discussed above).

Herz, R. H., The Photographic Action of Ionizing Radiations, Wiley-Interscience, New York, 1969, pp 140-141.This is due to the smoothing effects resulting from wider spread functionsof larger apertures.

14

13M 27A4 57A

1. 1.04 24 19.0 11.32. 1.55 38 32.0 14.53. 1.80 33 28.0 14.34. 1.90 31.6 14.55. 2.75 29.3 22.0 12.3

Table 4. RMS Granularity R (SC), 8 keV

Aperture DiameterFilm Density Micro-Analyzer

13m 2 7 / 57m

1. 1.18 25 25.3 142. 1.46 26.6 20.6 12.03. 1.60 29.0 23 13.04. 2.06 33.0 24 14.65. 2.60 35 26 14.0

Table 5. RMS Granularity R (DC), 8 keV


1314 27. 5 7,

1. .99 26.6 19.3 11.02. 1.46 29.3 22.0 12.63. 2.06 37.0 25.3 13.04. 2.18 30.3 24.3 13.3

Table 6. RMS Granularity M, 8 keV

15

B. QUANTUM MOTTLE

1. RMS Granularity: The sixty-five radiographs generated in this phaseof the program were analyzed under varying scanning aperture conditions. Allwere analyzed for RMS granularity with 27 um and 57 pm aperture scans usingthe Micro-Analyzer and most were analyzed using a 13upm aperture. As was pre-viously noted several of the radiographs were digitized on the Optronics scannerusing the 12.5upm, 25 pum, and 50 pm scanning apertures. The RMS granularityfor the Micro-Analyzer generated data is seen in Tables 7 through 20. Whenappropriate the associated RMS granularity values as computed from the Optronicsscanner are included.

The number preceding the slash in the Optronics sections of Tables 7through 20 is one thousand times the standard deviation of the digital valuesread from the Optronics scanner. The number following the slash is the digitalstandard deviation multiplied by a constant and relates the standard deviationto an effective RMS granularity value. The constant is one thousand times theslope of the effective calibration curve. The effective calibration of the Optronicsscanner for x-ray's is illustrated by the curve in Figure 5.

To more readily demonstrate the findings of the quantum mottle investi-gation data selected from Tables 7 through 20 have been presented graphicallyin Figures 6 through 14. In Figure 6 the effect of scanner aperture size andimage density is demonstrated for type R (SC) film. As was expected the mea-sured film granularity increased with decreasing spot size. For photographicfilms this increase in granularity with decreasing aperture size follows SelwynsLaw. For x-ray film Selwyn's Law is known not to hold. This figure alsodemonstrates that for all three apertures there is a nearly linear increase ingranularity with increasing density over the density range from 1.0 to 2.0 andthat the granularity takes a slight drop at densities in excess of 2.0.

In Figures 7 and 8 results similar to those found for R (SC) are demon-strated for type R (DC) and type M film. Both films show an increase in granu-larity with a decrease in aperture size. For all three apertures with both filmsthere is again a nearly linear increase in granularity with increasing densityover the 1.0 to 2.0 density range followed by a drop in granularity for densitiesgreater than 2.2.

Figure 9 demonstrates the relationship among the granularities measuredfor the three film types with the 13 micrometer spot. Type R (DC) demonstratesthe lowest granularity over the entire density range investigated. Type R (SC)film demonstrates a substantially higher granularity than R (DC) over the densityrange. Since both R-type films are made with the same emulsion, the differencein granularity measured for the two films is interesting to say the least. Thedifferent granularity for the two films is most likely the result of the difference inthe arrangement of the emulsion layers. Because of the complexity the situationwhere the scan aperture interogates emulsion layers separated by a distanceseveral times greater than the diameter of aperture, no model has been developedto explain the difference between the R type film in the granularity measurements.

16

Aperture Diameter

Film Density Micro-Analyzer Optronics13M 27m 57 M 12.5m 25m 50,

1. 1.10 49.0 33.6 21.0 11/39.6 8/28.8 5.0/182. 1.30 53.0 36.3 21.3 12/43.2 9/32.4 6.0/21.63. 1. 69 58.0 42. 3 26.0 11/39. 6 10/36.0 6. 5/23.44. 1.95 61.0 44.3 29.0 11/39/6 7.0/25.25. 2.35 56.0 41.3 25.0

Table 7. RMS Granularity M, 100 keV, Fe (. 25 inch)

Aperture DiameterFilm Density Micro-Analyzer Optronics

13A 2 7M 57M 12 . 5M 2 5M 50m

1. 1.03 46.3 34.6 21.0 6.5/2.34 7/25.22. 1.39 61 44 27.2 4/14.4 4/14.43. 1.70 65.0 46.6 29.7 8/28.8 8/28.8 6.0/21.64. 2.02 66.3 48 29.7 4.0/14.4 4.2/15.15. 2.58 48 36.6 22.0



13M 27, 57,u 12 .5p 25, 50M

1. 0.98 46.6 33.0 20.4 4/14.4 4/14.4 4/14.42. 1. 35 56.3 39.3 25.7 12/43.2 10/36.0 6.5/23.43. 1.81 59.0 40.6 27.4 11/38.6 11/39.6 7/25.24. 2.18 64.0 46.0 28.8 7/25.2 10. 5/37.8 6/21.65. 2.36 62.0 43.0 27.7


17

Aperture Diameter

Micro-Analyzer13/p 27m 571

1. 0.90 45.6 32.0 20.32. 1.56 58.6 43.3 26.33. 2.39 61.0 45.0 28.04. 2.93 35.0 26.3 15.6

Table 10. RMS Granularity M (with screens), 200 keV, Fe (. 25 inch)

Aperture DiameterFilm Density Micro-A nalyzer

27, 57m

1. 1.04 31.3 19.02. 1.45 41.0 23.03. 1.68 46.6 27.04. 2.08 56.3 33.05. 2.61 44.3 29.0

Table 11. RMS Granularity R (SC), 100 keV, Al (.25 inch)

Aperture DiameterFilm Density .Micro-Analyzer Optronics

13, 2 7m 57, 12. 5 m 2 5m 50 M

1. 1.00 40.3 33.0 19.02. 1.30 48.0 41.3 22.0 15.1/54.3 10.4/37.4 6.0/21.63. 1.79 63.0 52.0 30.6 13.6/48.9 12.1/43.5 14.1/50.44. 1.99 67.0 57.3 33 14.2/51.1 14.2/51.1 8/28.85. 2.66 63.0 53.0 30.6

Table 12. RMS Granularity R (SC), 100 keV, Al (. 50 inch)

18


27p 57m

1. 1.11 31.3 19.62. 1.31 39.6 24.33. 1.63 45.6 27.04. 2.10 58.0 405. 2.59 48.0 30



13p 274 57M

1. .97 38.0 33.0 19.72. 1.30 45.0 39.3 21.53. 1.77 58.0 52.6 29.34. 1.91 58.0 54.0 30.65. 2.30 65.0 59.0 33.1



27. 57m

1. 1.00 24.6 15.02. 1.41 29.3 16.63. 1.60 30.6 17.04. 1.95 31.6 20.65. 2.64 26.7 17.0

Table 15. RMS Granularity R (DC), 100 keV, Al (.25 inch)

19


13m 27A 57.

1. .99 35.6 23.0 14.62. 1.31 41.0 26.6 16.63. 1.67 46.0 28.6 17.64. -2.12 47.6 30 205. 2.41 46.6 29.0 19

Table 16. RMS Granularity R (DC), 100 keV, Al (. 50 inch)

Aperture Diameter

Micro-Analyzer

27,, 57P

1. .94 25.6 14.02. 1.38 31.0 18.03. 1.74 33.0 18.04. 1.94 34 22.05. 2.42 29.6 19.0

Table 17. RMS Granularity R (DC), 100 keV, Al (.75 inch)


13, 27, 57, 12.5 P 25,E 50,LA

1. .94 22.3 12.3 6/21.6 5/18.0 4/14.42. 1.27 26.6 14.6 9/32.4 7/25.2 5.0/18.03. 1.60 29.6 16.6 6/21.6 5.0/18.0 4/14.44. 1.97 31.3 17.0 8/28.85. 2.31 27.6 19.0

Table 18. RMS Granularity R (DC), 100 keV, Fe (.25 inch)

20


13M 2 7 M 57,

1. 1.07 35.6 24.6 15.92. 1.39 41.3 29 18.03. 1.72 44.3 32 19.24. 2.21 46.3 31.6 20.15. 2.45 44.0 33.0 18.3



1 3 m 27M 57A

1. 1.00 35.0 25.0 14.72. 1.40 43.6 30.3 19.73. 1.77 46.2 32.3 20.34. 2.14 47.6 33.6 20.15. 2.48 45.0 31.6 17.7


21

60-

OPTRONICS RANGE: 0 TO 3.OD

50

0 40

-r

-3

20-

0

40-

0 8 .

MACBETH - FILM DENSITY

Figure 5. Effective Calibration of MSFC's Scanner

-I-u-I

Figure 5. Effective Calibration of MSFC's Scanner

70- 13 am

60. 27 um

RMS 50Granularity 40

57 um30

20

10

1.0 1.5 2.0 2.5

Film Density

Figure 6. Comparison of RMS Granularity by Aperture for R (SC)100 keV, Al (. 50 inch)

50 , 13 pm

40RMS 27 m

Granularity 30_

57 pm20

10

01.0 1.5 2.0 2.5

Film Density

Figure 7. Comparison of RMS Granularity by Aperture for R (DC)200 keV, Al (. 50 inch)

23

60-13 pm

50 -RMS 27 gm

Granularity 40

30

20

10

1:0 1. 5 2.0 2. 5

Film Density

Figure 8. Comparison of RMS Granularity by Aperture for M,200 keV, Fe (. 25 inch)

7070- R (SC)

60 - M

50 R (DC)RMS

Granularity 40

30

20

10

01.0 1.5 2. 0 2:5

Film Density

Figure 9. Comparison of Film Types at 100 keV, 13 pm Aperture

24

The type M film, which is also a double coat film, is shown to have aslightly higher granularity than the R (SC) film at the lower image densities. How-ever, at the higher densities R (SC) has the higher granularity. Figures 10 and11 show similar results for the three films when measured with the 27 and 57micrometer apertures.

The data displayed in Figure 12 demonstrate that for radiographs pro-duced with x-ray energies varying from 100 to 300 keV on the MG-150 devicethere was no appreciable change in the granularity. Although only data forR (DC) film is displayed, similar results were found for other two films as canbe seen from the tables. Figure 13 illustrates that there is also no appreciableeffect upon the film granularity due to the specimen type or thicknesses underconsideration. Finally, the effect of lead intensifying screens is demonstratedin Figure 14. Although there were only a limited number of samples employingscreens (three samples below 2.5 density) there apparently is no significantchange in the granularity due to the lead screens.

2. Power Spectral Density Computations: The noise properties of selec-tive radiographs were investigated by means of the PSD. The PSD estimatesdepict the spatial frequency power density for the noise processes examined. Anin-depth examination of PSD can be found in Appendix III. The radiographs chosenfor PSD analysis include all three film types, type M, R (SC), and R (DC). Foreach film type radiographs taken at 100 keV, 200 keV, and 300 keV were examined.For film types R (SC) and R (DC) radiographs at two low density levels wereselected for analysis, and for film type M a range of density levels was chosen foreach of the three energy levels. Those radiographs under consideration arelisted in Table 21.

In presenting the results of the computations, the logarithms to the base10 of the PSD's' were computed and graphed as a function of spatial frequency.This was done so that the confidence intervals for the results could be given asstraight lines on the graph. The upper and lower confidence interval curvesare obtained by sliding the confidence interval lines over the computed resultswith the zero positioned on the curve and registering the values indicated atthe endpoints of the arrows. For all PSD's presented, the resolution intervalor bandwidth was approximately 5 cycles/mm. The results are presented inFigures 15-22. It should be noted that the PSD's are by definition normalizedby the variance of the process so that the area under the curve is one. (Inpractice the value will be close to one.)

The shapes of the curves in Figures 15-22 are similar regardless of filmtype, energy level, or density level. However, the absolute magnitude of thepower does change with the variance of the process. There is an apparenttrend showing up as a decrease in the PSD values with increasing spatial fre-quency. This can be explained by the data collection scheme. The radiographswere scanned on the Mead Micro-Analyzer with a 12.5 micrometer aperture and12 micrometer sampling interval.

25

60- R (SC)

50M

40RMS (DC)

Granularity 30

20

10

01.0 1.5 2.0 2.5

Film Density

Figure 10. Comparison of Film Types at 100 keV, 27 um Aperture

60

50

40RMS R (SC)

Granularity 30.M

20 - R (DC)

10-

01.0 1.5 2.0 2.5

Film Density

Figure 11. Comparison of Film Types at 100 keV, 57 um Aperture

26

o - 100 keVo - 200 keV

60 - A - 300 keV

5050 -

RMS 40 -Granularity 13 A"

20 -57pm

10 -

0

1.0 1.5 2.0 . 2.5Film Density

Figure 12. Comparison of RMS Granularity as a Function of keV for FilmType R (DC), for Aperture 13 um, 27 pm, 57 um

A - .25 inch Alo - .50 inch Al

50 o - .75 inch AlX- .25 inch Steel

40

RMSGranularity

30

10 57

1.0 1.5 2.0 2.5Film Density

Figure 13. Comparison of RMS Granularity for Type R (DC) 100 keV as aFunction of Specimen Material and Scanning Aperture

27

0 - Non-Screeno - Screen

70

60

RMS 50Granularity 4

40 -

30 a

20

10

1.0 1.5 2.0 2.5Film' Density

Figure 14. Effect of Intensifying Screens on RMS Granularity

28

Film Type keV Density Level

R (SC) 100 1.00R (SC) 100 1. 30R (DC) 100 .99R (DC) 100 1.31M 100 1.10M 100 1.30M 100 1.69M 100 1.95R (SC) 200 .97R (SC) 200 1.30R (DC) 200 1.07R (DC) 200 1.39M 200 1.03M 200 1.39M 200 1.70M 200 2.02M 200 2.58R (DC) 300 1.00R (DC) 300 1.40M 300 .98M 300 1.35M 300 1.81M 300 2.18

Table 21. Radiographs Selected for PSD Analysis

29

0.0

Confidence Intervals Density Level Variance

-. 4 _

95% 99% -1.30 .33

-. 8-

-1.2

Log PSD

-2.0- ".

-2.4-

0.0 4.17 8.33 12. 5 16: 67 20: 83 25. 00 29. 17 33'.33 37.'50 41. 0

Frequency (Cycles/mm)

Figure 15. PSD R (SC), 100 keV

0.0-

Confidence Intervals Density Levels Variance

-.4 _ *.99 .16

95% 99% 1.31 .22

-. 8

Log PSD

-1.2-

-1: 6" -----

-2.0

-2.4

-2.8I I I I I I I

0.0 4.17 8.33 12.5 16.67 20.83 25.00 29.17 33.33 37.50 41.67


Figure 16. PSD R (DC), 100 keV

0.0


-. 4 1. 10 .29

1.30 .3595% 99% 1.69 .47

1. 95 .47

-. 8 ........... 35 .45

Log PSD

-1.2

-1.6 - " - "0 ..... ""' .

-2.0

-2.4

-2.80.0 4.17 8.33 12.5 16.67 20.83 25.00 29.17 33.33 37.50 41.67


Figure 17. PSD M, 100 keV

0.0


_ _ _ _ 4,.97 .1795% 99% 1.30 .28

-1.

Log PSD

-1. '

-2.

-2.

-2.8 , , I ,0.0 4.17 8.33 12. 5 16.67 20.83 25.0 29.17 33.33 37. 50 41.67


Figure 18. PSD R (SC), 200 keV

0.0-

Confidence Interval Density Level Variance

-. 4A 1.07 .16

95% 99% 1.39 .21

-. 8

-1.2

Log PSD

-1.6-o

-2.0-

-2.4

-2.80.0 4.17 8.33 12.5 16.'67 20 . 83 25.0 2. 17 33.33 37.50 41. 6t



0.0


-. 4 - 1.03 .29

1.39 .5195% 99% 1.70 .58

... - - 2.02 .57

-. 8 ......... ..... 2.58 .28

-1.2 -

-2.04-2.0

-2.4

0.0 4.17 8.33 12.5 16.67 20.83 25.0 29.17 33.33 37.50 41.67



0.0-


-. 4

SA 1.00 .1795% 99% - -- - - - 1.40 .23

-. 8

-1.2

Log PSD

-1. 6 -

-2.0

-2.4

-2.80.0 4.17 8.33 12.5 16.67 20.83 25.0 29.17 33.33 37.50 41.67



0.0 -


-. 4 - .98 .29

95% 99% 1.35 .40

1.81 .482.18 .50

-. 8

-1.2

og PSD

-1.6

-2.0

-2.4

0.0 4.17 8.33 12.5 16.67 20.83 25.0 29:17 33.33 37.50 41. 67



The 12.5 micrometer aperture has a transfer function associated with itof the form

a2J 1 (21e a)

where 9 is the spatial frequency,a is the radius of the aperture, and

J1 is a Bessel function of order 1.

The cutoff frequency (zero of the Bessel function) occurs at approximately1200.0

1.2/2a x 1000 mm or 12.5 = 96.0 cycles/mm. At 40.0 cycles/mm, the

transfer function will decrease to approximately 0.7 the value at 0 cycles/mm.The PSD of the radiograph noise process, after smoothing by the aperture, isthe product of the PSD of the radiograph process before smoothing and the squareof the magnitude of the transfer function of the spot:

G s (f) = G (F) IT (f)I 2

where G (f) is the PSD of the process after smoothing,GS (f) is the PSD of the process before smoothing, andT (f) is the transfer function of the spot.

The square of .7 is approximately .5. An examination of the smoothedPSD estimates shows that the values at 40 cycles/mm vary from approximately.2 to .7 times the values of the PSD's at very low frequencies. Taking theindividual sample records as a group, the value of .5 probably represents agood approximation. In fact, using the log PSD values of -1.5 and -1.8, theratio of the high to the low frequency PSD values is .5. Based on these results,it can reasonably be determined that before smoothing by the aperture the radio-graphic noise structure is indicative of white noise which is represented by aflat spectrum.

C. SCANNER NOISE

The noise properties of MSFC's Optronics scanner were investigatedby operating the scanner with neutral-density (ND) filters as image specimensand recording the resulting digital information on magnetic tape. Both the magni-tude and the power spectrum of the noise process present in the digital datawere computed.

1. RMS Granularity Computation : The digital data collected from thescanner by digitizing the ten ND filters with apertures of 12.5 um, 25pm, and50pm were statistically analyzed to compute the mean digital step level and thestandard deviation of the digital readouts per filter. The standard deviation wasmultiplied by one thousand in order to give a measure comparable to RMS granu-larity. Also, since this scanner records only sixty-four distinct density levels,whereas the Micro-Analyzer records two hundred-fifty-six levels, the standarddeviation of the scanner noise process was converted to an effective RMS granu-larity number by multiplication by 4.25. This constant represents the slope ofthe scanner response curve, as the scanner response is compared against theND filters.

38

The results of the statistical analysis of the scanner noise process foreach aperture setting for each ND filter are shown in Table 22. The RMS granularitynumbers generally follow the trend of decreasing with increased aperturegiven density level and increasing with increasing density for a given aperture.The exceptions of these trends can generally be regarded as points of indecisionfor the scanning device. That is, points where the ND filter level coincides withthe boundary of one of the sixty-four selectable densities and where the devicemust make a continuing choice between two equally likely levels. The trendspossessed by the scanner noise process can more readily be seen in the graphicalrepresentation of Figure 23.

2. PSD Computation: The noise properties of the scanner were addi-tionally investigated through the computation of the power spectral density ateach ND level. The results obtained by these computations are shown in Figures24 and 25. It is obvious that there are large fluctuations present in the curves;and that some points of the graphs are off the scale. An examination of the PSD'salso indicates large power at low frequencies, 0-5 cycles/mm. An examinationof the results clearly shows that frequency periodicities were present in thedata. These periodicities were not only showing up as large PSD values in thelow frequency range but were also probably responsible for some of the extremefluctuations in the PSD's. The density values of one of the records analyzedwere examined and a dominant low frequency component was found. With thisinformation Optronics scans of one of the granularity patches that was made atMSFC were analyzed for PSD. The parameters selected were film type M, energylevel 100 keV, and density level 1.69D. The resulting PSD curves for threerecords of data can be found in Figure 26.

The same area on the granularity patch had previously been scanned bythe Mead Micro-Analyzer and analyzed for PSD. The results of that analysisare shown in Figure 17. Comparison of these results with the ND filter analysismakes the following facts apparent: there is a similar large power at low fre-quencies in the PSD's obtained from an analysis of the Optronics noise charac-teristics and the PSD's obtained from the Optronics scan of the granularitypatch but not in the PSD's obtained from the Mead Micro-Analyzer. This simi-larity in behavior rules out the possibility that the cause of the high power atlow frequency is due to the ND filters. The only remaining feasible alterna-tive is that the Optronics scanner was not operating correctly and was contamin-ating the data by generating low frequency periodicities.

D. IMAGE DEGRADATION

The image degradation of the entire x-ray/scanner system was computedby examining the system's MTF as measured by edge gradient analysis. Acomplete description of the techniques employed in this type analysis can befound in Appendix IV.

1. Edge Selection: The edge target provided for seven edges as candi-dates for analysis. These edges can be identified by a lower density step leveland an upper density step level. The identification chosen for these edges ispresented in Table 23.

39

Optronics Digital 1000 x Standard RMSFilter Density Mean Value Deviation Granularity

12.5.um 25u m 50,u pm 12.5 pm 25 um 50 pm 12.5,um 25 um 50,u m

0.18 0.8 2.0 3.0 2 0 0 9.5 0 00.48 6.8 8.4 9.0 3 1.75 0 12.75 7.44 00.82 14.6 16.7 17.0 2 1.5 1 9.5 6.38 4.250.98 17.9 19.8 20.3 3 2.25 3.25 12.75 5.06 13.81.16 20.4 22.9 24.0 3-3/4 3 1 15.94 12.75 4.251.50 28.2 30.5 2.5 4.25 14.9 19.131.76 35.5 37.0 38.0 4.5 3.3 3.0 19.13 14.03 12.751.94 41.1 42.5 43.0 4.0 3.0 2.0 17 12.75 8.52.19 44.7 47.4 48.1 3.3 3.7 3.9 14.03 15.73 16.582.39 51.1 53.1 53.0 3.0 3.7 4.0 12.75 15.73 17

Table 22. MSFC Optronics Noise Process RMS Granularity

12.5 ,um

18

25 m14

10 -50

RMSGranularity

6

2

0.1 0.5 1.0 1.5 2.0

Neutral Density

Figure 23. Graphical Display of MSFC Scanner Noise Magnitude

0.0Density Level Variance

Confidence Intervals .18 .02-.4 .48 .13

.82 .48

95% 99% 1.12 .16................ 1.50 .29

-. 8

-1.2 i

Log PSD

-1.6 I

-2.0

-2.4

-2.80.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40.0


Figure 24. PSD MSFC's Scanner Density Levels, .18, .45, .82, 1.12, 1.50(12. 5 unm Aperture)

0.0

Density Level Variance

Confidence Intervals - 1.76 .292.47 .25

-. 4 1 2.33 .33

95% 99%

-. 8

-1.2

og PSD \

-2.0

-2.4

-2.80.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40.0


Figure 25. PSD MSFC's Scanner Density Levels 1.76, 2.47, 2.33(12. 5 um Aperture)

0.0

Confidence Intervals

-. 4

95% 1 99% Density Level = 1.67

-.8

-1.2 -

-1.6 -

Log PSD

-2.0 -

-2.4

-2. 80.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40.0


Figure 26. PSD of Type M Film, 100 keV Scanned at MSFC- - (12.5 pm Aperture)

Edge No. Lower Step Upper Step

1 1 22 2 33 3 44 4 55 5 66 6 77 7 8

Table 23. Edge Identification

The nine exposures generated for MTF analysis presented a total of sixty-three edges for analysis. However, many of the edges had density differencesso low that the Optronics scanner was unable to detect the edge response. Theresult was twenty-six analyzable edges. The particular edges which weredetected by the scanner are found in Table 24.

Exposure Edge No.Identification 1 2 3 4 5 6 7

100-2-2.5 X X X70-3-2 X X X200-1-1 X X X100-1-4.2 X X X100-3-1 X X X200-1-1.5 X X X80-3-2 X X X100-1-4 X X100-3-1.5 X X X

Table 24. Edge Responses Suitable for Analysis

2. MTF Computation: From each of the twenty-six edge responses arepresentative MTF was computed. Each representative MTF was the averageof six MTF's computed from various locations on the edge. Approximately onehundred digital data records were collected from three positions on the edge.Each of the sets of one hundred were divided into two groups of fifty adjacentrecords. Each group of fifty records presented data for one MTF estimate.Hence, for each edge six MTF estimates were computed. The six MTF estimatesper edge were then averaged yielding a representative MTF for the edge. Thetwenty-six representative MTF's can be found in Table 25.

All of the computed MTF's fell into the narrow envelope shown in Figure27. Thus, regardless of the film type, energy level, exposure time or formatposition under consideration, the MTF for the system can be found to lie in theenvelope. The narrowness of the envelope indicates that the MTF of the x-ray/scanner system is relatively independent of the variables considered in theexperiment.

45

Modulation Modulation ModulationFrequency Transfer Transfer Transfer

Edge No. 3 Edge No. 5 Edge No. 7

0 1.00 1.00 1.002 0.91 0.93 0.894 0.71 0.80 0.676 0.52 0.66 0.448 0.33 0.48 0.24

10 0.16 0.25 0.1412 0.07 0.07 0.0814 0.06 0.14 0.0316 0.06 0.13 0.05

25a. Exposure 80-3-2



0 1.00 1.00 1.002 0.91 0.95 0.914 0.73 0.82 0.916 0.54 0.67 0.408 0.34 0.49 0.19

10 0.16 0.29 0.0912 0.06 0.14 0.1114 0.03 0.12 0.1016 0.04 0.12 0.07

25b. Exposure 100-2-2.5



0 1.00 1.00 1.002 0.93 0.94 0.944 0.77 0.78 0.816 0.58 0.58 0.658 0.36 0.37 0.47

10 0.15 0.19 0.2612 0.07 0.09 0.0914 0.08 0.10 0.1216 0.08 0.10 0.12

25c. Exposure 70-3-2

Table 25. MTF's for Twenty-Six Edges(1 of 3)

46



0 1.00 1.00 1.002 0.88 0.87 0.884 0.71 0.59 0.616 0.53 0.35 0.408 0.32 0.25 0.30

10 0.16 0.19 0.1812 0.13 0.20 0.0814 0.08 0.19 0.0916 0.08 0.15 0.13

25d. Exposure 200-1-1



0 1.00 1.00 1.002 0.91 0.90 0.924 0.70 0.70 0.726 0.46 0.56 0.468 0.27 0.45 0.24

10 0.14 0.26 0.1012 0.06 0.08 0.0714 0.06 0.11 0.0716 0.06 0.08 0.06

25e. Exposure 100-1-4.2



0 1.00 1.00 1.002 0.94 0.91 0.934 0.78 0.68 0.736 0.55 0.40 0.488 0.31 0.19 0.25

10 0.11 0.11 0.1112 0.06 0.10 0.0914 0.08 0.08 0.1016 0.06 0.08 0.08

25f. Exposure 100-1-4


47



0 1.00 1.00 1.002 0.92 0.90 0.904 0.71 0.70 0.666 0.50 0.56 0.458 0.31 0.44 0.32

10 0.16 0.26 0.1812 0.07 0.18 0.1214 0.06 0.21 0.0816 0.09 0.16 0.07

25g. Exposure 100-3-1



0 1.00 1.00 1.002 0.90 0.90 0.904. 0.68 0.68 0.696 0.46 0.46 0.388 0.29 0.29 0.29

10 0.16 0.16 0.1312 0.08 0.08 0.0914 0.05 0.05 0.1016 0.04 0.04 0.09

25h. Exposure 200-1-1. 5



0 1.00 1.00 1.002 0.92 0.93 0.884 0.73 0.77 0.636 0.50 0.61 0.418 0.29 0.44 0.25

10 0.15 0.24 0.1212 0.05 0.10 0.0814 0.07 0.12 0.0716 0.06 0.11 0.05

25i. Exposure 100-3-1. 5


48

1.0

0.8

0.6

0.4

Modulation

0.2

I I I I I I I I

2 4 6 8 10 12 14 16

Figure 27. Envelope Curve of MTF's

SECTION IV

CONCLUSIONS & RECOMMENDATIONS

A. CONCLUSIONS

This program investigated four areas to be considered by MSFC whenengaging in digital image enhancement of weld flaws through the applicationof digital filtering techniques to radiographic imagery scanned on the Optronicsmicrodensitometer. Three of these areas film granularity, quantum mottleand the scanning process were considered as potential noise sources andquantified by RMS granularity numbers and/or the process' power spectrum(PSD). The fourth area of investigation was the image degradation introducedby the combination of x-ray source, film and scanning device. The performanceof these combined elements was measured through computation of the system'sMTF.

Quantitative analysis of these four areas has indicated the following:

1. For a fixed film type and density level, the film graininessdecreased as the aperture size increased.

2. For a fixed film type and aperture size, the film graininesswas seen to increase as density increased up to approximately2.0D where graininess began to decrease.

3. Type R (DC) film demonstrated the lowest RMS granularitylevel over the specified density range.

4. No appreciable variation in RMS granularity of quantummottle (including film grain) was detected for x-ray energiesin the 100 to 300 keV range on the MG-150 device.

5. RMS granularity for quantum mottle (including film grain)at a specific density level did not change as a result of varyingspecimen thickness or material type.

6. The quantum mottle (including film grain) noise processfor each film type was approximately "white".

7. Trends or low spatial frequencies were introduced intodigital data by the Optronics scanner.

8. The sixty-four levels of digital output from the Optronicsscanner were not sufficient to distinguish the low modula-tion edges or flaws.

9. The family of MTF's generated under varying conditionsfell into a relatively narrow envelope indicating generalindependence of the system MTF on film type, energy leveland format position.

50

B. RECOMMENDATIONS

For a particular application of filtering to radiograph enhancement,itmay be possible to reduce the effect of the film grain noise by bandpass spatialfiltering. This could be accomplished by suppressing high spatial frequenciesin the data. Great care should be taken since such "low-pass" spatial filteringwill also blur the image, which is usually undesirable. However, it is quitepossible to suppress noise from a radiograph by deleting a selected band ofspatial frequencies while leaving enough at high frequencies to maintain sharp

edges. Since there is no dominant band of spatial frequencies in the filmgrain noise process investigated, the effect of this type of filtering wouldhave to be related to the image of interest and type of information desired.

The trends or low frequencies introduced by the Optronics scanner present

special problems. Any attempt to filter out these low frequency components would

distort most images beyond recognition. Removal of such trends, if possible,would have to be done spatially on a point by point basis. This, in turn, re-quires exact knowledge of the trend introduced by the scanner.

The MTF curve envelope indicates that image enhancement through arestoration process could be accomplished through the application of a singlefilter independent of the film type, energy level or format position within the

ranges considered in this study. A representative MTF belonging to the gener-

ated family could be employed for restoration.

51

SECTION V

BIBLIOGRAPHY

1. Blackman, R. B. and Tukey, J. W., The Measurement of Power Spectra,

Dover Publishing Co., New York, 1958.

2. Clark, G. L., Applied X-Rays (Fourth Edition), McGraw-Hill Book Co.,

New York, 1955.

3. Radiography in Modern Industry (Third Edition), Eastman-Kodak Co.,Rochester, 1969.

4. Herz, R. H., The Radiographic Action of Ionizing Radiations, Wiley-Interscience, New York, 1969.

5. Hinsley, J. F., Nondestructive Testing, Gordon and Breach Science

Publishers Inc., New York, 1959.

6. Jenkins, G. M. and Wahs, D. G., Spectral Analysis and Its Applications,Holden-Day, San Francisco, 1968.

7. Morgan, R. H. and Corrigan, K. E., (editors), Handbook of Radiology,The Leer Book Publishers Inc., Chicago, 1955.

8. Rickmers, A. D. and Todd, H. N., Statistics: An Introduction, McGraw-

Hill Book Co., New York, 1967.

9. Scott, W. G. (editor), Planning Guide for Radiologic Installations (Second

Edition), The Williams and Wilkins Co., Baltimore, 1966.

52

APPENDIX I

MEAD MICRO-ANALYZER LENS AND FOCUS SELECTION

A. FOCUS CRITERION

In order to establish the scanning conditions for the granularity analysis,test scans were made on a single piece of type M film which was processed atMSFC in an automatic film processor. As part of the test two lens sets were used,specifically Ul/L1 and U3/L3, where U1 is a Bausch and Lomb (B&L) 40 mmfocal length; .08 numerical aperture lens; L1 is a B&L 48 mm, .08 lens; U3 is aB&L 215 mm, .25 lens; and L3 is a B&L 215 mm, .20 lens. The U3/L3 combina-tion provided a depth of focus less than the film thickness while U1/L1 provideda depth of focus on the order of the film thickness. Both lens sets were employ-ed in making several scans of a specific area on the film. Each scan was per-formed under various focusing conditions consisting of:

Focus on lower emulsionFocus on upper emulsionFocus midway between emulsionsFocus for maximum response

The criterion used to evaluate the optics and focusing techniques was theRMS granularity determined from scanning the control radiograph. The opticsand focusing technique which yielded the maximum RMS granularity was assumedbest for the analysis of the graininess and quantum mottle radiographs.

B. LENS COMBINATION SELECTION

For the short depth of focus lens combination U3/L3, the maximum RMSgranularity was found to occur with the focus on the top emulsion layer. Lenscombination U1/Li, with its greater depth of focus, was unable to distinguishbetween the two emulsion layers. However, the Ul/L1 lens combination pro-duced maximum granularity values equal to the maximum values achieved bythe U3/L3 lens combination. Since the larger depth of focus of the U1/L1 lensprovided a greater range of focus positions where maximum response could beobtained, it was selected for scanning of granularity and quantum mottle ex-periments.

To assure proper scanning of each radiograph the Micro-Analyzer shouldbe re-focused for each radiograph. Because of the large number of radiographswhich had to be scanned, a rapid method was required for accurate focusing.From several tests it was found that the optimum focus could be located by ob-serving the analog output of the Micro-Analyzer on a paper chart recorder. Thisprocedure was implemented in scanning the radiographs for the film grain andquantum mottle examinations.

53

APPENDIX II

RMS FILM GRANULARITY

A. MEAN AND VARIANCE OF A RANDOM VARIABLE

Given an experiment with a set S of outcomes, if we assign to each element

a of S , according to some rule, the number X , we obtain the function X(a)

The functional relationship between elements of S and the corresponding numbers

X is called a random variable. One of the most important properties of a random

variable X is its central tendency, called the expected value or mean of X , de-

noted by PX and defined by

oo

X = E[X] =f xf(x)dx (1)

where flx)is the probability density function (PDF) of the random variable X and the

PDF at x represents the relative frequency of occurrence of the event a such

that X (a) = x . This definition of the mean is essentially equivalent to the concept

of average given byN

lir 1/N E X(c.)N- i=1

Another important quality of a random variable X is the spread of values about

the mean, called the variance of X , denoted by a2 , and defined by

2 = E[(X - pX) 2 ] (2)

The standard deviation of the random variable X , denoted by oX , is defined to be

the square root of the variance:

aX X (3)

Given an experiment and the associated probability space S of outcomes, to

every outcome a we can assign, according to some rule, a time or distance function

denoted by X (t, a). Note that for a fixed a., X(t, a)is a single time or distance function.

The ensemble or family of all these time or distance functions, for all elements a of

S , is called a random process or stochastic process and is denoted by X(t).

54

One of the most important properties of a random process is its central tendency,

called the expected value or mean of the process. The mean of a process is generally

dependent on time and is defined by

X (t) = E[X (t) ] (4)

Thus, the mean of the process at t is defined to be the mean of the random variable

X(t) defined by Eq. (1). The variance of the random process X(t) , denoted by

2X(t) , is defined by

o'(t) = E{[X(t) - X (t)]2} (5)

Hence, the variance of the process at t is defined to be the variance of the random

variable X (t) , defined by Eq. (2).

Another important property of a random process X lt) is the correlation between

values of the process at two different times. The autocorrelation of X (t) , denoted by

p (t1 , t2) is defined by

P(tlt 2 ) = E{[X(tl) - 1X(t1)][X(t2) - PX(t2)11 (6)

A random process X (t) is said to be "stationary" in the wide sense, or weakly

stationary, if the mean of the process is constant and the autocorrelation depends only

on time differences. This essentially means that the common first-order statistics

of the random variables X (t) are the same for all t . Thus, the mean and variance

of a weakly stationary process X(t) can be calculated using Equations (4) and (5),

respectively, with only a single value of t .

In general, a stationary random process is said to be "ergodic" if all of its statis-

tics can be determined from a single time record X (t, 0O) . The mean and variance

are often defined for a single time record by

T9 = lim 1/2T f X(t)dt

T -T

T

02 = lim 1/2T f [X(t) - p] 2 dtT- -T

55

Although p and G2 are actually random variables, it can be shown that

E[] = pX, E[(, - X 0, E[2 = a, E[(02 - G2)] = 0

In the discrete case, p and 02 are estimated from a single time record by

N1/N Z x.

i=1 (7)

N2 (I/N-i) Z [(x - )2]

i=1 (8)

where N is the number of samples in the record and x i = X (iAt)

56

B. DATA SAMPLING

1. Independence of Sample Points:

In order to obtain the statistics of an ergodic random process, we must gener-

ally be satisfied with estimates of the statistics. These estimates are based on analyz-

ing one or more discrete sample functions. For example, in order to estimate the

mean of the process, we use the formula given by Equation (7). One of the most im-

portant properties of such an estimate is its accuracy, i. e., the numerical proximity

of the estimated statistic to the actual statistic.

Generally the accuracy of an estimate improves with increasing sampling size.

But particular caution must be exercized in utilizing this concept in the estimation of

the first-order statistics (mean and variance). The accuracy of these statistics is a

function of the number of statistically independent samples.

With respect to the grain noise process, statistical dependence of sample points

arises in two ways. First, if the distance between two adjacent samples is smaller than

the diameter of the scanning aperture, these samples will be dependent, since each of

the densities arose partly from the same area on the film sample. Second, if the dis-

tance between two adjacent samples is smaller than the width of the grain noise auto-

correlation function, these samples will be dependent, by definition of the autocorrelation

function.

In determining the sample size for the desired accuracy of first-order statistical

estimates, the two sources of statistical dependence noted above must be considered.

For example, if N independent samples are required and the sampling interval is

half the aperture diameter, approximately 2N such samples must be taken. Further,

if the sampling interval is one-fifth the width of the autocorrelation function, at least

5N such samples must be taken.

2. Sampling Distributions:

If we consider a discrete sample function (time series) from an ergodic random

process, we may look upon this sample of size N as observations of N random

57

variables. Moreover, any quantity computed from these sample values is also a ran-

dom variable. For example, the estimate of the mean, given by Equation (7), is a

random variable, since this estimate represents the sum of N random variables. The

probability distribution of a sample statistic such as the sample mean is called a

sampling distribution.

A number of sampling distributions naturally arise in practice. One of these is

the well-known normal (Gaussian) distribution. Another is the Chi-Square distribution

with n degrees of freedom, x , defined as the sum of the squares of n+1 indepen-

dent random variables, where each random variable has zero mean and unit variance.

Another common sampling distribution is the Student t distribution with n degrees

of freedom, tn , defined by

t n = ZI'I-n

where Z and Y are independent random variables such that Y has a Xn2 dis-

tribution and Z is normally distributed with zero mean and unit variance.

3. Sampling For the Mean:

Given the grain noise sample X = {x,x 2 ,.. . ,xN } , we can obtain estimates

of the mean and of the variance, OX and 2 , by equations (7) and (8). Let

us define

tn = X - x)N/X

where pX is the actual mean of the grain noise process. Since pX and 8X are

random variables, tn is also a random variable. Further, drawing on the Central

Limit Theorem, 1X has a normal distribution; hence, tn has a Student t dis-

tribution. The probability density function of tn as a function of n and a , has

been tabulated in such a way that

prob {St n tn < St 1 = 1 - a (9)no1-b/2 -5n - n; /2

58

In Eq. (9) prob denotes probability of occurrence and St denotes the tabulated Student

t distribution. The interval denoted in the braces of equation (9) is called the

(1-a) x 100% confidence interval on t n , since the probability that t n lies in this interval

is 1-a,

Since the probability density function of the Student t distribution is symmetric,

we may denote the C1--a) x 100% confidence interval by

-k(n,a) < t < k(n, a) (10)-n-

wherek (n, a)=Stn; 1-a/2=Stn; a/2" Manipulating the inequalities in (10), we obtain

(Ix -__x)___

-k (n, a) < X < k (n, a)

OX

k (n, a) X k (n,a)oXX TX < X +

(11)

Further, if we assume n (and hence N ) is large, k (n, a) becomes a function of a

only; so we may write (11) as

kl) k) x (12)1X -V X < 1X + (12)

The inequality in (12) means that the probability is 1- a that the true mean will be

contained in the interval

k(a) a X1IX ±

The computation required to determine the sample size N necessary to give a

reasonable confidence interval for the mean can now be illustrated. For this purpose

we will assume that aX = aX and that we seek an interval of length 0.01, centered

at "X , such that this interval will contain 1X at least 90% of the time. Thus,

N satisfies the inequality

k (0. 1) yx< 0.005 (13)

59

Solving (13) for N , we obtain

N > (1.645)20/0.0052

wherek (0. 1)=1. 645 For example, with a grain noise process where aX=O. 05

at X , N > 271 is required for a 90% confidence interval of 0.01 about iX

4. Sampling for the Standard Deviation:

Let us define the parameter X2 byn

2= nX/oX

where again n=N - 1. Note that 4 has a Chi-Square distribution with n de-

grees of freedom. It can be shown that the parameter V27 tends to a normal

distribution for n > 30 with mean v72 - 1 and unit variance. Thus, the variable

Gn , defined by

2na2X

G v= - n- 1

n X

has a normal sampling distribution with zero mean and unit variance. Thus, utilizing

the tabulated values of the normal probability density function, we may say that

prob {-k(o) < G < k()} = 1 -a (14)- n-

where k (c/2)

1//2/ f exp(-t 2/2)dt = 1 --k (a/2)

Thus, the interval denoted in (14) represents the (1-a) x 100% confidence interval

on G . Rearranging the description of this interval we obtain

X

Y2n - 1 - k(a) <X 2 -n- 1 + k ()< - < (15)V' - aX - -

60

The computation required to determine the sample size N necessary to give a

reasonable confidence interval for the standard deviation can now be illustrated. As

before, we let X=0. 05 . We further assume that n is so large that V2--1 " 2-n

Lastly, we assume that it is desired to have a confidence interval about CX such

that oX will be within 3% of o*X 90% of the time. Hence the value of N (or n) re-

quired satisfies the inequality

0.97 < k(0.1)

0.97 < 72n - 1.645--

0.97 2n < 2n - 1.645

1.645 < 0. 03 2

2-n > 55

2n > 3025

n > 1513

61

C. PRACTICAL CONSIDERATIONS

In the estimation of RMS film granularity, care must be taken in preparing

the sample. The test film should be uniformly exposed and the processed film should

be free of scratches, pinholes, and developer mottle. In practice, some deviations

from such an ideal sample often occur. At Mead Technology Laboratories, any bias

in granularity estimates induced by flaws in the film sample is minimized by the

techniques employed by the analysis program (GRANSTAN). The theory behind the

techniques used by GRANSTAN are detailed below.

1. Density Wedging:

One of the most frequently occurring defects in granularity samples is density

wedging, whereby a variation in exposure across the sample causes a trend in density

across the sample. Although careful exposure of the sample can minimize the magni-

tude of this wedging, some residual trends invariably occur. The technique employed

by the GRANSTAN program minimizes the effects of such residual trends. This techni-

que is based on estimating the variance of the grain noise process from the variance of

the process defined by each independent adjacent difference (IAD) in the time record.

Let us denote a grain noise process by X , where we seek jX and o 2

Suppose we are estimating pX and oX from a time record R={x 1 ,x 2 ,... , N } ,

where each x. is a realization of X x . and xi+1 statistically independent

for each i=1,2,...,N-1 .

We assume that linear density wedging occurs, so that Ad.=k(i-1) is the bias11at each i . Hence, the actual time record available is S={x.+Ad.:i=,2, ... ,} ,

where k is unknown and is the slope of the density variation across the sample.

Let us define a new process Y , where each realization of Y is the IAD of

realizations of X . Hence, given the set of realizations S of X , the set of

realizations T of y is given by T={D.:i=1, 2,....,N/2} , where

D.i = (x2i + Ad) - (x 2 + Ad 2i )

62

We will show that the variance of Y is precisely twice the variance of X , re-

gardless of the magnitude of k.

According to Equation (1),

Iy = E[Y]

Further,

i = x2i - x2i-1 + Ad2i - Ad2i-1 (16)

and

Ad2i - Ad2i-1 = k (17)

Substituting (17) in (16) and noting that the expected value of each realization of X is

PX , we have

yP = IX - "X + k = k (18)

According to Equation (2),

c2 = E[(Y - pX)2] (19)

Drawing on Equation (16) and (17) and the result in (18), we have02 = E[ (xi - x2i_1 + k - k) 2 ]

= E[x 2 i - 2x 2ix2i1 + x 2i -

= E[x2] - 2E[x 2i.x2i l] + E[xw2i]

(20)S= 2E[X2 ] - 2X

where we have used the assumption that adjacent samples are statistically independent.

Equation (20) can be manipulated in the following manner

63

= 2[E(X 2) - 2 +p2

= 2[E(X 2 ) - 2E(X)iX + p2]

= 2E[(X - IX)2 ]

Y X= (21)

Thus, the GRANSTAN program actually estimates aX by 2 Y

2. Random Flaws:

Random flaws, such as scratches and pinholes in the sample, while havinglittle effect on the computed mean, can have significant effect on the computed var-iance. These flaws generally manifest themselves as density values which wouldotherwise not occur. Hence, the GRANSTAN program, if so directed, will discardtypical densities in the sample. This editing is done on the basis of a histogram ofthe sample densities. If sampling was done in accordance with the guidelines pre-scribed above, the histogram of the sample will be more or less continuous. TheGRANSTAN program searches for "holes" in the histogram on either side of the mode.Three adjacent densities which do not occur in the sample are used to identify thedensity range of the sample material. Densities which occur outside this range arediscarded by the program before the sample statistics are estimated.

64

APPENDIX III

POWER SPECTRAL DENSITY

A. DEFINITIONS

The power spectrum of a stationary random process is defined as the

Fourier transform of the autocovariance of the process. The power spectral

density function for the process is defined as the power spectrum normalized

by the variance of the process. In mathematical terms the autocovariance of a

stationary random process X (t) is:

gXX(u) = E [(X(t) - p) (X(t+u) - U)] (1)

where E denotes the expectation operator. The power spectrum of the process

X(t) is:

G (f) = gXX (u) e -j2 fu du (2)

There is also an inverse transform relation given by:

gXX (u) = f GXX (f)e j2 rfu df (3)-MD

Setting u = o in (3) gives:

gXX (o) =J G (f) df

and shows how variance or 'power' of the X(t) process is distributed over

frequency.

The spectral density function is:

G (f)XX R (u -j2 r fu (4)X2X

-0

where RXX (u) is the autocorrelation function of the random process X(t).

Since GXX(f)/ XX2 is non negative and GXX(f)/ a XX2 = 1 the term

density function is appropriate.

65

In the previous discussion we considered an ideal case in which the

stochastic process X (t) was defined for all time - - < t <- . Suppose weT

only have available a record x(t), - - t _ T/2, as one of many time series

that might have been observed, that is, as a realization of the process X(t). The

sample power spectrum may be defined as the Fourier transform of the sample

autocovariance function. The sample autocovariance is defined as:

ST- lulC ) = I (x(t)-x) (x(t+u)-x) dt o < lul < T

0, Jul > T (5)

The sample power spectrum is:

Pxx (f ) = xx (u) e -j2 rfudu, - - f < (6)

-T

The inverse Fourier transform of (6) may be written:

Cxx (u) = Pxx(f) eJ 2 7rfu df, -T < u T (7)

In discrete time, the sample spectrum is:

(x- 1)-j2 r fkP (f) = A C (k)e , -A<I f <JA (8)xx -C-xx

The inverse transform of (8) is:

1/2ACxx(u) = Pxx(f)e j2r fu df, -nA _ u < nA (9)

-1/2 A

B. SAMPLE SPECTRUM STATISTICS

We have regarded the record x(t), -T/2 5 t _ T/2 as a realization of a

stochastic process X(t). The variability in the sample record is characterized

by rv's X(t) for -T/2 5 t 5 T/2. The sample spectrum P (f) can be regard-

ed as a realization of the random variable PXX (f), just as the sample covariance

function CXX (u) is normally regarded as a realization of the random variable

66

CXX (u). An examination of the variability in the sample power spectrum indi-

cates some undesirable characteristics. Before discussing the nature of the prob-

lem, eq. (8) will be expressed in a different form. For the discrete case the

sample spectrum is often defined as:

P (f) = Xtexx tN

= C Xtcos 2 7 ft)) + ( Xtsin 2rft2

1/2A - f < 1/2A (10)

where A is the sampling interval, x(t) is a signal observed at times

t = -n& , -(n-1)A , ... , (n-1)A , and N = 2n.

It can be shown that eq's. (8) and (10) are equivalent. Eq. (10) is

usually referred to as a sample spectrum estimator. For a purely random dis-

crete process (discrete white noise) it can be shown that the variance of the

sample spectrum estimator is independent of the number of observations N.

However, the average of the sample spectrum is close to the theoretical value

of the spectrum. These results follow from the sampling properties of the sample

spectrum. To describe the sampling properties, we first consider the random

variables associated with the real and imaginary Fourier components of a dis-

crete purely normal process Zt , -n < t < n-1. These are

X-i

R(f) = Zt cos 2 ftA (11)e-'

I(f) = j Zt sin 2 rftA , -1/2A f < 1/2A (12)

The sample spectrum estimator is then:

PZZ (f) = A/N [R2 (f) + 12 (f)] , -1/2A f < 1/2A (13)

It can be shown that the following results hold:

67

1) The random variables

Y(fk) = 2PZZ (fk) , k = +1, +2, ... , +(n-1) (14)

2are distributed as 2 (chi square) random variables, where fk = k/NA .

2) When fk = 0 or fk = -1/2A , the random variables

Y(fk ) = ZZ (fk) (15)A oZ2

2are distributed as a x

1

3) The random variables Y(fk ) are mutually independent for K = 0,

+1, +2, ... , +(n-l), -n.

It follows from 1) - 3) that:

E PZZk)] 2 = G (fk) (16)ELPzz(fk)J = Z = GZZfk

which means that the sample spectrum at the harmonic frequencies is an unbiased

estimator of the spectrum for white noise. It also follows that:

Var PZZ 4 2 = GZZ 2 k) (17)

This shows that the variance of the estimator is a constant independent of the

sample size. It should be noted that even if the Zt process is not normal, the

random variables R(f) and I(f) will be very nearly normal by the Central Limit

Theorem. The distribution of Pzz (f) will be close to a X 2 regardless of

the distribution of the Zt process.

C. SMOOTHED SPECTRAL ESTIMATES

A general method has been developed for reducing the variance of the

sample spectrum estimator by smoothing the spectral estimator. The smoothed

spectral estimator is of the form:

PXX(f) = w(u) Cxx(u) e -j2fu du (18)

68

The function w (u) is referred to as a lag window and satisfies the conditions.

1) w(o) = 1

2) w(u) = w(-u)

3) w(o) = o, I u >T

In practice condition 3) is replaced by:

4) w(o) = o, lul >M, M<T

The Fourier transform W (f) of the lag window w (u) is referred to as a spec-

tral window. Using a convolution property, Eq. (18) may be written as:

PXX(f) = W(g) PXX (f-g) dg

which is a smoothing of the sample spectrum PXX (f) by the spectral window

W(f). Analogous to the properties for the lag window, W(f) satisfies the condi-

tions.

1) W (f) df = 1,

2) W(f) = W(-f),

3) W(f) is a slit with base width of order 2/M.

D. STATISTICS OF SMOOTHED SPECTRAL ESTIMATOR

We now examine the mean and variance of the smoothed spectral estimator.

Taking expectations in Eq. 19 gives:

0

E [Pxx (f)] = W(g) E [P X (f-g)] dg

For large T, E CPXX (g)] GXX (g), so

E [ XX (f)] " f W(g) GXX(f-g) dg = XX(f) (20)

GXX (f) is called the mean smoothed spectrum. The bias B (f) associated with

the smoothed spectral estimator is defined to be:

69

B(f) = E [PXX] - GXX = GXX ( f ) - GXX (f) (21)

It can be shown that to keep the bias small, M must be large. We will now show

that this is contrary to the requirement to keep Var [ XX ( f ) ] small. The

variance of a smoothed spectral estimator can be approximated by:

Var [ T f W (g) dg (22)

or

Var GX 2 (f) f W2

Var [ PXX T (u) du

For the Bartlett window this becomes:

Var [ x(f) ] GXX M ) ( 23)

This shows that the variance of the smoothed spectral estimator can be reduced

by making M small. However, reducing M increases the bias and causes

more distortion of the theoretical spectrum. In order to partially resolve this

conflict between bias and variance the level of discernible detail desired in the

smoothed spectral estimate must be decided upon. In the next section of this

report a method for choosing M is described based on the desired detail or

resolution.

Previously, it was stated that the sample spectrum estimator PXX ( f) is

such that 2PXX(f)/Gx (f) is approximately distributed as X2 . The corres-

ponding result for the smoothed spectral estimator is that:

PXX ( f ) / G XX ( f )

is approximately distributed as X where Y > 2. Since Pxx (f)/Gxx(f)2

is distributed according to a X , it follows that:

70

x (-) < G(f) v (1--) = - (24)

where

P X X ( -) -- and v =Pr 7 2 2 JW 2 (u) du

The interval between:

SP (f) Pxx (f) (25)x (1-(a/2)) ' x (a/2)

is a 100 (1- a ) % confidence interval for GXX (f). It is convenient to plot spectra

on a logarithmic scale, since then the confidence interval for the spectrum is

simply represented by a constant interval about the spectral estimate. It follows

from Eq. (25) that the confidence interval for log Gxx (f) is:

log P (f) + log ( log P (f) + log /) (26)xx (()) xx (a/2)

E. BANDWIDTH

TjW2 (u) du pro-It was shown in eq. (22) that the term S = -- (u) du pro-

vides a measure of the reduction in variance of the estimator due to smoothing

by a spectral window. In order to obtain a small variance it is necessary to make

S small. For a given window W this can be done by making M small.

Another useful characteristic of a window is its width. It can be shown that in

order to obtain a good estimate of a peak in a spectrum, the 'width' of a spectral

window must be of the same order as the width of the peak. The question arises

as to what is meant by width of a spectral window since it is non zero for most

f in the range -- < f < - . One way to define the width of a spectral window

is to make use of the motion of width or bandwidth of a 'bandpass' spectral

window. Considering the 'bandpass' spectral window:

W(f) = 1/h, -h/2 < f < h/2

we can see that the spectral window is rectangular in the frequency domain and

has a unique width h, that is, its bandwidth b = h. Using Eq. (22), the var-

iance of a smoothed spectral estimator based on this window is:

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Var Pxx(f) ] = GXX 2 (f)/Tb

Some authors have defined the bandwidth of the window as the width of the rec-

tangular window that gives the same variance. That is,

G xx(f) G (f) J W2 (u duVar [P xx ( f ) ] b w (u) du

which results in:

b = 1/ W2 (u) du

F. SPECTRAL ANALYSIS DESIGN

The basic requirements to be met in designing a spectral analysis calcu-

lation in advance of collecting the data are discussed below. The four basic re-

quirements are:

1) The sampling interval a must be chosen small enough so

that the spectral estimate can be obtained in the range of

interest o 5 f < f . Based on the sampling theorem this

requirement means that A _ 1/2f .

2) Aliasing must be avoided. This can be done in one of the

following two ways.

a) Choose A small enough so that the spectrum G(f)

is effectively zero for f > 1/2A . This method

requires initial knowledge of the spectrum and also

if fo is much less than the frequency beyond which

G (f) is effectively zero it may be necessary to collect

data at a much finer sampling interval than is necessary

to satisfy (1).

b) The 2nd method is to filter the signal before sampling

so that all frequencies above f are effectively removed.

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3) This requirement is concerned with the resolution desired

in the spectral estimate. If it is desired to "detect" detail of

width or more in the spectrum, then the truncation point

M should be chosen so that the bandwidth b is less than

a. For the Tukey window this means that b = 1. 33/M -< a

or M=LA _ 1.33/a or L 1.33/a .

4) For finite records the variance of the estimator will influ-

ence the amount of fine detail in the spectral estimate. Peaks

may be due to the variability of the spectral estimate. In

order to tie the estimate down to a given stability, it is

necessary to specify the number of degrees of freedom Y

desired with each estimate. The record length necessary to

obtain the specified value of Y is given by the formula:

T = /2a

or

N = /2a

where a was defined in (3). A confidence interval based on

the chi-squared distribution can be determined for the given

number of degrees of freedom. It should be noted that it may be

necessary to process very long records or large amounts of data

to obtain a narrow high confidence band (95%) about the spectral

estimate.

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APPENDIX IV

EDGE GRADIENT ANALYSIS

A. SYSTEM PERFORMANCE MEASURES

If an imaging system is linear, it can be characterized by its optical

transfer function (OTF) defined by:

OTF (f) =Ti(X) Cf (1)

where f is spatial frequency;

{ ( (f) denotes the Fourier transform operator defined by

acx) I(f) = (CX)exp(-27rifX) dX

i(x) represents an input function to the system (target);

8 (x) represents the corresponding output function of the system (image).

The OTF is generally a complex function. The system performance measure

usually reported is the (normalized) modulation transfer function defined by:

OTF (f)MTF (f) = (2)OTF(o) I

where denotes the absolute value function.

There are several items of interest in Equation (2). First, the MTF is

scaled so that MTF(0) = 1. Hence, at some spatial frequency fo, MTF (f o)

answers the following question:

Given a sine-wave input to the system of the form i(X) = a + b sin(2fo x),

i.e., with spatial frequency f and modulation b/a, by what factor

will the modulation be reduced by the system?

Second, if the system is truly linear, Equation (2) is valid for any target func-

tion having frequency content. But if we are interested in the MTF out to a

specific spatial frequency, the Fourier transform of the target should have positive

magnitude at that frequency.

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B. APPLICATION OF EDGES AS TARGETS

One of the most common target functions is the edge target, where

i(x) is the Heaviside function:

H (x) = 0 for X _0

1 for X > 0 (3)

Equation (2) cannot be used directly for edge analysis, since the Heaviside

function does not have a Fourier transform. In this case, we draw on the funda-

mental result of linear system theory, namely that 0 (X) is the convolution

of i(X) with the system impulse reponse (spread function), s(X). Equation

(2) may be written as:

MTF (f) = ( f (4)

where i (X) is some function possessing a Fourier transform and * is the

convolution operation defined by:

a--Mf( P) (- )dy (5)

Equation (4) may be written as:

MTF (f) = ;(X) (f) (X) (f)

MTF (f) = s (x) ) (f)l (6)

In the case of the edge target, 8 (x) is the convolution of s with the heavi-

side function. Therefore,

0(x) = s(x) * H(x) (7)

After some manipulations, Equation (7) may be written as:

6 () = x s () du (8)

And, differentiating both members of Equation (8), the following is obtained:

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And, differentiating both members of Equation (8), the following is obtained:

O'(X) = s(X) (9)

Hence, edge gradient analysis involves the use of Equations (9) and (6); that

is, differentiating the edge reponse to obtain the spread function, then Fourier

transforming the spread function to obtain the MTF. The differentiation step in-

volved in using these equations is generally avoided by the use of integration

by parts:

MTF (f)= I 0'(X) I (f) I

I 8O'(x) exp (-27ri fx) dx j (10)

I (x) exp (-2 ri fX) I + 2r ff (x) exp (-2rif x) dx

MTF (f) = ( exp (-2rifA) + 2rif f (x) exp (-2ri fx) dx I

where the interval from O to A contains the edge response 8(x). Equation

(10) is the fundamental equation of edge gradient analysis.

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, o , - i ''im par , 1, i,,,,;t .- '111 1,, .i, , 1 1 1 ... · , o , - i...

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