Pixel classification by morphological granulometric features

Rochester Institute of Technology

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5-1-1991

Pixel classification by morphological granulometric

features

John T. Newell

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Recommended Citation

Pixel Classification by Morphological Granulometric Features

by

John T. Newell, ill

Rochester Institute of Technology

Center for Imaging Science

May 1, 1991

A thesis submitted in partial fulfillment of the requirements for the degree of Master of

Science in the Center for Imaging Science in the College of Graphic Arts and Photography

of the Rochester Institute of Technology

Signature of Author _

Center for Imaging Science

Approved by

Mendi Vaez-Pravani

THESIS RELEASE PERMISSION FORM

ROCHESTER INSTITUTE OF TECHNOLOGY

COLLEGE OF GRAPHIC ARTS AND PHOTOGRAPHY

CENTER FOR IMAGING SCIENCE

Pixel Classification by Morphological Granulometric Features

I, JohnT. Newell, III, hereby grant permission to the Wallace Memorial Library of R.I.T.

to reproduce my thesis in whole orinpan. Any reproduction will not befor conunercial

use or profit.

Abstract

Pixelclassificationsystems_rely on a certain set offeaturesthatare sufficientto

classify a givenpixelintoa classdefinedwithinadatabase. Unlike brightnessand spectral signaturefeatures commonlyusedinremote_sensingapplications,texture-basedfeatures

cannotbe definedfora single pixel andmustbe derived froman area or window

surroundingthatpixel. Inthis research,allfeaturesarederived from_binarymorphological granulometries. Once_{generated, these}featurescomprise adatabasewhich can beusedto

classify images. A Gaussian Maximum Likelihood Classifier is trainedwiththisdatabase forsubsequentclassification ofboth dependentandindependent data. Severalaspects of

thesetexture-basefeaturesrequireinvestigation inordertodeterminetheir_abilityto

distinguish imagetextures. Three importantaspectsareaddressedinthis study; the effects

of maximumnoise,theoptimalsize ofthelocalizedwindow, andtheminimum number of optimalfeaturesrequiredforaccurate classification. Astatistical approachhas beentaken todetermine theclassification_accuracywith_varyingwindow size,varyingnumberof

features,and_varyingamounts offourtypesof maximumnoise _usinggranulometric

features. Analysisoftheseinvestigationsindicate fourmainresults. _First,classification

accuracy in theabsence of noiseis_{extremely high.} Second,forthese texturesatthespatial resolutionof75dpi,classification_{accuracy decreasesdramatically}belowawindowsizeof

11x11 pixels. _Third,thenumberoffeaturesneededfor highclassification_accuracycanbe

reducedtoa_fairlysmall number ontheorderof6features. _Finally,thesefeaturesare

generallyrobustinthepresenceofmaximumnoiseifthetypeand amount of noise canbe

Acknowledgements

I wouldliketo thankJeff B. Pelz for his _helpand computer programswhich were of

tremendous_helpin_completingthis thesiseventhough_theywere writtenin Pascal.

I wouldalsoliketo thank_WendyRosenblumfor heralgorithms,code and afineexample

of a well writtenthesis.

Finally, Iwishto thank Kaleen _Moriartyfor her immeasurable_friendship which hasseen

me throughtheheavenandhellofRIT. Thanks for_keepingme _smilingthrough thetough

Dedication

Thisthesisis dedicatedtoJohn T. _Newell,Jr.andAnne W. Newell fortheir_love,

confidence andsupportthroughout_my collegeeducationandfor_instillinginmethepride

Table of _Contents

TableofContents vii

ListofFigures ix

ListofTables xi

1.0 Introduction 1

1. 1 MorphologicalGranulometries 1

1.1.1 _Opening 1

1.1.2 Granulometries 3

1.1.3 LocalGranulometries 7

1.2 Image Texture 9

1.3 Image SegmentationUsingGranulometric Features 10

1.3.1 Segmentation 10

1.3.2 UseofGranulometricFeature for Segmentation 10

1.3.3 Higher OrderMoment Features 1 1

1.3.4 S_tructuringElementsandDerivationofOther

Granulometric Features 1 1

1.4 Image ClassificationandDiscriminant Analysis 13

1.4.1 Classification 13

1.4.2 Feature_ProbabilityDistributions 1 4

1.4.3 Maximum Likelihood Classification 1 5

1.4.4 Gaussian Maximum LikelihoodClassification 16

1.5 Minimal WindowSize 19

1.6 Optimal Feature Selection 20

1.6.1 FeatureReduction 20

1.6.2 Mahalanobis-LikeDistance Measure 20

1.6.3 Divergence Measure 22

1.7 Noise 27

1.7.1 MaximumNoise 27

1.7.2 Point Noise 27

1.7.3 OcclusionNoise 28

1.7.4 Scratch Noise 28

1.7.5 Spaghetti Noise 29

2.0 Statement of Work 30

2.1 SelectionofTexture Images 30

2.2 _ThresholdingofTexture Images 33

2.3 GenerationofNoise 36

2.4 GenerationandSelectionofLocalGranulometric Features 42 2.5 ClassificationofDependentandIndependent Data 44

3.0 Analysis of Results 46

3.1 Dependent Classification 46

3.2 Independent Classification 47

3.3 Minimal Window Size Determination 50

3.4 Optimal Feature Selection 52

3.5 ClassificationwithMaximumNoise 56

3.5.1 Dependent Classification 56

3.5.2 Independent Classification 57

3.5.3 CombinationsofNoise Models 63

3.6 Noise Estimation 75

4.0 Conclusions 79

4. 1 Suggestions for Future Work 81

5.0 References 83

Appendix A 86

List of _Figures

Figure 1: ImageS and_structuringelementE 2

Figure 2: Open _{(S,E); Opening}of_{image S by}

structuringelementE 2 Figure 3: Simulated_{binarygranulometry}resultantimages 5 Figure 4: _^(k), <P(k)and_d<D(k)fromthesimulatedimage_granulometry 6

Figure 5: Example_{dOx(k) probability distribution 8}

Figure 6: Feature Zvaluedistribution fortwoclasses 14

Figure 7: Maximum likelihooddecision_boundary 16

Figure 8: Feature setsforclass _separability 24

Figure9: Textureimages 31

Figure 9: _Binarytextureimages 34

Figure 10: Examplesof_binary noiseimages 40

Figure 12: Classification_Accuracyvs.Window Size 50 Figure 13: Classification_Accuracyvs. NumberofOptimalFeatures 53

Figure 14: Classification_Accuracyvs. % Point Noise 58 Figure 15: Classification_Accuracyvs. % Spaghetti Noise ₅₈

Figure 16: Classification_Accuracyvs. % OcclusionNoise 59

Figure 17: Classification_Accuracyvs. % Scratch Noise 59

Figure 18: Classification_Accuracyinthepresence of_Horizontal,Fixedand

Random Scratch Noise 62

Figure 19: Classification_Accuracy withcombinations ofnoise models 64

Figure 20: Feature distributions for Circular PSSD 67

Figure 22: Feature distributions forPositiveDiagonal PSM 68

Figure 23: _Probabilitydistributions for Circular PSSD 69

Figure 24: _Probabilitydistributions for Negative Diagonal PSSD 69

Figure 25: _Probabilitydistributions for Positive Diagonal PSM 70

Figure 26: Optimal Feature Classification in Point Noise 73

Figure27: Optimal Feature Classification in Spaghetti Noise 73

Figure 28: Optimal Feature Classification in Occlusion Noise 74

Figure 29: Optimal Feature Classification in Scratch Noise 74

List of Tables

Table 1: Classificationofdependentdata 46

Table 2: Classificationofindependentdata 48

Table 3: Classificationofindependent_{data using}pooled covariance 49

Table 4: Optimal_{Feature Sets using Rosenblum Optimization} 55

Table 5: Classificationofdependenttexture-plus-noisedata 56

Table 6: Classificationofindependent data in 5%pointnoise 66

Table 7: Classificationofindependent data in 10% pointnoise 66

Table8: Classificationofdatawith 10%point noise after_trainingwith5%

point noise 76

Table 9: Classificationofdatawith5% point noise after_trainingwith 10%

1.0 Introduction

1.1 Morphological Granulometries

Morphologicalgranulometries wereconceived_byMatheron _[1975] asatypeof"sieving"

operation for_binaryimagesinwhichparticlesinthe imagestructurearefiltered according

to theirsize. Quantificationoftherate atwhichanimageisalteredin the_sievingprocess

producesa numericalsize_{distribution containing image} textureinformation. _Binary

granulometriesaregenerated_{bysuccessively opening}a_binary image_byan _increasing

sequence of convex_{binarystructuring}elements. The imageswhich make_upa setthe

structuringelementsequence are of a specific shape (i.e. linecircle, square,etc.)andthe

texturalinformationwhich can begatheredfroma_{granulometry is}specific to theshape of

the_structuringelement sequence.

1.1.1 _Opening

The openingofa_binaryimage S _bya_{binary structuring}elementE is definedtobethe

unionof all translationsofEwhich are subsets ofS. Rigorously,x e _OPEN(S,E)ifand

only ifthereis sometranslate_(E+z)ofEsuchthatx e _(E+z) c S. Considertheexample

ofa_binarydigitalimageSandthe threepixelhorizontal structuringelementErepresented

Image S _{StructuringElement E}

* _{\ \ \} * * *

***** i i

11111111 111

* _{1 1} * _{1 1} *

1111*1*

11*11*1

Figure 1: Image S and _structuringelementE.

Theones represent activated pixelsandthestars areconsideredundefined or non-activated

pixels. Allpixels outside animageare also considered non-activated. Toopenimage S _by

structuringelement_E, theorigin ofE istranslated toeach pixelin S. Wherever E entirely

fitsoveractivated pixelsinS,all pixelsinthe_{resulting image Open(SE)}areactivated. See

Figure 2.

Open(S^)

* _{1 i} |****

*****jjl 11111111

********

********

Figure 2: Open(S,E); OpeningofimageS_{bystructuring}elementE

Since Ewillfitover allpixelsinthe third row, theentire rowisactivatedinOpen(S).

Noticethat thelastpixelinthefirstrow isactivatedin imageS butnotin imageOpen(SE)

rows. Becauseofthe size andthe shapeofthe _structuringelementinthisparticular

example, _{any horizontal}runlengthof3or more pixelswillbeactivated.

1.1.2 Granulometries

From thedefinition of an_opening, itfollowsthatwhen _OPEN(F,E)=_F,

OPEN(SE) is a

subimageof OPEN(S,E). Asa_result, if_{Ei, E2, E3,}... isan increasingsequenceof

structuringelements suchthat_{OPEN(Ek+1, E^}=

Ek+1 , then thefiltered imagesforma

decreasingsequence

OPEN(S,Ei) z> _OPEN(S,E2) 3 ...

Countingthenumber ofpixels_{remaining in} each_{succeeding opening}resultsina

decreasingfunction^(k),such thatforsome _K,*F(k)=0 for k>_K.

Dependingon the

shapeofthe_structuringelements,varioustexturalinformation isrevealed_bystudyingthe function^(k). Theimage sequence _{OPEN(S,Ek)} iscalled a_granulometryandthe

resulting function^(k) iscalledthesizedistribution. Inpractice,Ex consistsof asingle pixel sothat₍₁₎givesthe totalnumberof activatedpixelsin S.

Since_4*(k) is_decreasing,thenormalizationof^(k)isa_{probability distribution function}

given_byEquation 1.

Thediscretederivative, dO(k), isadiscrete_{probabilitydensity}function. It has become

populartorefertothisnormalizedgranulometric-_{sizedistribution}

densityasthepattern

spectrumoftheimages. Thisdistributionrevealstheparticle sizedistributionoftheimage fromwhichit iscalculatedand canbedescribed_byitsmoments. Themoments ofthe

patternspectrum canthen beusedtodescribetextureinformation.

Figure 3ashows asimulated_binary imagemade_upoffoursizedisksofdiameters _{4, 7,}

15,and31 pixels. Whentheimageisopenedwitha seriesof circular_structuringelements

Ekofdiameter 1 through_4, theresultantimage isunchanged. _However, whentheimage is

opened with acircular_structuringelement ofdiameter_5,thedisksofdiameter4arefiltered

out ofthe image, leavingtheimage shownin Figure 3b. _Opening thisimagewith elements

ofdiameters 6and7 produce nofurtherchangeintheoutputimages. Whenthe image is

opened withdiameter_8,thedisksofdiameter 7arefilteredout ofthe_{image resulting in}

Figure 3c. Again,there isnochangeintheoutputimageuntilthe_structuringelement

diameterreaches 16pixelsandthedisksofdiameter 15arefilteredoutasshownin Figure

3d. Finally, when the_structuringelement sequencereaches32, allthedisks have been

filteredout_{resulting in}anullimage. (Itshouldbenotedthat there_{may be} somedigitization

Figure3: Simulated_binarygranulometryresultantimages

a) originalandOPEN(S^i) throughOPEN(S34)

b) OPEN(S^5) through_OPEN(S7)

c) OPEN(S,E8) throughOPEN(S,Ei5)

6000

<D(k)

dtD(k)

The^(k),O(k)and_{dO(k)distributionsfromthe simulatedimage}

granulometryare shown

in Figure 4. Allthree parametersare_functionsofthe_{diameter, k,}ofthecircular_structuring

elements. It isimportanttonotethat thesedistributionsarebasedon a pixel count ofthe filtered_image,ratherthana particle count.

1.1.3 LocalGranulometries

A _{local granulometry is}an extension ofthisconcept_describingtheparticlesizedistribution in a given neighborhood or window aboutsomepixel x. _^(k)is thenthepixel count

withina windowcenteredon pixelx, ratherthan thepixelcountovertheentireimage. In

ordertomaintainlarge-scaletextural_information, theimageisopened_globallyandthe pixel countisperformedlocally. Inthe samemannerdescribed fora global_{granulometry,} thenormalized_{probability distribution <Dx(k) is}calculatedfromthelocalsizedistribution

<Dx(k) = _{l-x(k)/x(l) (2)}

foreach point x intheimage. Thediscretederivative,dOx(k),definesthe_probability

densityaboutthepixel x. d$x(k)isthen thelocalpatternspectrum atx. Theresultofthe

binarylocal granulometrywith a given windowsizeisa one-dimensional_probability

densityateach(ij)pixellocation intheimage. These probability densitiesserve as

0.20-1

0.15

-dO

0.10-Figure 5: Example_{d<Dx(k)probabiUty distribution}

This distributionisa_robust,but impractical descriptorofthe localtexture. However,the

moments canbe usedtodescribe thedistributionand canbeused as a much more practical

descriptorofthelocalimagetexture. The localgranulometric_mean,standard_deviation,

varianceand skewness canbeusedasvaluable texturedescriptorsforimagesegmentation

and classification_[Doughertyet_al, 1990]. Since thesemoments arederived fromrandom

1.2 Image Texture

Image texture andtexturalinformation have been studiedfor manyyears. Lewis _[1971]

illustrated howtexturerelatesto_{geomorphology using K-band}radar_imageryof_plains,

low_hills,highhills, and mountainsinthe PanamaandColumbiaarea. Haralickand

Anderson_[1971] illustratedhow texturerelatestolandusecategories. Suttonand Hall

[1976] usedtexturemeasuresforautomatedclassification of_{pulmonary disease.}

Rosenblum[1990] demonstratedtheclassification_{accuracy increase}of aerial_imageryby

the addition oftextural featurestoa multi-spectral classificationdata base.

Textureisadescriptionofthespatialdistributionand spatial_{dependence among}the_grey

tones _[Rosenblum, 1990]. Itcan be described_byperceptualdescriptorssuch as _"fine",

"smooth", "coarse", "mottled", "lineated"or "irregular". It may also be described interms

of a pattern made_upof repeatedtextureprimitives[Nevatia, 1982]. Atexture image Jcan

therefore be thoughtofas atransformfromonebandof a spectralimage I inwhich_J(i,j) is

afunctionof_I(i,j)and_neighboringpixels_[Haralick, 1979]. Atexturemeasureata point of

animageissome functionoftheobserved values within alocalneighborhood aboutthe

point_[Ahuja, 1983]. Granulometriesusea structural approachtoanalyze visual scenesin

termsof organizationandrelationships_{among its} substructures[Haralick, 1986].

Granulometric features describe imagetexturesintermsofthesizedistributionsofthe

1.3 Image Segmentation _Using Granulometric Features

1.3.1 Segmentation

Oneofthereasonsbehindthedevelopmentofimage processing has beentheneedto

identifydifferentobjectsorregionswithina givenimage. Withinthe _studyofimage

texturehas beenthedevelopmentofalgorithmsforsegmentationbasedonimagetexture.

The intuitive ideabehindimagesegmentationistodividetheimageintosegmentssuchthat

each segmentis homogeneous insomesense andtwo_neighboring segmentsdiffer from

one anotherin thesamesense [Kashyap,1986]. Segmentation isaccomplished_by

separatingtwoor morehomogeneousregions whichhaveasignificantstatistical

difference. Sincethepixel valuesof a_binarytextureregions are_inherently

non-homogeneous, texture measures needtobeassignedtoeach pixel forsubsequent

segmentation.

1.3.2 UseofGranulometric Feature for Segmentation

DoughertyandPelz _[1989]developed bothadeterministic andanondeterministic model of

image segmentation_usingtexturemeasuresderived frommorphological granulometries.

Usingthedeterministicmodel, an imagecomprisedoftwodifferentsizediscswas

segmented_byusingthemeanofthelocalcircular granulometry. Agranulometric-mean

imagewas generated_byassigning thislocalcircular pattern spectrum mean_(PSM) toeach

pointx of animage. Eachpixelinthe_{resulting image}wasthereforea measureofthelocal

Thisgrey-scaleimageof_{mean values wasthen}

successfullysegmented_{bythresholding}the

image.

1.3.3 Higher Order MomentFeatures

Ifthelocal PSMoftwo textureregionsisnot_sufficientlydifferenttoallow_{segmentation,}

higherorder momentsofthelocalpattern spectrum_{may be}employed. The localpattern

spectrumstandarddeviation(PSSD), variance(PSV),and skewness_(PSS)can beusedas

texture measurestosegmentanimage. _DoughertyandPelz_[1989] used thepattern

spectrumvariance_(PSV)tosegmentanimageinwhichtwo textureregionshadsimilar

PSMs. _Byviewingahomogeneoustextureas a population ofpixels, alllocal

granulometric moments can be interpretedasrealizationsof random variables which are

characteristic oftheimage texture. Theserandom variablespossess_probability

distributions indicativeof an imagetexture.

1.3.4 _StructuringElementsandDerivationofOtherGranulometric Features

In additionto_being texturedependent,all moments of a patternspectrum are specificto the

structuringelements usedtogeneratethespectrum. _Manydifferenttypesof_structuring

elementshave beenusedtogenerateapattern spectrum ofanimage. _Circular,elliptical and

linear structuringelementshave commonly beenemployedtogenerate granulometric

texturemeasures. Linearandnon-linear combinationsofthemomentscanalsobeusedas

localtexture measures. Dougherty, Kraus, andPelz_[1989] introducedthreesuch

four lineargranulometries; horizontal,vertical,positive-diagonal₍₄₅₎and

negative-diagonal(135),andMaxLinasthemaximumPSMofthe samefour lineargranulometries.

AveLinisan exampleof alinearcombination whereasMaxLinisanexample ofanon

linearcombination. _Linearityis ascale-invariantfeature definedasMaxLindivided_bythe

PSMofthecirculargranulometry. Thisratio will resultina value of1 for anycircular

imageelement regardless ofdiameter. Elongated imageelements will producehigher

1.4 Image Classification and _{Discriminant Analysis}

1.4.1 Classification

Imageclassificationistheprocessof_assigningapixeltooneof a number ofpossible classes onthebasisofsomeobservationsmade onfeaturesofthatpixeland/orits

surround. It isa_{decision making}process which uses statisticaldecision _theorytomake an intelligentestimateoftheclass towhich a pixelbelongs [Schowengerdt,1983]. In

supervisedclassification,asampleof eachclassis takenforeach observationandthe

statisticaldistributionofthe elementsineach class areanalyzed. Fromthat_information,the

classification algorithm reaches adecision abouthowtoassignpixelsnotinthesampleto

theappropriateclasses. Schowengerdt_[1983] recommendsthat 10to 100pixels be includedper_trainingclasswithmorepixelsforthoseclasses with highervariability.

Pixelclassificationhas_longbeenanintegralpart of remote _sensingandotherimage

processingapplications. Spectraland radiometricdata fromaerial and satelliteimages have

been usedasfeaturesto_classify specificregionsoftheseimages_accordingtosome

predescribedcriterion. Linearcombinations ofthisdata, suchastheredtogreenbandratio

inmultispectralaerial _images,have alsobeenfoundtoemphasizedifferences inground

cover typesandcharacteristics of particularinterest Manyapproachestoclassification

basedontexturehave been developedovertheyears. Someofthese approachesinclude

the useoffeatures derived from firstorderstatistics,spectral power_{densityfunctions,}

autocorrelation_functions,and grey-tonerun-lengthdistributions. Oneofthemost

measuretherelativefrequencieswith which twopixel_values,withacertain separation,

occurinanimage [Haralickand_{Anderson,1971].}

1.4.2 Feature_ProbabilityDistributions

Anynumber oftheseor otherfeaturescanbeusedto_classifyan image intosomepreset

number ofclasses. Eachclasswillhavethe samenumberoffeatures inafeaturevector.

The distributionof valuesfor anyonefeaturefora givenclasshasacertaindistribution

which can beusedtodecidetheclassification of an unknown pixel. Considerthe two

distributionsof somefeature Z in Figure 6. Each distributioncorrespondstoa separate

class.

class 1

P(x_{I i)}

featureZ

Figure6: Feature Zvaluedistributionfortwoclasses

Thearea under eachofthesedistributioncurvesisnormalizedto 1.0and_theyare assumed

toapproximatethefeatureprobabilitydensityfunctionsof each class. Thesefunctionscan

belongstoclassi. The probabilityof_findinga_featurevalue of xgiventhatweare

sampling fromclassi is givenas p(x Ii). Thediscriminant functionisdefinedasthe

probabilityof a pixel _belongingto classigiventhatithasafeaturevalue x or p(ilx).

p(ilx) =

p(xli)p(i)/p(x) ₍₃₎

where_p(i) isthea priori_probabilitythatclassiexistsintheimageand_p(x)isthe

probabilityof_findingapixel_{from any}class.

Assumingthateachclasshasan equal_probabilityof_{occurring, p(i)}willbeequalforeach

class. Thevalueforp(x) is simplya_{normalizing factor}andthereforeaconstantforeach

class. TheaUscriminantfunction isthen_simply a calculation of p(xI_i)foreach class.

1.4.3 Maximum Likelihood Classification

Maximum likelihoodclassification comparesthediscriminantfunctionvalueforeach

featurevalue xcalculatedforeachclass and assignsthepixelto theclasswhichproduces

the_{highest probability}value. Forexample,considerapixel with afeatureZvalue ofx, as

shownin Figure6. Sincethecalculated value ofthediscriminantfunction is greaterfor

class 1 thanforclass _2, thepixel wouldbeassignedorclassifiedintoclass 1.

The decision _boundaryforclassification,d,liesatthepointatwhichthe twodistributions

class 1

P(x_{I i)}

feature Z

Figure 7: Maximum likelihood decision_boundary

Anypixel with a feature Zvalueless thandwouldbeassignedtoclass _1, sincethereisa

higher probabilityofthisvalue_{coming from}class 1 than_{coming from}class2. _Likewise,

anypixel with afeature Zvalue greaterthandwouldbeassignedtoclass2. Thetotalerror

inthisclassification isrepresented_bythe_overlapofthe twodistributionswhichis shown

astheshaded region. Thiserroris minimized_byplacingthedecision _boundaryat thepoint

at which p(xI ₁₎isequaltop(xI _2),This decision_boundaryisrepresented_byd in Figure

7. Notice thatifthedecision_boundary wasmovedineither_direction,theerror would

increase.

1.4.4 Gaussian Maximum Likelihood Classification

This is thesimplest exampleofclassificationsincethere_{is only}onefeature and_only two

classes. _However, thesame principles canbeextendedtomore complicated classification

modelsin whichthereare_anynumberoffeaturesand classes. Themost_commonlyused

mean vectors and covariancematricesoftheclasses are requiredtocompute the

class-conditional_densityfunctions. Thisclassifier requires thedistributionoffeatureswithin

eachclass to_{be approximately}multivariatenormal. _However,theclassifieris_"relatively

tolerant"

of_{deviations from normality [seeSwain, 1986].}

Adiscriminant function is developed_usingameasure ofthegeneralizedsquareddistance

fromthemean vector. Theclassificationcriterioncanbe basedon eithertheindividual

within-class covariance matrices or a pooledcovariancematrix. Aswiththesinglevariable

case,each unknownpixelisclassifiedintotheclassfromwhichit hasthesmallest

generalizedsquareddistance.

Thegeneralized squareddistance fromx toclass tis

Dt2(x) =

gi(x,t) + _{g2(t) (4)}

where xis thevector_containingthefeaturevalues of an unknown pixel and

tisasubscripttodistinguishtheclasses.

Ifthewithin-class covariance matricesareusedthen

gl(x,t) = (x

-mO' Sf1 (x

-mt) + lnlStl (5)

where mt isthevector_containing themeans ofthefeaturesof an unknownpixel

Stisthe covariance matrixwithin-class t.

Ifthe pooled covariance matrixisusedthen

gl(x,t) = _(x

-mO' _S-1 (x

-mt) (6)

where S isthepooledcovariancematrix.

Ifthea priori probabilitiesareallequal,g2(t)iszero. However,if_theyare not all equal

g2(t) =

-21n(qt) (7)

1.5 Minimal Window Size

The sizeofthewindowforthelocalgranulometriescan havea significant effectonthe

distributionsofthe granulometricfeatures. _Dougherty,PelzandNewell_{[ 1990]}

demonstratedthat thevarianceofthe granulometric_{feature distributions decreases}with

increasingwindow size. _Decreasing thevariancedecreasestheamount of_probability

overlap betweenclasses soclassification_accuracycanbeimproved_{by increasing}the

windowsize. _{However, increasing}thewindowsizecan also makeit hardertodetermine

theborder between adjacenttextureregions andleadtomisclassificationofpixelswhose

surroundincludes 2ormoretextureclasses. _{Generallyspeaking,}largerwindowsdecrease

variabilityofthegranulometricfeaturesatthecost ofless detailedsegmentation.

Afeaturevaluefromalocal granulometry is onlyan accurate representation of a given

imagetextureiftheentire window usedtogeneratethatfeaturelieswithinthe texture

region. _Otherwise,thefeaturevalue canbeaffected_byotherimagetexturesandtherefore

represent a combination of anumber ofimagetextures. Apixel_lying neartheedgeoftwo

textureregions shouldthereforebeunclassified. Giventwoadjacenttextureregionsinan

image, and awindowan oddnumber,x,pixelsin_length,the number of unclassified pixels

1.6 Optimal Feature Selection

1.6.1 Feature Reduction

The_{keystep in any}classification problemis thechoiceofaset offeatureswhichreduces

thedimensionofdatatoa_{computationally}tractablelevel while_preservingmuch ofthe

classifying information presentintheactualdata[Kashyap, 1986]. Thenumber offeatures

used intheclassification shouldstillgive aminimal_probabilityofmis-classification _[Fu,

1976]. Featureswhichdonotaddtoclassification_accuracyrepresent a cost_since,witha

maximumlikelihoodclassifier, the timeneededtomake a calculation_{increases quadratically}

with theaddition offeatures _[Richards, 1986].

Inrecent yearstherehasbeen much attention paidto_determiningan optimal set of m

features outof atotal set ofNfeatureswithout_{significantly}degradingtheclassification

abilityofthealgorithm. Thesetechniquesfor featureselection attempttomeasurethe

separabilityoftheclassesforeach combination of mfeaturesout ofthetotal set offeatures.

The subset withthemostpotentialforcorrect classificationis_subsequentlyselectedforuse

intheclassifier.

1.6.2 Mahalanobis-LikeDistance Measure

Thesimplesttechniquesoffeatureselectionusetheseparationofthefeaturemeansin

multidimensionalspace. However,thisapproach_mayresultina set offeatureswhichare

developed_{by Schott, Salvaggio,}andKraus _[1988], usesthe Mahalanobis-Likedistanceas

a measure ofthe separationofclasses. TheMahalanobis-Likedistancemeasureis defined

as

dst =

(ms

-mO' _S"1

(ms

-mt) ₍₈₎

where _msand_mtarethemean vectors ofclasses s andtand

S isthepooledcovariancematrix.

Comparedto_calculatingtheexact_{probability overlap betweentwo classes, this}methodis

very fastsinceitrequirestheinversionof_onlythepooledcovariancematrix. _Usingthe

pooledcovariance hastwomajordrawbacks: ₁₎theassumption of equalcovariance

matrices isnot_usuallytrueand₂₎ it doesnot accountforthe_variabilityoftheindividual

classes. Onesolutiontothisproblemistousetheindividualcovariance matricesinplace

ofthepooled covariance matrix _[Robert, 1989]. TheMahalanobis-Like distance between

classes correctedfortheindividualcovariancematricestakes theformofEquation ₍₉₎

[Richards, 1986].

dst = [(ms

-mt)' _St"1

(ms

-m,)] + lnlStl ₍₉₎

where Stisthecovariance matrixwithin-classt

However, thisintroducesother problemssincetheresultdependsupon whichcovariance

1.6.3 DivergenceMeasure

Richards _[1986] describes a_way to_quantifythe separation betweentwoclasses_bythe

degreeof_overlapoftheclass_{distributions. Theoptimal setoffeaturescanthen be found}

by findingthe setwiththeleastamountof_probabilityoverlap. _{The divergence between}

two classes, dvst,is definedinEquation₍₁₀₎asa_separability measurewhichtakesinto

account the_variabilityofboth classes.

-Jt

dvst = J_{[ p(X} I s)

-p(X I t) ] In _[ p(X I s)/p(X I t) ] dX ₍₁₀₎

X

where _dvst isthe divergence betweenclass sandclass_t,

p(X I _s) isthe _probabilityof_findingthefeaturevectorXwhen

sampling fromclass s and

p(X _{I t)} isthe_probabilityof_findingthe featurevectorXwhen

sampling fromclasst.

Iftheclassesare assumedtocomefrommultidimensional normal_{distributions,}the

divergence becomes

dvst=_(l/2)Tr[(Ss

where _msand_mt are themeanvectors ofclassess andtand

Ssand_St arethe covariance matrixwithin-classs andt respectively.

UnliketheMahalanobis-Like distancemeasureinEquation_(9),dvstissymmetric (i.e. _dvst

=

dvts)because bothclassdistributionsaretakenintoaccount.

Thisdivergencecan thenbesummedover allclass pairstogivea measure oftheoverall

divergence. Thesetoffeatureswhichresultsinthegreatestoverall_divergenceshouldgive

the greatest classification_accuracywhen aGaussianmaximumlikelihoodclassifierisused.

Mausel,KramberandLee. _[1990]transformed thedivergence betweentwoclassesin the formofEquation(12)toemphasize smallchangesinthedivergence_{resulting from}

significant changesintheclass separability.

tdvst= _{2000[l-exp(-dvst/8)] (12)}

Thisvaluehasalimitof2000which wasdesignedto_{limit extremely high}divergence

values whichdonot_necessarilycorrespondtocompleteclassseparation.

1.6.4 _SeparabilityMeasure

Rosenblum_[1990]developedasimilar methodtoenhancethe_accuracyofthe

Mahalanobis-Like distanceseparationmeasuregivenin Equation (9). Theoverall _separabilitymeasure

otherclasses. Figure 8shows two feature setsofthe same threeclasses. In feature set_A,

classesx and_y are_{poorly separated and class} z_{is greatlyseparatedfrom these two.In}

feature set_B, allthreeclasses are_fairlywellseparated.

featuresetA

/\

feature setB

Figure 8: Featuresetsforclass_separability

It isobviousthatfeaturesetB doesabetter jobof_separatingthe threeclassesthanfeature

setA. _However,the _separability measurefor feature setAwillbegreaterthanthe

separabilitymeasure for featuresetB. Since alarge separation of_anyoneclassfroma

groupofotherscaninflatetheoverall_separabilitymeasurefora set of_features,adistance

thresholdwasdevelopedtonormalize andlimitthevalues. Withoutthat_inflation,a set of

featureswhichseparate all oftheclasseswillbechoseninsteadof a setwhichseparates one

class_verywell.

The_probabilityof_findingthemean ofclass tinasamplefromclass sisdefined as

P(mls) =

[

"

"V Ss'1

<mt

where k isthenumberof_featureswhich aretobeoptimized.

UsingEquation₍₁₃₎ and_{assumingmultivariate normal class}

distributions,itcanbeshown

that

[(mt

-ms)' _Ss"1

(mt

-ms)] + lnlSsl = _-2ln[P/

2tc^2]. (14)

(See Appendix A). The left handsideofEquation(14)representstheMahalanobis-Like

distance betweentwoclass_means,dts,when theindividualclass covarianceforclasssis

used. The right hand side oftheequationistheminimum_{distance, dthresh,}which must

separatethe twoclassmeansforthe_probabilityofmisclassification tobe P.

dthxesh = _{-2 1n[P/2K^].}

(15)

ThevalueofPshouldbesettoa_sufficientlysmall valuetoassure nearcompleteseparation

oftheclasses.

Thereare now twodistancemeasures which needtobecalculated: theactualdistanceas

calculated_byequation _(9),

dSt= t(ms

-mO' _St"1

(ms

-mO] + lnlStl ₍₁₆₎

andthe thresholddistancefora preset_{probability P}given_byequation(15). After both_dts

and_dthreshhave beencalculated, theratiooftheMahalanobis-Likedistancetothe threshold

dratio = _dst/dthresh

(17)

Sincethe_separabilitymeasureisnot_symmetric,theratiomustbecalculated twiceforeach

pair of classes. Asa resultof_calculatingtwo_separabilitymeasuresforeach pair of

classes, a matrix mustbe usedtorepresentallrelativedistancemeasures. Thesum ofthis

matrix can thenbeusedas a measure oftheoverall_separabilityoftheclasses. Before this

summation_{however, any dratio}values whichexceed 1.0are setto 1.0topreventthe

inflationoftheoverall_separabilitymeasure. Theoverall_{separability is}thencomputedfor

all permutations offeaturesubsetsfromthewholeset. Thesubset withthehighestoverall

1.7 Noise

1.7.1 MaximumNoise

Everyaspect of_{image processing}andclassificationisaffected_bynoise. Althoughthere

are_manytypesofnoise associatedwith_images,becauseofthenatureofgranulometric

based_features, we willconcern ourselves_onlywith maximum additivenoise. Sincethis

typeofnoise adds tothe activatedpixel countofathresholdedimageneededfor_binary

granulometries,it is readily apparentthatadditivemaximumnoisewill skewthe

granulometricdistributions usedtogeneratethesefeatures.

Thereare _manytypesof maximum noise which are inherenttodigital images. Of_these,we

will examinetheeffects offour basiccategories: point_noise,occlusion _noise,scratch noise

and spaghetti noise. _Dougherty, PelzandNewell _{[1990] briefly}examinedtheeffect of

maximum point noise and spaghetti noise on granulometricbased features. Although it

was concludedthatthefeatureswere_generallyrobust, furtherexaminationisrequiredfora

deterministic analysis oftheeffect of additional noise onimagetextureclassification.

1.7.2 PointNoise

Point noiseis definedas singlerandom activated pixels. It may becaused_byflaws

inherentto the_detector,bydustanddirton _anydigitallyscannedimage,or_byelectronic

noiseat_{any level}ofthesystem Sinceuniform response of_{array detectors is virtually}

"push"

some pixelvaluespast agiventhreshold Thesignal_may alsofluctuatefromother

electronicsinthedetectorsuch as_{photomultipliers}andamplifiers.

1.7.3 OcclusionNoise

Occlusionnoiseis definedas noisewhichoccludesorcoversan _underlyingsignal. Inan

image, thiscanbethoughtof asparticleswhich arelargeenoughtoaltertheapparentshape

or_boundaryofimage structures and substructures. Thiscanbecaused_bylarger dirtand

dustparticles on_digitallyscannedimagesor anundesired_intersectingobjectwithinthe

originalimage.

1.7.4 Scratch Noise

Scratchor streak noiseischaracterized_bylong,thinstraightlineswhich propagateina

singledirectionina particularimage. Physical scratches on aphotographic negativecan

appear asmaximum scratch noiseonaphotographic print. These scratches are_commonly

caused_byphotographicequipment and_{processing machinery}asthe negativeispulled

though. Singleelementflawsina"push

broom"

typedetectorcanhavea similar effecton

digitized images. Astheone-dimensionaldetector arraymoves acrossan_image,one

defectiveelement or an elementimpaired_bydirtcancause a streakintheresultantdigital

1.7.5 SpaghettiNoise

Spaghetti noiseischaracterized_byathin"curly" or

"windy"

lineof connected pixels.

Dependingontherelativesizeofthedetectorelements andimages_beingscanned, this type

2. 0 Statement of Work

2.1 Selection of _{Texture Images}

Ten textureimageswere chosenforthestudy. AlltenweretakenfromBrodatz'

collection

ofphotographictexture images[1966]. Thesephotographswere scannedat75 dpiintoan

8-bitgrey-scaledigitalformat.Thetenimageswereselectedtorepresent alargerange of

textural complexity. Thistextural_complexitycanbethoughtofastheamountofstructure

inthe _underlyingtextureprimitives andthevariationinthatstructure. Figure 9showsall

ten_{images along}with

Brodatz'

originaldescriptions.

Theoriginaldigitalimageswere512x512pixels. _However,the 132x 132pixelimages

in Figure9werecroppedfromtheoriginalimages before processingtolimitthe

computationtime. Acomplete descriptionofthereasonsforthisexact sizeare statedin

section2.4. Eachtextureimagerepresentsa separatetextureclass. Throughoutthis paper,

eachtextureclass willbereferred to_bythedescriptionnumber given_byBrodatz (i.e.

a) dl02 Cane

llllllllllllllllll

llllllllllllllllll

llllllllllllllllll

llllllllllllllllll

llllllllllllllllll

liiilliiiliiiniiii

liiiillllliiiiiini

limiiiiiiiiniU!

Iniiiiiiiiimiiii

c) d20 Frenchcanvas

b) d!03 Loose_burlap

....J.%*-*'^ '

I*

III_{t^i| J|}||J

....

v"^..., mfc

d) (152 Oriental Strawcloth

e) d64 Orientalrattan f) d65 Orientalrattan

g) d67 Plastic pellets

(inverted greyscale)

i) d75 Coffee beans

h) d68 Woodgrain

wm

lAit

jr-v^--" j^rtyr^T *_*>

*i

j) d84 Raffialooped

toahighpile

2.2 _Thresholding of Texture _Images

Before thelocal_binary granulometrieswere_calculated,these 8-bitgrey-scale imageswere

reducedto _binaryimages. Theuse of athresholdprovidedthesimplestmethodforthe

gray levelcompression. Thechoice ofthethresholdvaluecould _{significantly}changethe

results ofthe granulometries_bychangingthe activatedpixelcountinthe_{image. In many}

casesit isprofitable tochooseathresholdwhich results_{in approximately half}thepixels

being activated. _However,when_dealingwithimage_{texture, maintaining}the_underlying

textural structureandsubstructuresis themostimportantaspect. Althoughthe textural

structure of eachimagecouldbe bestmaintained_bychoosinga separatethresholdvaluefor

each _image,a singlethresholdvalue was chosen whichmaintaineda_majorityofthe

underlyingstructureinall the imagesand more_closely simulated reallifeconditionsfor

textureclassification. Figure 10showsthe thresholded versionsoftheimages in Figure 9.

These_binaryimagescould_{be greatly}affected_{bynonuniformity}oftheimages. Intra-image

nonuniformityofthemean _{gray level}ofthe 8-bitimagescould causethesize and shape of

the textureprimitivesto_{vary significantly}withina_{supposedly homogeneous image. This}

may have inflatedthevariance and couldhaveskewedthe granulometricdistributions.

Skewingthe_{distributions may have}caused ashiftinthefeaturemeans. Theinflated

variance ofthedistributions_{may have}resultedinadecreaseofclassification_accuracydue

J

>- 1 f' kIII II k4 ha.

> - ,. . ..

.,, ... . . .

a a a

a) dl02 Cane

HllllllTlllllUM lltilllllllDlllll i^iiifnirM^Mi tfipliipifiiiumi

^HUilili^ikii

^iilii|fifirii.i<

fcplpfcpiiirliipiUpi

c) d20 Frenchcanvas

faifail^

Hft

_llfiMJI

tSUiSG

CMrfll

nmrnrnm

fcMCM*

b) dl03 Loose_burlap

.t! ""

^'>tw *-*

d) d52 Oriental Strawcloth

e) d64 Orientalrattan f) d65 Orientalrattan

g) d67 Plasticpellets

(inverted _greyscale)

i) d75 Coffeebeans

h) d68 Woodgrain

iilPtl

^T7)~*^~l

iH

&1fl^l

j) d84 Raffialooped

toahighpile

Figure 10(g-j): _Binarytextureimages

Inter-image nonuniformityofthemean_{gray level}could cause a significantdifference

betweentheapparentimagetextureofthe8 bit imagesand the textureofresultant_binary

images. As aresult, the most significanttexturalinformationina_grayscaletextureimage

maynothave beenrepresentedinthe_binarytextureimage. Thegranulometricfeatures

2.3 Generation of _Noise

As_{previouslymentioned,}fourcategoriesofadditivemaximumnoise wereinvestigated.

Thefournoise modelssimulatedwere point_noise,occlusion noise,scratch noise and

spaghetti noise. Thissimulation wasaccomplished_{by directlyoverlaying}noise imageson

theten_binarytexture images. Noiseimageswere created_byplacinga number ofparticles

of_noise, "noiseelements", ontoablank image. Eachnoiseelement was described_byits

length, width, "straightness",andinitialangleofpropagation.

Eachofthefournoise models musthaveacertainmean andrangeforeach ofthefour

descriptiveparameters mentioned above. Foreachnoise model,appropriate valuesforthe

mean and range ofthe lengthand width weredefined. Beta distributions were usedto

determine thelengthand width of eachindividualnoise element. Sincethebeta distribution

is only defined between 0and _1.0, a_{scaling factor}and/or a shiftfactorwasappliedtoalter

therange ofthedistributionsothat thepredefined mean number of pixels wouldliewithin

thisrange. The beta distributionparameters,r\and_y, were_subsequentlysetto the

appropriatevalues todeterminetheshapeofthedistributionand_therebythevariationofthe

length and width of allthenoiseelementina noise image.

The initialangle of propagationcouldeitherbe set or_randomlychosenfroma uniform

distribution foreach noise element. A beta distributionwas usedtodeterminehowstraight

or_curlya noise element wouldbe. Themaximumchangeintheangleofpropagationwas

set_bythemean ofascaledbeta distribution. Thevariationofthedirectionof alineof

wasactivatedintheimage. Anadjacent second pixelwasthenactivatedattheinitial

propagation angle. Achangeinthe angleofpropagationwascalculated_usingthebeta

distributionfortheangle. Thenew angleofpropagation thenbecame theinitialangle plus

the angular change.

Pointnoise was the simplest of all themodels. The lengthandthewidth of each noise

element was aconstantof 1 pixel. Thestraightness and angleof propagation ofthenoise

elementswerethereforeirrelevant. Therange_{scaling factor for}thelengthand width

distributionswas set to2pixels. This forcedthemidpointoftherangeto 1 pixel. _{Both r\}

and_yweresetto 10E+15sothat the final lengthand widthdistributionswere_effectively

deltafunctions at 1 pixel.

Theocclusion noise model wasspecified so thatthelengthandwidth of each individual

noiseelement were equal. _Again,thestraightness andangleof propagation ofthenoise

elements wereirrelevant. Therange_{scaling factor for}thelengthandwidthdistributions

wassetto 8 pixels sothat themidpoint occurred at4pixels. Both_t| and_ywere setto3.0

sothat thefinal lengthandwidthdistributionsweresymmetric,centered at4pixelswith a

standarddeviationof_{approximately}2pixels.

Inthescratchnoisemodel,thewidthwas againsetto 1 pixelforall noise elements_by

settingtherange to2pixels and_t\and_yto 10E+15. Therangeofthelength distribution

was setto40pixelstocenterthedistribution about20pixels. Parameters_r\and_ywere set

to3.0and _1.5,respectively, inordertoskew thedistributiontohighernumbersofpixels

ordertoproducescratchesinthe same_{direction, theinitialangleof propagationfor} allthe

noiseelementsin asingle noiseimagewas settoarandom constant. Thiswas accomplished_bysettingtherangeoftheangle_{distributionto2n}

radians, therangeshift constant toa random numberbetween0and_2k,andthe_r|and yofthedistributionto

10E+15toensure straight propagation.

The spaghetti noise model alsohadthewidthsetto 1 pixelforall noiseelements. The

range ofthelength distributionwas setto80pixelstocreate noise elements _{approximately} twice thelengthofthescratch noise. Aswith thescratch_noise,r\ and_{y for}thelength distributionwere setto3.0and _1.5,respectively. The initialangleof propagationforeach

noise elementwas settoarandomangle. Theappropriate valuesfor hand_gofthe angle distributionwerefound_byvaryingtheseparameters untilthenoise elementshadthe

desiredcurliness.

The initial image position_(i,j)of each noise element(i.e. theposition ofthe firstpixel of

the_element) was chosenfromatwo-dimensionaluniformdistributionthesame sizeasthe

originaltextureimages. A 132x 132non-activated pixelimagewas created asatemplate

forthe additionofthenoiseelements. Aftergeneration,each noise elementwas addedto

this image inordertocreatea noiseimage. Thisadditionoperationallowedfor_overlapping ofthenoise elements. Athresholdof1 was thenappliedto thisimage inordertocreatea binarynoiseimage.

Sixnoise conditions were createdforeachnoise model. These noise conditions varied_by

chosenforeach ofthefournoise models were_{0%,5%, 10%, 15%, 20%,} and25%

activated noisepixels. Examplesofthenoisemodelsunderdifferentconditions are shown

a. 5% Point

^*

7<.* ."*;

''. *.:_-'

* i

^^>.-J.v.--|i^i^V

b. 15% Point c. 25% Point

d. 5% Spaghetti e. 15% Spaghetti f. 25% Spaghetti

9

. : _<

5

g. 5% Occlusion h. 15% Occlusion i. 25% Occlusion

j. 5% Scratch k. 15% Scratch 1. 25% Scratch

Figure 1 l(g-l): Examplesof_binarynoiseimages

Foreach ofthe ten_binarytextureimages,anindependentrandom noiseimageof each

noise model andconditionwasgenerated. Thenoiseimages werethenaddedtoeach ofthe

binary textureimages. Thetexture-plus-noise imagesweresubsequentlythresholdedata

2.4 Generation and Selection of Local Granulometric Features

In ordertocreate the granulometricfeaturesneededfor_{classification,} localgranulometries

were run withfivetypesof_structuringelements. _{Four linearelement sequence}

granulometries: _{horizontal,vertical,}positive-diagonal₍₊₄₅₎ andnegative-diagonal_(-45)

as well as asequenceofcircular elements were run on all240 images. Foreach_{pixel, the}

local _{PSM, PSSD,} andPSS were calculatedforallfive_structuringelement granulometries.

The PSMofthe MaxLinand_Linearitymeasures werealso calculated_{resulting in}atotalof

17granulometric features foreachimage.

There weretwomain concernsaboutthe selection ofthefeature data: ₁₎ theneedforgood

estimates oftheclassdistributionsand₂₎ theneedtolimittheamountofdatatosome

computationally tractable amount Inaccordance withSchowengerdt's_[1983]

recommendation, 100pixels fromeach class were usedinthestudy. Sinceeach ofthe

pixelsin aclass wastoberepresented_by 17 features,atotalof1700realdatavalues were

neededforeach ofthe240 _binaryimages.

Toensurethesepixels would_accuratelyrepresentanentiretextureclasswithorwithout

noise,all pixels were_randomly selectedfrom 100x 100pixel "featureimages". The

feature imagesconsisted of real numbers_representing somelocalgranulometric statistic

about each pixelinthe_binarytextureimage. Sinceeach _binaryimagewas assumedto

representahomogeneous texture,thefeature images resulting fromthe localgranulometries

were assumedtobe wide sense stationary. The 100 datavaluesfromeachfeature image

Nopixel wasallowedtobechosena secondtimetoensureaccurate estimates ofthemean

and variance ofthedistributions.

A33 x 33pixel windowsizewas usedtogenerateeach ofthefeature images. Edge effects

may becausedwhenthiswindowdoesnot_{lie entirely}withina givenimage. Sincethe

local granulometric statisticsforareas_lyingneartheedgeof animagecan_{significantly}

differ fromthosefor interior imageareas, a132x 132pixel_binarytextureimagewas

2.5 Classification of _{Dependent and Independent Data}

The initial step inall_{classification algorithmsis}

trainingtheclassifier. Supervised_training

isusedto_identifyanarearepresentativeof eachclass. Inmost_cases,greatcare mustbe

takento include_onlypixelsordatawhich_belongtoagiven class. _However,inthis case,

data fromeachtextureclasswas_easilyseparated sincethe granulometries wererun

separatelyon eachclass. Thissupervised_trainingisconducted_{by inputting}the 17 features

foreach ofthe 100 datapointsofeach class intothe classifier. Themean vectorand

covariance matrixforeachclassisthen_calculated,and adiscriminant function is developed

fromthesemeans and covariances.

Dependent data is definedas thesetofdatausedtotrain theclassifier. Classificationofthe

dependent datacanbeused asaninitialmeasure ofthegoodnessoftheclassifier. A low

degreeof classification_accuracyofthedependent datacan _implyaninadequatestatistical

difference amongtheclasses. However,ahigh degreeof_accuracyofdependent data

merely impliesa reasonable statistical_{difference among}theclassesinthe_trainingdata.

Furtherexaminationisneededtodetermine theoverallgoodnessofthefeaturevectorsfor

classification ofdatanotincluded in the_trainingset.

Aftertheclassifierhas beentrained,independentdatacanbeclassified_usingthemaximum

likelihood discriminant function developed fromthedependent data. This independentdata

typicallycontains some orallofthesameclasses asthedependentdata. Inthiscase,any

data. _Anyotherset offeaturevaluesfrom 1 toall 10textureclassescanthenbeused as the

independent data.

The classification_accuracywasdetermined_{by dividing}thenumber of_correctlyclassified

pixels_bythe total number ofpixels classified Thiscould thenbeused as a measure ofthe

abilityofthe granulometricfeaturestodiscriminate betweenthe textureclasses. The

minimum windowsize _{for generating}thegranulometricfeaturescouldbe found_by

determiningthepoint at whichtheclassification_{accuracy became}unacceptable. The

minimum number of optimalfeaturescouldbefound inasimilarmanner. The

classification_accuracycould alsobeusedas a measure oftherobustness ofthefeatures in

3.0 Analysis of Results

3.1 Dependent Classification

The initial indicationofthepower ofthegranulometric featureswasfound_{byclassifying}

thedependent datausedto train theclassifier. All 17featureswere employedinthefeature

vectorsforeach class. The granulometries were run ontheoriginal 10_binarytextureclass

imageswithout_anyadditional noise. The 17 features from 100random pixelsfromeach of

the 10textureclasseswere usedtotrain aGaussianmaximumlikelihoodclassifier. These

same 1000pixels were _subsequentlyclassified_usingthediscriminant function developed.

Theresults ofthisdependentclassificationareintheformoftheconfusion matrixin

Table 1.

Table 1: Classificationofdependentdata

dl02 dl03 d20 d52 d64 d65 d67 d68 d75 d84

dl02 100 0 0 0 0 0 0 0 0 0

dl03 0 100 0 0 0 0 0 0 0 0

d20 0 0 100 0 0 0 0 0 0 0

d52 0 0 0 100 0 0 0 0 0 0

d64 0 0 0 0 100 0 0 0 0 0

d65 0 0 0 0 0 100 0 0 0 0

d67 0 0 0 0 0 0 100 0 0 0

d68 0 0 0 0 0 0 0 100 0 0

d75 0 0 0 0 0 0 0 0 100 0

d84 0 0 0 0 0 0 0 0 0 100

Thisconfusion matrixshows thatall ofthedependent datawere_correctlyclassified. The

rowsofthe matrix representtheoriginal classofeach pixel. Thecolumnsrepresentthe

class intowhich _{each pixel was classified. Since therewere 100}

pixelsfromeach_{class, the}

values inthematrix representboththe number andthepercentage ofpixelsclassifiedinto

theclassdesignated_bythecolumn.

Althoughthedata faileda_homogeneitytestforequalcovarianceof_{the classes, the}

classifierwastrainedasecondtimewiththesame_{data using}apooled covariance to test the

statistical separationofthemeans. Inordertoachieveahighclassification _{accuracy using}

thepooled_covariance,thefeaturemeanshadtobe_sufficiently separatedtominimize the

probability distributionoverlap. Theresultsof_classifyingthe _{dependent data using}a

pooled covariancewere identicalto theresults_usingwithin-class covariance. This

demonstratesthat themean vectors of alltenclasses were wellseparated andindicatesthat

thegranulometricfeatures sufficientlyrepresentedthebasic texturaldifferences betweenthe

classes.

3.2 Independent Classification

Independentdatawasemployedtodetermine theoverallgoodnessoftheclassifier. After

trainingwithfeaturevaluesfromtheoriginalsetof 1000dependentpixels, a second setof

100pixels was_randomly selected_usinga uniformdistribution. _Again,all 17 featureswere

included in thefeature vectorsforeachclass. Thiswas considered anindependentsetof

datasincethe_probabilityofarepeat pixel was_only0.01 _usingtheuniformdistribution.

magnificationalrobustnessofthefeaturessincethemost of_{the granulometric features}

inherently size and_{direction dependent. Theresultsof}

the classificationare giveninthe

confusionmatrixinTable2.

were

Table2: Classificationof_{independentdata}

dlUZ dl03 d20 d52 d64 d65 d67 d68 d75 d84

dl02 100 0 0 0 0 0 0 0 0 0

dl03 0 100 0 0 0 0 0 0 0 0

d20 0 0 100 0 0 0 0 0 0 0

d52 0 1 0 99 0 0 0 0 0 0

d64 0 0 0 0 100 0 0 0 0 0

d65 0 0 0 0 1 99 0 0 0 0

d67 0 0 0 0 0 0 100 0 0 0

d68 0 0 0 0 0 0 0 100 0 0

d75 0 0 0 0 0 0 0 0 100 0

d84 0 0 0 0 0 0 0 0 0 100

Overallclassification_accuracy=99.8%

Table 2showsthat_onlytwooftheindependentpixels weremisclassified. Theoverall

classification_accuracyof99.8%indicatesthatassumptionof within-class _homogeneityof

the 17featureswasjustified. Thisalsoindicatesthat the basictextural_{differences between}

theclasses were wellrepresented_bythesegranulometricfeatures.

Aswiththedependentclassification, theclassifier wastraineda secondtime_usingapooled

covariance matrixtoassurethat thefeaturemeans were well separated. Theresultsofthis

Table 3: Classificationofindependent_{data using}pooled covariance

dl02 dl03 d20 d52 d64 d65 d67 d68 d75 d84

dl02 100 0 0 0 0 0 0 0 0 0

dl03 0 98 0 0 0 0 0 0 0 2

d20 0 0 100 0 0 0 0 0 0 0

d52 0 0 0 100 0 0 0 0 0 0

d64 0 2 0 0 98 0 0 0 0 0

d65 0 0 0 0 0 100 0 0 0 0

d67 0 0 0 0 0 0 100 0 0 0

d68 0 0 0 0 0 0 0 100 0 0

d75 0 0 0 0 0 0 0 0 100 0

d84 0 0 0 0 0 0 0 0 0 100

Overallclassification_accuracy=99.6%

Notethat theoverall classification_{accuracy decreasedbyonly 0.2%} when comparedto

classification_usingwithin-classcovariance. The difference inclassification_accuracywas

dueto thedifferenceoftheestimatedfeaturedistributions foreachclass. Sincethepooled

covariance matrix was an estimate oftheaveragecovariance ofthe tenclasses,the estimates

ofwide within-class variancestendedtobenarrower_usingpooled covariance. _Likewise,

theestimatesof narrow within-class variancestended tobewider_usingpooled covariance.

Ingeneral,thiscaused anincreaseofthe_{probabilityoverlap}andintroduced some

3.3 _{Minimal Window Size Determination}

Overallclassification_accuracywas usedtodeterminetheminimumlocalwindow size

neededforclassification. Six localgranulometrieswere run oneachoftheten texture

images_using square windowswith sides oflength7, 11, 15, 19, 25, and33 pixels. Two

sets offeaturedatawere collectedinordertodeterminetheeffect of window sizeonboth

dependentandindependent data.

Figure 12shows resultsof classification with the6 differentsize windows. The side

lengthofthewindow isreferredtoasthe window size. Notethattheclassification

accuracy axison thisgraph ranges_{only from}80% to 100%.

100_-\ u S 95 < s o a u 90-5 85 80 10 1 15 WindowSize T 20 25 dependent independent -J 30 -1 35

Over 99%classification_accuracyofthedependentdatawasachievedforall window sizes

greaterthan 1 1 pixels. Althoughtheclassification_accuracyoftheindependent datawas

lessthan thatofthedependentdata,theclassification wasstill94.6% accurate_using a

window size of 1 1 pixels. It alsoshouldbenotedthat theclassification_{accuracy for both}

thedependentandindependent data fell_dramaticallybelowthewindowsize of11 pixels.

This indicatesthatmostofthe _underlyingtextureprimitiveswhichdistinguishtheseimage

textureswere no smallerthan 1 1 pixels. _However,itshouldbe kept in mindthat the

3.4 Optimal _{Feature Selection}

Anumber ofavailable methodsfor_determininganoptimalfeatureset were applied.

Richards'

[1986] methodfor_determiningan optimalfeatureset_bythe degreeof_overlapof

theclassdistributionsisconsideredthemostaccurate sinceitusesthecovariance ofall

classes and requires_only theassumption ofGaussiannormaldistributedfeatures ineach

class. _However,this methodisalsothemost_{computationally}intensive. The totalnumber

ofcalculations neededtodeterminethedivergence is determined_bythenumberof

permutationsofoptimalfeaturestochoose outofthe totalnumber offeatures. For

example,tochoosetheoptimal6 featuresout of atotalof 17 forall 10classes,thenumber

of calculations wouldbe:

[17!/3!

[10!/2!

Foreach ofthe 30600divergencemeasures, 2matrixinversesmustbecomputed.

The next viable optionforoptimalfeature selection wastheclass separation method

developed_byRosenblum [1990]. Theresults shouldbesimilartoRichards'method since

thecovariancematrices of allclasses wereincorporatedintotheseparation measure. This

methodhadtheadvantage of_beingfaster because onlyone matrixinversion isrequiredfor

each separationmeasurebetweentwoclasses.

Inordertodeterminetheoptimal number offeaturesneededforadequateclassification,the

optimalfeatureswas_{found fromthe 17features,all otherfeatures}

were removedfromthe

featurevectors. Theoptimal_{featuredata fromthe ten imagetextureswas then}

usedtotrain

theclassifier. Bothdependentandindependent datawere_subsequentlyclassified andthe

overallclassification_accuracy wasdeterminedforeachoptimalfeatureset. Theresults of

classification withtheoptimalfeature sets are giveninFigure 13.

S3

<

100-i

90

-2 80

60

8 10

#ofFeatures

-O

dependent

independent

-1

12

t

14 16

Figure 13: Classification_Accuracyvs.NumberofOptimal Features

Notethat theclassification_accuracyaxis onthisgraph rangesfrom60%to 100%. Over

99%classification_accuracyofboth thedependentandindependent datawas achieved with

6optimalfeatures. Additional featurescontributed_verylittleto_improvingthisaccuracy.

Thefirst6optimal featuresetsusedintheseclassifications canbe foundinTable 4. A

Noticethatthe classification_{accuracyusing 5}optimal featureswas_{slightly less} thanthe

classification _accuracy

using 4optimalfeatures. Theaddition ofmorefeaturestothe

feature vectorsdoesnot_necessarilycorrespondtohigherclassificationaccuracy. The

classification_{accuracy may}evendecrease ifthe_{probability overlap between}the classesis

increased_bytheadditionof morefeatures. Inthiscase,therewas more_{probability overlap}

betweenthe tenclasseswith _any set offive featuresthantherewas withtheoptimalsetof4

features.

Although, as_{previously stated,}most ofthese granulometricfeatures used weresizeand

direction_dependent,it is_interestingtonotethat thecircular_PSM,which is rotationally

invariantwasthemost significant of all 17featuresand appearedineach ofthefirst four

optimalfeature sets. _{LinearityPSM,}whichis invarianttoboth directionandscale,also

appearedinthesetof3optimalfeatures. Althoughtheseoptimalfeaturesets aredependent

ontheimagetexture classes,giventhe diverserange ofimagetextureclasses inthis study,

an optimal set of6features for anygiven setoftextureclasses canbeexpectedtogive

Table 4: _{Optimal FeatureSets}

usingRosenblumOptimization

1 feature: circularPSM

2features: circularPSM

horizontalPSSD

3 features: circularPSM

horizontalPSSD LinearityPSM

4features: circularPSM

horizontalPSM

negative-diagonalPSSD

negative-diagonalPSS

5features: horizontal PSM

negative-diagonalPSM

negative-diagonalPSSD

negative-diagonalPSS

positive-diagonalPSM

6 features: horizontal PSM

negative-diagonalPSM

negative-diagonalPSSD

negative-diagonalPSS

positive-diagonalPSM

verticalPSM