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Description and Simulation of Visual Texture

Carstensen, Jens Michael

Publication date:1992

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Carstensen, J. M. (1992). Description and Simulation of Visual Texture.

https://orbit.dtu.dk/en/publications/64482701-d678-4ce7-afda-2300556a7247

DESCRIPTION

AND

SIM

ULATION

OF

VISUAL

TEXTURE

By

JensMichaelCarstensen

LYNGBY1992

Ph.D.THESIS

NO.59

ISSN

0107{525X

c

Copyright1992

by


Printedby

,TechnicalUniversityofDenmark

ii

ThisdocumentwasformattedwithL ATEX.

HIPSandHIPS-2aretrademarksofSharpImageSoftware,NewYork,and

TheTuringInstitute,Glasgow,UK.

CARTisatrademarkofCaliforniaStatisticalSoftware,Inc.,Lafayette,

California.

SASisaregisteredtrademarkofSASInstituteInc.,Cary,NorthCarolina.

ConnectionMachineandC�

areregisteredtrademarksofThinkingMa-

chinesCorporation.CM-200isatrademarkofThinkingMachinesCorpo-

ration.

HP-UXandHPApollo9000/750areregisteredtrademarksofHewlett-

PackardCompany.

Sun-4isaregisteredtrademarkofSunMicrosystems,Inc.

Someoftheworkinthisthesishaspreviouslybeenpublishedin:

Carstensen,J.M.&

Conradsen,K.(1992)Spin- ipalternativestospin-

exchangeMarkovrandom

�eldsimulationimplementedonaSIMDmas-

sivelyparallelcomputer,SubmittedtoIEEETransactionsonPatternAnal-

ysisandMachineIntelligence.

Carstensen,J.M.,Grunkin,M.&Conradsen,K.(1991)Measurementof

enzymatictreatmente�ectontextileusingdigitalimageanalysis,IMSOR

Technicalreport.

iii

iv

Preface

ThisthesishasbeenpreparedattheInstituteofMathematicalStatisticsand

OperationsResearch(IMSOR),TechnicalUniversityofDenmark,inpartial

ful�llmentoftherequirementsforthedegreeofPh.D.inengineering.

Thegeneralframeworkofthethesisisstatisticsanddigitalimageanalysis.

Itisimpliedthatthereaderhasabasicknowledgeoftheseareas.

Thetreatmentofthesubjectsisbynomeansexhaustive,butisintended

toimprovetheknowledgeontexturedescriptionandtexturesimulationby

goingthroughselectedtheoryandexamples.Hopefullythiscanleadtoan

improvedtextureunderstanding.

Lyngby,April1992


v

vi

Acknowledgements

TheauthorwouldliketothankProfessorKnutConradsenforguidanceand

encouragementduringthecourseofthisworkandforprovidingexcellent

researchfacilitiesfortheimagegroupatIMSOR.

ProfessorBrianD.Ripleyisthankedforinspiringdiscussionsduringmy

visittotheUniversityofOxfordandforprovidingmanyusefulreferences.

IwishtothankmycolleaguesatIMSORforcontributingtoapleasant

andinspiringscienti�candsocialenvironment.EspeciallyIwouldliketo

thankmyoÆcemateMichaelGrunkinforhelpandmoralsupportandfor

copingwithmyfrustrationsinapleasantandenjoyableway.Dr.Bjarne

Kj�rErsb�llprovidedusefulhelpandcomments,forwhichIamthankful.

NielsChristianKriegerLassen,NielsJacobCarstensenandAllanAasbjerg

Nielsenwereveryhelpfulinthelastcriticalmomentsofthepreparation

ofthisthesis.IappreciatethehelpandcontinuoussupplyofL ATEX-styles

fromDr.CarlM.Bilbo.Dr.NielsKj�lstadPoulsenisacknowledgedforhis

expertcommentsoncontroltheory.Iamalsoverygratefultothemembers

oftheimagegroup,notmentionedabove,forhelpandinspiration.

vii

ImagedatawaskindlyprovidedbyNovoNordiskandImperialCancerRe-

searchFund.IwishtothankPeterRosholm

ofNovoNordisk,andDrs.

RichardMottandHansLehrachofImperialCancerResearchFund,Lon-

donfortheirsplendidcollaboration.

Dr.PeterFrykmanoftheGeologicalSurveyofDenmarkisacknowledged

forhiscollaborationonourattempttomakeareservoirsimulationprogram

thatprovesusefultogeologists.

Iam

gratefultotheViggoJarlfoundationforsupportingme�nancially

duringmystudies.

ThisresearchwaspartiallysponsoredbytheDanishTechnicalResearch

CouncilandtheDanishNaturalScienceResearchCouncilundertheMOBS

andCAPprograms.

viii

Summary

Theproblemoftextureanalysisisconsideredwithintheframeworkofdigi-

talimageanalysis.Anextensivesetoftexturestatisticsisreviewedand

explained,andtheirperformanceinmeasuringenzymatictreatmente�ect

ontextileandinclassi�cationofamoregeneralsetoftexturesisstudied.

Wefoundthatbothproblemsweresolvedsatisfactorilywiththesetof

texturestatisticsused.

Markovrandom�eldsarereviewedandinvestigatedasmodelsoftexture.

Resultsfrom

the�eldofstatisticalphysicsarereformulatedinastatis-

ticalsetting.StandardMarkovrandom

�eldsdonothavetheabilityto

modelmorphologicalpropertiesoftextures,andthisleadsustoformulate

anextensioninthetermsofmathematicalmorphology.Thepropertiesof

morphologicalMarkovrandom

�eldsareillustrated.Wegothroughthe

problemofMarkovrandom�eldparameterestimationandsuggestanex-

tensionoftheasymptoticmaximumlikelihoodestimator(Pickard,1987)to

theanisotropic�rst-ordermodel.

ix

Markovrandom�eldsimulationisdescribedandanew,fast,parallelal-

gorithmforsimulationconditionalonthe�rst-orderstatisticsispresented.

ThisalgorithmandthemorphologicalMarkovrandom�eldsarethenused

forthesimulationofthegeometricalstructureofoilreservoirs.

Markovrandom�eldsinaBayesiansettingareusedsuccessfullytoanalyze

hybridization�ltersautomaticallyforthehumangenomeproject.

x

Resum�e

Teksturanalysebetragtesindenforrammerneafdigitalbilledanalyse.Et

omfattendeantalstatistiskestikpr�vefunktionertilteksturbeskrivelseer

gennemg�aetogforklaret,ogderesevnetilatm�alee�ektafenzymbehand-

lingp�atekstilerogtilatklassi�cereetmeregenerelts�tafteksturerer

unders�gt.Beggeproblemervistesigatkunnel�sestilfredsstillendemed

deanvendtestikpr�vefunktioner.

DergivesenoversigtoverMarkovfelter,ogderesanvendelighedsomtekstur-

modellerudforskes.Resultaterom

dissemodellerfrastatistiskfysikre-

formuleresistatistiskterminologi.S�dvanligeMarkovfelterkanikkemodel-

lereteksturersmorfologiskeegenskaber,ogdettef�rerosfrem

tilatfor-

mulereenudvidelsevedbrugafmatematiskmorfologi.MorfologiskeMarkov-

feltersegenskaberbliverendvidereillustreret.Derredeg�resforproble-

merneiforbindelsemedestimationafparametre,ogderforesl�asenudvidelse

afdenasymptotiskemaximumlikelihoodestimator(Pickard,1987)tildet

anisotropetilf�lde.

xi

TeorienforsimulationafMarkovfelterergennemg�aet,ogderpr�senteres

enny,hurtig,parallelalgoritmetilsimulationafMarkovfeltergivetden

marginalefordelingafpixelv�rdier.Dennealgoritmeogdemorfologiske

Markovfelterbliverderefteranvendttilsimulationafdenrumligestruktur

ioliereservoirer.

MarkovfelterbliverienBayesiansksammenh�nganvendttilautomatise-

ringafanalysenafhybridiserings�ltreindenforhumangenomeprojektet.

xii

Contents

Preface

v

Acknowledgements

vii

Summary

ix

Resum�e(inDanish)

xi

1

Introduction

1

1.1

Texture

..............................

1

1.2

Textureanalysis

.........................

5

1.3

Outlineofthethesis.......................

6

xiii

2

Texturestatistics

9

2.1

First-ordergraylevelstatistics

.................10

2.1.1

Multi-resolution�rst-orderstatistics..........12

2.1.2

Histogrammatching...................13

2.2

Second-ordergraylevelstatistics................13

2.2.1

Graylevelcooccurrencematrices............13

2.2.2

Grayleveldi�erencehistogram

.............19

2.2.3

Graylevelsumhistogram

................20

2.2.4

Haralickfeatures.....................23

2.2.5

GLCM

asacontingencytable..............24

2.2.6

Multi-resolutionGLCM

.................25

2.2.7

GLCM

performance...................25

2.3

Higher-ordergraylevelstatistics

................27

2.3.1

Graylevelrunlengthmatrix

..............27

2.3.2

Neighboringgrayleveldependencematrix.......29

xiv

2.4

Statisticsforbinaryimages...................32

2.5

Fourierfeatures..........................34

2.6

Measurementofenzymatictreatmente�ectontextile

....36

2.6.1

Background........................36

2.6.2

Imageacquisition.....................36

2.6.3

Descriptionofvisualproperties.............39

2.6.4

AnalysisintheFourierdomain.............41

2.6.5

Spatialdomainfeatures.................45

2.6.6

Conclusion

........................52

2.7

GLCM

featureperformance...................53

2.7.1

Imagematerial......................53

2.7.2

GLCM...........................58

2.7.3

CARTclassi�cation

...................62

2.7.4

Classi�cationsummary

.................71

2.7.5

Conclusion

........................73

xv

3

Markovrandom

�elds

75

3.1

Random�elds...........................76

3.1.1

2Dgrids..........................76

3.2

Gibbsrandom�elds

.......................78

3.2.1

Historicalperspective

..................78

3.2.2

Generalproperties....................79

3.3

Markovrandom�elds

......................80

3.4

BinaryMarkovrandom�elds..................84

3.4.1

Isingmodelrevisited...................84

3.4.2

Morphologicalextension.................94

3.5

Pottsmodels

...........................105

3.5.1

Phasetransitions.....................106

3.5.2

Morphologicalextension.................107

3.5.3

Otherextensions.....................107

3.6

GaussianMarkovrandom�elds.................108

xvi

3.6.1

Alternativegrayleveldistributions...........109

4

Markovrandom

�eldparameterestimation

111

4.1

Introduction............................112

4.2

Codingestimation

........................112

4.3

Pseudolikelihoodestimation...................114

4.4

BinaryMRF

...........................115

4.4.1

Maximumpseudolikelihood

...............115

4.4.2

Asymptoticmaximumlikelihood

............118

4.4.3

Otherestimationmethods................118

4.5

Pottsmodel............................119

4.5.1


...............119

4.6

GaussianMRF

..........................120

4.6.1


...............120

4.6.2

Maximumlikelihood...................121

5

Markovrandom

�eldsimulation

123

xvii

5.1

Introduction............................124

5.2

Iterativesimulation........................124

5.2.1

TheMetropolisalgorithm

................126

5.2.2

Spin- ipalgorithms

...................127

5.2.3

TheMetropolisspin-exchangealgorithm

........129

5.2.4

Swendsen-Wangalgorithm................130

5.3

The�-controlledspin- ipalgorithm

..............134

5.3.1

Introduction

.......................134

5.3.2

Thefeedbackloop

....................135

5.3.3

Relationtoimportancesampling............137

5.3.4

Parallelimplementation.................138

5.3.5

Results

..........................139

5.3.6

Conclusion

........................145

5.4

Simulationofgeologicalstructures...............147

5.4.1

Introduction

.......................147

xviii

5.4.2

Modeltypes........................148

5.4.3

AMarkovrandom�eldreservoirmodel........149

5.4.4

Simulationresults

....................151

5.4.5

Conclusion

........................153

6

Bayesianparadigm

157

6.1

Introduction............................158

6.2

Priordistribution.........................158

6.3

Observationmodel........................159

6.4

Maximumaposteriori(MAP)estimates............160

6.4.1

Simulatedannealing...................160

6.4.2

Iteratedconditionalmodes(ICM)

...........161

6.5

Marginalposteriormodes(MPM)................161

6.6

Hybridization�lteranalysis...................163

6.6.1

Background........................163

6.6.2

Robotdynamics

.....................164

xix

6.6.3

Imageanalysisproblem

.................164

6.6.4

Digitization........................166

6.6.5

Preprocessing.......................168

6.6.6

Spotlocalization.....................172

6.6.7

Spotclassi�cation

....................179

6.6.8

Results

..........................179

6.6.9

Conclusion

........................188

7

Conclusion

189

7.1

Summary

.............................189

7.2

Acomment............................191

A

Developedsoftware

193

B

GLCM

forallBrodatztextures

199

References

213

Index

223

xx

Chapter1

Introduction

1.1

Texture

Thetermvisualtextureinthetitleofthisthesisemphasizes,thatthede�-

nitionoftextureusedhereiscloselyrelatedtoperception.

Atextureisaregionin2Dor3Dthatcanbeperceivedasbeing

spatiallyhomogeneousinsomesense.

Thisde�nitionisverybroad.Itincludesastexturethetotallyuniform

region,whichinthedailylanguageissaidtohavenotexture.Indeedthe

interestingthingabouttexturesisthestudyofthespatialvariationsover

thetexturedregion,andthesevariationsoftenbecomesynonymouswith

1

2

Chapter1.Introduction

texture.Weemphasizethattexturesonlydi�eringinluminancearecon-

sidereddi�erenttexturesaccordingtoourde�nition.Brodatz(1966)isa

photographicalbum

with112textures.Thisalbum

hasbecomeastan-

dardreferenceintextureanalysisandsubsequentlythesetexturesshallbe

referredtoastheBrodatztextures.

Figure1.1showsastrictlyrandomtexture.Thepixelsareuncorrelated.

Figure1.2showsastrictlydeterministictexture(acheckerboard).Itisa

strictlyorderedpattern,thatisfullydeterminedfromtheknowledgeofa

smallsubpattern.Observabletexturesaresomewherebetweenthesetwo

extremes.Figures1.3and1.4showexamplesofarandomtexture(hand-

madepaper)andadeterministictexture(abrickwall).Thewordtexture

comesfromtheLatinwordtextura,thatmeanstextilefabric,andtextile

fabricisanotherexampleofadeterministictexture.Rao(1990)classi�es

alloftheBrodatztexturesinthreeclasses:disordered(random),weakly

ordered,andstronglyordered(deterministic).

Thequestionofscaleorresolutionisfundamentaltotextureperception.If

wezoominonthebrickwallof�gure1.4weseethetextureoftheindividual

bricks.Ifwezoomoutwemayseeatextureofwallshadingoratextureof

wallandwindows.Thustheremaybeseverallevelsofcompletelydi�erent

texturesinthesameimagebutatdi�erentscales.Atexturewithmorethan

onetextureleveliscalledahierarchicaltexture.Todistinguishbetween

di�erenttexturelevelswecanusethetermsmicrotextureandmacrotexture.

Subsequentlythetermsscaleandresolutionwillbeusedinterchangeably.

Researchersintextureperceptionhaveinvestigatedpreattentive(e�ortless

orinstantaneous)texturediscriminationinthehumanvisualsystem.The

famousiso-second-orderconjecture(Julesz,1975)statedthattextureswith

1.1

Texture

3

Figure1.1.Astrictlyrandomtexture

Figure1.2.Astrictlydeterministictexture

4


Figure1.3.Handmadepaper(D57fromBrodatz)

Figure1.4.Brickwall(D95fromBrodatz)

1.2

Textureanalysis

5

thesamesecond-orderstatistics(graylevelstatisticsofpairsofpixels)can

notbedistinguishedeveniftheyhavedi�erentthird-orhigher-orderstatis-

tics.Thisconjecturehaslaterbeendisproved(Julesz,1981)andreplaced

byatextontheory(Julesz&Bergen,1983).Textonsaresmallconspicuous

featureslike

�Elongatedshapes,suchasellipses,rectanglesorlinesegments.

�Endsoflinesegments.

�Crossingsoflinesegments.

Thetextonconjecturearguesthatpreattentivetexturediscriminationis

basedondi�erencesinthedensityoftextons.

1.2

Textureanalysis

Themaingoaloftextureanalysisistoextractusefultexturalinformation

fromanimage.Historicallytherehasbeentwomajorapproaches,astruc-

turalandastatistical.Thestructuralapproachdescribesatexturebya

subpatternorprimitiveandthespatialdistributionofprimitives,theplace-

mentrule.Theprimitivesarealsocalledtextureelements.Ifweconsider

thebrickwalltheprimitiveisabrickandtheplacementrulespeci�esthe

arrangementofbricksinthewall.Thestatisticalapproachismoregenerally

applicable,becauseitdoesnotpresumethatthetexturecanbedescribed

intermsofprimitivesandplacementrules.Itdrawsonthegeneralsetof

statisticaltools.Thisthesisisprimarilybasedonthestatisticalapproach.

6


Theextractionoftexturefeaturesisessentialtoapplicationssuchastex-

turemeasurement,texturesummarization,textureclassi�cationandtexture

segmentation(Texturedescriptiondenotesalloftheseareas).Thegoalof

texturemeasurementistocharacterizeatexturewithonefeature,e.g.a

featurefortextilewearassessment.Intexturesummarizationwegivesum-

mariesre ectingthevisualpropertiesoftextures.Textureclassi�cation

usuallyservesoneoftwogoals.Wemaywanttoassignaclasstoanentire

texture,e.g.acceptorrejectinindustrialqualitycontrol.Wemayalso

wanttoassignatextureclasstoeverypixelinanimageandthusobtaina

partitioningofthisimage.Texturesegmentationcorrespondstopixelwise

textureclassi�cationwithnoaprioriknowledgeofthenumberoftexture

componentsorthepropertiesofeachtexturecomponent.

ForgeneralreviewsontextureanalysisthereaderisreferredtoHaralick

(1979),vanGool,Dewaele,&Oosterlinck(1985),Tomita&Tsuji(1990),

Rao(1990).

1.3

Outlineofthethesis

Chapter2givesanoverviewoftexturestatisticsusedintextureanalysis.

Thisoverviewisfollowedbytwocasestudiesthatevaluatestheperformance

ofthesestatisticsinaccuratelymeasuringtexturalproperties.Firstwe

wanttomeasurethetexturalchangesthatcottontextilesundergoduring

cellulaseenzymatictreatment.Thenweusesecond-orderstatisticsforthe

classi�cationofBrodatztextures.

1.3

Outlineofthethesis

7

Chapter3dealswithparametricdescriptionoftexturebasedonaclass

ofmodelscalledMarkovrandom

�elds.ThetheoryofMarkovRandom

�eldsisreviewedtogetherwiththetheoryoftheassociatedGibbsrandom

�elds.ThetheoryofGibbsrandom�eldswerefoundedinstatisticalphysics

(Ising,1925)andsomerelevantresultsfromthisareaispresentedinanew

statisticalsetting.AvarietyofMarkovrandom�eldsisreviewedwithan

emphasisondiscretemodels.Furtherweintroduceasetofmorphological

Markovrandom�elds,thatextendsthestandardsetofmodelsbyusingthe

operatorsofmathematicalmorphology(Serra,1982).

FormostpracticalapplicationsofMarkovrandom�eldsitisessentialthat

wehaveaccurateandfeasiblealgorithmsforparameterestimation.

In

chapter4aselectionofestimationmethodsisreviewed,andsomeofthese

methodsareappliedinchapter5.Anextensionoftheasymptoticmaximum

likelihoodestimator(Pickard,1987)totheanisotropiccaseisproposedin

section4.4.2.

Inchapter5wereviewasetofiterativesimulationschemesforMarkov

random�eldsimulation.Wethenpresentafastnewparallelalgorithmfor

simulatingMarkovrandom�eldsconditionalongiven�rst-orderstatistics.

WeinvestigatetheuseofthisalgorithmandamorphologicalPottsmodel

inthesimulationofgeologicalstructures.

TheBayesianparadigmisaframeworkforincorporatingstochasticmodels

ofvisualphenomenaintoaverygeneralsetoftasksfromimageprocessing

andimageanalysis.Inchapter6wegiveashortreviewofBayesianimage

analysisandpresentanapplicationthatmakessuccessfuluseofMarkovran-

dom�elds,theMetropolisalgorithmandsimulatedannealinginaBayesian

framework.

8


Chapter2

Texturestatistics

Texturestatisticsisfrequentlyclassi�edinto�rst-order,second-orderand

higher-orderstatistics.First-orderstatisticsrefertothemarginalgraylevel

distribution.Second-orderstatisticsrefertothejointgrayleveldistribution

ofpairsofpixelsandhigher-orderstatisticsrefertothejointgraylevel

distributionofthreeormorepixels.

Thischaptergivesanoverviewoftexturestatisticsusedintextureanalysis.

Thisoverviewisfollowedbytwocasestudiesthatevaluatetheperformance

ofthesestatisticsinaccuratelymeasuringtexturalproperties.Firstwe

wanttomeasurethetexturalchangesthatcottontextilesundergoduring

cellulaseenzymatictreatment.Thenweusesecond-orderstatisticsforthe

classi�cationofBrodatztextures.

9

10

Chapter2.Texturestatistics

2.1


The�rst-ordergraylevelstatisticscanbederivedfromthegraylevelhis-

togram

fhi g.hiisthenumberofpixelsinanimagewithgrayleveli.If

NisthetotalnumberofpixelsandGisthenumberofgraylevelsthen

PG�1

i=0

hi=N.Thenormalizedhistogram

fHi gwithHi=hi =N

isthe

empiricalprobabilitydensityfunctionforsinglepixels.Statisticscomputed

fromHiinclude:

1.Themeangraylevel

�=

G�1

Xi=0iHi

�measurestheaverageintensityintheimage.

2.Thegraylevelvariance

�2=

G�1

Xi=0

(i��)2H

i

where�isthestandarddeviation.Thevarianceandthestandard

deviationmeasurestheglobalcontrastintheimage.

3.ThecoeÆcientofvariation

cv=��

ThecoeÆcientofvariationisinvariantunderachangeofscale,i 0=Ai,

thusiftheintensityscalehasanaturalzero,thenthecvwillbeascale

invariantmeasureofglobalcontrast,

4.Thegraylevelskewness

1=

1�3

G�1

Xi=0

(i��)3H

i

2.1


11

Skewnessmeasurestheextenttowhichoutliersfavoronesideofthe

distribution.Skewnessisinvariantunderalineargrayscaletransfor-

mationi 0=Ai+B.

5.Thegraylevelkurtosis

2=

1�4

G�1

Xi=0

(i��)4H

i �3

Kurtosismeasuresthepeakednessortailprominenceofthedistribu-

tion.Itis0:0fortheGaussiandistribution.Kurtosisisinvariant

underalineargrayscaletransformationi 0=Ai+B.

6.Thegraylevelenergy

e=

G�1

Xi=0H2i

whereG�1�e�1.Energymeasuresthenonuniformityofthehis-

togram.

7.Thegraylevelentropy

s=�

G�1

Xi=0Hi logHi

where0�s�logG.Entropymeasurestheuniformityofthehis-

togram.Thisquantityiswidelyusedinimagecompression.Ifthe

logarithmisofbase2,itisthelowerboundontheaveragelengthof

thebinarycodewordsusedinerror-freecompressionofindependent

datasamples.

12


00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Normalizedvariance

Scale

D43

3

3

3

3

3

3

D16+

+

+

+

+

+

+

D292

2

2

2

2

2

2

D53�

�

�

�

�

�

�

D774

4

4

4

4

4

4

Figure2.1.Normalizedvarianceversusscalefor�veBrodatztextures.

2.1.1

Multi-resolution�rst-orderstatistics

First-orderstatisticscomputedatseveraldi�erentscales(resolutions)will

provideuswithinformationaboutsecond-andhigher-orderstatistics.As

anexamplewehavetaken�veBrodatztextures,andsuccessivelylowpass

�lteredandsubsampledthem�vetimes.Wehavecomputedthevariance

ofeachimageandthendividedbythevarianceofthefullresolutionimage.

In�gure2.1weseethattheresultis�vecurves,thatcanbedistinguished.

Thusmulti-resolution�rst-orderstatisticscontainimportanttexturalinfor-

mation.

2.2

Second-ordergraylevelstatistics

13

2.1.2

Histogram

matching

The�rst-orderstatisticsarehighlydependentonthelightingconditions.It

isthereforecommonpracticetotrytoeliminatethein uenceof�rst-order

statisticsintextureanalysisbymakingthegraylevelhistogrammatcha

speci�cdistribution.Amatchtoauniformdistributioniscalledhistogram

equalization,andthisisbyfarthemostusedmatch.AmatchtoaGaussian

distributionisanotherpossibility,andinsection2.7weseethatthisisa

moregentlematchespeciallyforstochastictextures.

2.2


Theautocorrelation(orthecloselyrelatedvariogram)isprobablythebest

knownsecond-ordergraylevelstatistic.Wewill,however,considerthe

second-ordergraylevelstatisticsinamoregeneralsetting:graylevelcooc-

currencematrices(GLCM).Theautocorrelationcanbecomputedfrom

thesecooccurrencematrices.

2.2.1

Graylevelcooccurrencematrices

Thegraylevelcooccurrencematricesareafullrepresentationofthesecond-

ordergraylevelstatistics.AGLCM,c,isde�nedwithrespecttoagiven

(row,column)displacementh,andelement(i,j),denotedcij ,isthenumber

oftimesapointhavinggrayleveljoccursinpositionhrelativetoapoint

havinggrayleveli.LetNh

bethetotalnumberofpairs,thenCij=cij =Nh

denotestheelementsofthenormalizedGLCM,C.

14


Themeaningoftheabovede�nitiongetsmoreapparentifweasanexample

computecfromthe4-colorimage

2

1

1

3

3

3

1

0

2

2

3

3

1

1

2

0

2

1

0

1

0

0

1

0

0

Ifh=(0,1),i.e.onestepinthehorizontaldirection,thencwillbe

0

1

2

3

0

2

2

2

0

1

3

2

1

1

2

0

2

1

0

3

0

2

0

2

andNh

willbeequalto20.

Itiseasilyseenthat

C(�h)=CT(h)

whereCT

isCtransposed.

AsymmetricGLCM,cs (h),canbeobtainedbypoolingthefrequenciesof

c(h)andc(�h).Hence

cs (h)=c(h)+cT(h)

and

Cs (h)=12

[C(h)+CT(h)]

2.2


15

Assumingisotropy(nodirectionality)wecanpoolthefrequenciesofcooc-

currencematriceswithdisplacementshofdi�erentanglesandapproxi-

matelythesamelengthh.ThisprovidesuswiththeisotropicGLCM,

ci (h),where

ci (1)=cs (0;1)+cs (1;0)+cs (1;1)+cs (�1;1)

and

Ci (1)=14

[Cs (0;1)+Cs (1;0)+Cs (1;1)+Cs (�1;1)]

Oneofthemainproblemsassociatedwiththeuseofcooccurrencematricesis

thattheyhavetobecomputedformanydi�erentvaluesofh,thusproviding

uswithanimmenseamountofdata.Datareductioncanbeaccomplished

bypoolingthematricesasshownabove,byreducingthenumberofgray

levelsorbycomputingtexturefeaturesfromeachmatrix.Thesefeatures

canthenbeusedfordescriptionandclassi�cationoftextures.

Let

Cxi

=G�1

Xj=0Cij

Cyj

=G�1

Xi=0Cij

andlet�x,�y ,�x

and�y

bethemeansandstandarddeviationsofCxi

and

Cyj

overiandj.Thenanumberoffeaturescanbecomputedfrom

the

GLCM

including:

1.EnergyorAngularSecondMoment

E=

G�1

Xi=0

G�1

Xj=0C2ij

16


whereG�2�E�1.EtakesthevalueG�2forauniformdistribution

overC,andthevalue1i�onlyonecellisnonzero.

2.Entropy

S=�

G�1

Xi=0

G�1

Xj=0CijlogCij

where0�S�logG2.StakesthevaluelogG2forauniformdistri-

butionoverC,andthevalue0i�onlyonecellisnonzero.

3.MaximumProbability

M=maxCij

whereG�2�M�1.MtakesthevalueG�2forauniformdistribu-

tionoverC,andthevalue1i�onlyonecellisnonzero.

4.Correlation(orAutocorrelation)

�=

G�1

Xi=0

G�1

Xj=0

(i��x)(j��y )Cij

�x�y

where�1��1.�takesthevalue1i�onlyvaluesonthemain

diagonalofC

arenonzeroandthevalue0i�thegrayvaluesare

uncorrelated.

5.DiagonalMoment

D=

G�1

Xi=0

G�1

Xj=0 ji�jj(i+j��x ��y )Cij

Thediagonalmomentbasicallymeasuresthedi�erenceincorrelation

forhighgraylevelsandforlowgraylevels.ItismentionedinLaws

(1980),buthasotherwisebeenleftoutinmoststudiesofGLCM.

2.2


17

6.InformationalCoeÆcientofCorrelation

r1= p1�e�2r0

where

r0=�

G�1

Xi=0

G�1

Xj=0CxiCyjlog(CxiCyj)�S

istheLogarithmicIndexofCorrelation(Linfoot,1957).Sincer0 �0

wehave0�r1<1.

7.MaximalCorrelationCoeÆcient.Thisfeatureisthesquarerootof

thesecondlargesteigenvalueofQwhere

Qij= X

k

Cik Cjk

CxiCyk

LetR

andSbeequaltoCwithrespectivelyrowsumsandcolumn

sumsnormalizedtounity,i.e.R

ij=Cij

Cxi

Sij=Cij

Cyj

then

Q=RST

R,ST

andQarestochasticmatrices,i.e.theirlargesteigenvalueis1.

IftheyareconsideredastransitionmatricesforaMarkovchainand

iftheyareirreduciblethenthehistogramvectorpwillbetheunique

invariantdistributionfortheMarkovchain.Therateofconvergence

totheinvariantdistributionisdeterminedbythesecond-largesteigen-

value,�2 ,where0��2<1(Seneta,1981).Qisthetransitionmatrix

foronejumpwithdisplacementvectorhandbackagain.Ifwestart

18


atonepixelwithaninitialgrayleveldistributionandthenmakes

successivejumpsbackandforth,thenthegrayleveldistributionwill

approachtheinvariantgrayleveldistribution.Thememoryofthe

grayleveldistributionretainedineachjumpbackandforthisdeter-

minedbythesecond-largesteigenvalue,�2 ,ofQ.Ifthepixelsoneach

sideofajumpareindependent,wehave�2=0.

Energy,entropyandmaximumprobabilityareuniformitymeasures.They

allhaveoneextremumfortheuniformdistributionandanotherextremum

whenoneprobabilityequalsunity.Thedi�erencebetweenthesemeasures

isdemonstratedfortwodistributionswith4possibleoutcomes.

p1

p2

p3

p4

Energy

Entropy

0.50

0.50

0.00

0.00

0.50(1)

0.69(2)

0.76

0.08

0.08

0.08

0.60(2)

0.81(1)

Theuniformityrankingsareshowninparentheses.Theenergymeasure

assumesthe�rstdistributiontobethemostuniformofthetwo,whilethe

entropymeasurechoosesthesecond.Weseethat,whenmeasuringunifor-

mity,energypenalizessinglehighprobabilities,whileentropypenalizeszero

probabilities.Ifweincreasethezeroprobabilitiesofthe�rstdistributiona

littlethenentropywillreversetheranking,andmakethisthemostuniform

distributionaccordingtobothmeasures.

p1

p2

p3

p4

Energy

Entropy

0.48

0.48

0.02

0.02

0.46(1)

0.86(1)

2.2


19

Maximumprobabilitymeasuresuniformitysolelyonthebasisofthehighest

probabilityandtherankingbythismeasurewilloftenagreewiththatbased

onenergy.

Thecooccurrencematrixitselfcanalsobeusedasafeature(Vickers&

Modestino,1982;Parkkinen&Oja,1986).

TheuseofGLCMintextureanalysisissometimesreferredtoasthespatial

grayleveldependencemethod(SGLDM).

2.2.2

Grayleveldi�erencehistogram

Thegrayleveldi�erencehistogram(GLDH)isahistogramoftheabsolute

di�erencesofgraylevelsfrompairsofpixels.ItiscomputedfromtheGLCM

bysummingthetwo-dimensionaldensityCijoverconstantvalueofji�jj.

TheGLDHcanberegardedasahistogramofthe"distance"tothemain

diagonalintheGLCM.

Dk=

G�1

Xi=0

G�1

Xj=0

| {z}ji�jj=k

Cij ;k=0;::;G�1

ThefeaturescomputedfromtheGLDHinclude:

1.Di�erenceEnergy

DE=

G�1

Xk=0

D2k

whereG�1�DE�1.

20


2.Di�erenceEntropy

DS=�

G�1

Xk=0

DklogDk

where0�DS�logG.

3.Inertia,ContrastorVariogram

I=

G�1

Xk=0

k2D

k=2�2(1��)

(2.1)

where�isthegraylevelvarianceand�isthecorrelation.

4.InverseDi�erenceMomentorLocalHomogeneity

IDM=

G�1

Xk=0

Dk

1+k2

5.Di�erenceVariance

DV=I�(

G�1

Xk=0

kDk )2

GLDHfeaturesareasubsetofGLCM

features,andthisrelationwillsub-

sequentlybeimplicit.Theonlyusefulwayofcomparingthetwosetsof

featuresistodeterminethelossofinformationwhengoingfromGLCMto

GLDH.FeaturecomputationfromtheGLDHisoftencalledthegraylevel

di�erencemethod(GLDM).TheadvantageofGLDM

isthelowerstorage

requirementsandlowercomputationalcomplexity.

2.2.3

Graylevelsum

histogram

Thegraylevelsumhistogram(GLSH)isahistogramofthesumofpairsof

pixels.ItiscomputedfromtheGLCM

bysummingthetwo-dimensional

2.2


21

densityCijoverconstantvalueof(i+j),i.e.

Sk=

G�1

Xi=0

G�1

Xj=0

|{z}i+j=k

Cij ;k=0;::;2G�2

WewillusetheSumAveragebelow

SA=

2G�2

Xk=0

kSk=�x+�y

ThefeaturescomputedfromtheGLSHinclude:

1.SumEnergy

SE=

2G�2

Xk=0

S2k

where(2G�1)�1�SE�1.

2.SumEntropy

SS=�

2G�2

Xk=0

SklogSk

where0�SS�log(2G�1).

3.SumVariance

SV=

2G�2

Xk=0

(k�SA)2S

k=2�2(1+�)

where�isthegraylevelvarianceand�isthecorrelation.

4.ClusterShade

A=

2G�2

Xk=0

(k�SA)3S

k

22


5.ClusterProminence

B=

2G�2

Xk=0

(k�SA)4S

k

GLSH-featureshasnotbeenusedaswidelyasfeaturesbasedonGLDH.

Conners,Trivedi,&Harlow(1984)foundthatclustershadeandcluster

prominencewasausefulsupplementtotheGLCM

andGLDHfeatures

mentionedabove.LiketheGLDH,theGLSHhaslowerstoragerequire-

mentsandlowercomputationalcomplexity,butnoauthorshavetoour

knowledgetriedtheGLSHfeaturesbythemselves.

ItisobviousthattheGLDHandGLSHcontainsalotoftheinformation

fromthecooccurrencematrices.Duetothelowercomputationalcomplexity

itis,asmentionedfortheGLDH,relevanttoinvestigateifanysigni�cant

informationislostwhengoingfromtheGLCM

totheGLDHandGLSH.

ThestructureoftheGLCM

isdiagonalandoftenatleastapproximately

symmetric.Unser(1986b)approximatedtheenergyandentropyfeaturesof

theGLCMfromtheGLDHandtheGLSH.Thisapproximationgaveonlya

slightdecreaseinclassi�cationaccuracy.Unser(1986b)notedthatthesum

anddi�erenceofpairsofpixelsaredecorrelated,butthisisnotgenerally

trueforthesumandtheabsolutedi�erence.Thediagonalmomentmeasures

thiscorrelationanditvariesfromtexturetotexture(Seesection2.7).

Unser(1986b)alsousedtheGLDHandGLSHthemselvesasfeatures.

2.2


23

2.2.4

Haralickfeatures

Mostofthefeaturesmentionedinthissectionwereintroducedintexture

analysisinapaperbyHaralick,Shanmugam,&Dinstein(1973),where14

di�erentfeatures(f1-f14)werepresented.Theyareallnaturaldescriptors

oftwo-dimensionaldistributions,althoughtheyseemtohavebeenselected

inaratheradhocmanner.Eventhoughitisrecognizedthatthesefeatures

donotdescribeallaspectsofthecooccurrencematricestheyhavebeenused

veryrigorouslyinmanypapers.Threeofthefeatureswerenotincludedin

thelistofGLCM

features.

�Variance(f4).Thegraylevelvariancebelongstothe�rst-orderstatis-

tics.

�SumAverage(f6)

f6=SA=

2G�2

Xk=0

kSk=�x+�y

Thisfeaturealsobelongstothe�rst-orderstatistics.

�InformationMeasuresofCorrelation(f12andf13).

f12=HXY�HXY1

max(HX;HY)

f13= p1�exp(�2(HXY2�HXY))

whereHXY=Sand

HXY1=�

G�1

Xi=0

G�1

Xj=0Cijlog(CxiCyj)

HXY2=�

G�1

Xi=0

G�1

Xj=0CxiCyjlog(CxiCyj)

24


AsmentionedinLinfoot(1957)HXY1=HXY2.f12andf13are

thuscloselyrelated,andonlyf13,theinformationalcoeÆcientofcor-

relation,isconsidered.

2.2.5

GLCM

asacontingencytable

Zucker&Terzopoulos(1980)interpretedthecooccurrencematrixCasa

normalizedcontingencytable(Seee.g.Bishop,Fienberg,&Holland(1975))

andusedthe�2statistictoselectmatricessuitableforclassi�cation.

�2=Nh

G�1

Xi=0

G�1

Xj=0

(Cij �CxiCyj)2

CxiCyj

=Nh(

G�1

Xi=0

G�1

Xj=0

C2ij

CxiCyj

�1)

The�2valuesandtheselecteddisplacementshcanbeusedasfeaturesfor

classi�cation.

The�2statisticmeasurestheassociationbetweenvariablesincontingency

tables,butdoesnotdiscriminateamongthetypesofassociation.Figueiras-

Vidal,Paez-Borrallo,&Garcia-Gomez(1987)pointedoutthatperiodicity

isindicatedinacooccurrencematrixbyaconcentrationofhighcounts

aroundthemaindiagonal.Theysuggestedtheinertiameasure(2.1)to

detectperiodicities.

The�measureofagreement(Cohen,1960)

�= P

G�1

i=0

Cii � PG�1

i=0

CxiCyi

1� PG�1

i=0

CxiCyi

wassuggestedbyParkkinen,Selk�ainaho,&Oja(1990)todetectperiodic-

ities.Itdirectlymeasurestheconcentrationonthemaindiagonalandthe

2.2


25

computationalcomplexityisO(G)insteadofO(G2)forthe�2

statistic.

The�statisticworksbestwithalimitednumberofgraylevels,e.g.from

4to32,and,aswewillseeinsection2.4,itcorrespondstothecorrelation

measureforbinarytextures.

Manyotherfeaturescanbeusedtoselectthedisplacement(s)thatgivethe

bestclassi�cation.

2.2.6

Multi-resolutionGLCM

Weszka,Dyer,&Rosenfeld(1976)concludedthatlarge-distancecooccur-

rencefeaturesgavebetterperformanceifaspatialaveragingwasdone�rst.

Thissuggeststhatcooccurrencematricesatseveraldi�erentscalesshould

beconsidered.

2.2.7

GLCM

performance

GLCMfeatureshaveanextensivehistoryasareferencefortexturefeature

performance.Weshallgiveabriefsummary.

Haralicketal.(1973)usedGLCM

featurestoclassifyphotomicrographs

ofsandstone,panchromaticaerialphotographsandmultispectralsatellite

imagery.Theyfoundthattexturalfeaturesisavaluablesupplementto

spectralfeatures.

Weszkaetal.(1976)comparedtheclassi�cationperformanceonaerialpho-

tographsandLANDSATimageryofGLCM

features,GLDHfeatures,ring

26


andwedgefeaturesinthespatialfrequencydomainandgraylevelrunlength

features.GLCMandGLDHfeatureswerefoundtobethemostusefuland

ofalmostequalperformance.

Conners&Harlow(1980)madeatheoreticalcomparisonofthesamegroups

offeatures,andtheresultsagreeverywellwiththoseofWeszkaetal.(1976).

Manyauthorshavesincethenintroducednewtexturalfeaturesandclaimed

thesetobesuperiortotheGLCMfeatures.

Laws(1980)claimedthathistextureenergyfeaturesperformedsigni�cantly

betterthanGLCMfeaturesinsegmentationofacompositeofeightBrodatz

textures.

Kashyap,Chellappa,&Khotanzad(1982)usethemaximumlikelihoodes-

timateoftheparametersinaSimultaneousAutoregressiveModel(SAR)

asfeaturesforclassi�cation.TheresultiscomparabletothatofGLCM

features.

Vickers&Modestino(1982)usedanisotropiccooccurrencematrixtoclas-

sifysubimagesof9Brodatztextures.

Fordistancesof1,3and5they

obtainedbetween95%and98%correctlyclassi�ed.Parkkinen&Oja(1986)

usedcooccurrencematriceswithahorizontaldisplacement.

Siew,Hodgson,&Wood(1988)usedGLCM,GLDH,graylevelrunlength

andneighboringgrayleveldependencefeaturestomeasurecarpetwear.

TheirdistinctionbetweenGLCM

andGLDHfeaturesisnoninformative

since2featuresarecommontobothgroups,andtheyuseastandardGLCM

andanisotropicGLDH.Theresultshowsthatfeaturesfromallfourgroups

cancharacterizetheappearanceofcarpets.Theirresearchindicatesthat

2.3


27

thesestatisticalmeasuresaresuperiortoatrainedpanelinreliablyranking

carpetsaccordingtowear.

duBuf,Kardan,&Spann(1990)compared7setsoffeaturesandfound

thatGLCM,Laws(Laws,1980)andUnser(Unser,1986a)featureswere

generallybest.

Berry&Goutsias(1991)madeacomparisonbetweenfeaturesbasedonthe

neighboringgrayleveldependencematrix(NGLDM)ofSun&Wee(1983)

andGLCM

features.OnsynthetictexturesNGLDM

featuresperformed

better.Onnaturaltexturestheyperformedequallywell.

2.3


Higher-ordergraylevelstatisticsweredeclaredunimportantfortextureper-

ceptionbythenowdisprovediso-second-orderconjecture(Julesz,1975),but

theyseemtohaveregainedtheirpopularityintheliterature.Wewillreview

afewapproaches.

2.3.1

Graylevelrunlengthmatrix

Agraylevelrunisasetofconsecutive,collinearpixelswiththesamegray

level.Thenumberofpixelsinarunistherunlength.Galloway(1975)

usedagraylevelrunlengthmatrix(GLRLM)tocomputetexturefeatures.

Element(i,j)oftheGLRLM,r,isdenotedrij ,andthisisthenumberof

runsofgraylevelihavinglengthj.ThetotalnumberofrunsisNr

The

28


GLRLM

canbecomputedinanydirection,butusuallyonlydirections0Æ,

45Æ,90Æand135Æareused.Fromtheimage

2

1

1

3

3

3

1

0

2

2

3

3

1

1

2

0

2

1

0

1

0

0

1

0

0

wecancomputerforthehorizontaldirection(0Æ).

GLRLM

1

2

3

4

5

0

3

2

0

0

0

1

4

2

0

0

0

2

3

1

0

0

0

3

1

2

0

0

0

whereNr=18.

Thefollowingfeatures,computedfromtheGLRLM,weresuggested:

1.ShortRunsEmphasis

RF1=

G�1

Xi=0

LXj

=1

Rij

j2


ofthecountsandthevalue1i�onlyonecellisnonzero.

2.3


29

2.LongRunsEmphasis

RF2=

G�1

Xi=0

LXj

=1j2R

ij

3.GrayLevelNonuniformityR

F3=

G�1

Xi=0

[LXj

=1Rij ] 2

where1=G

�RF3�1.RF3takesthevalue1=Gforauniform

distributionofthecountsandthevalue1i�onlyonecellisnonzero.

4.RunLengthNonuniformity

RF4=

LXj

=1 [ G

�1

Xi=0Rij ] 2

where1=L�RF4�1.RF4takesthevalue1=Lforauniform

distributionofthecountsandthevalue1i�onlyonecellisnonzero.

5.RunPercentage

RF5=Nr =N

where1=N�RF5�1.

TheGLRLM

featuresareverysensitivetonoise,andthisisprobablythe

reasonforthereportedbadperformance(e.g.Weszkaetal.(1976)).The

performancefordiscrete(e.g.binary)texturesislikelytobebetter.

2.3.2

Neighboringgrayleveldependencematrix

Theneighboringgrayleveldependencematrix(NGLDM)wasintroducedby

Sun&Wee(1983).Inthisapproachallneighborsofapixelareconsidered

30


atthesametime.Aneighborisapixelwithinacertaindistancedofthe

centralpixelandSisthenumberofneighbors.disusuallychosentobe

p2andthenS=8.Apixelanditsneighboraresaidtohavesimilargray

levelsiftheabsolutegrayleveldi�erenceislessthanorequaltoachosen

positivenumbera.Element(k,s)ofaNGLDM,q,isdenotedqks ,andthis

isthenumberofpixelswithgraylevelkhavingsneighborswithsimilar

graylevels.LetNd

bethetotalnumberofcountsinq,thenQ

=q=Nd

isthenormalizedNGLDM.Thenotationpresentedheredi�ersfrom

the

notationofSunandWee.Thisistokeepthede�nitionsalongthelinesof

theGLCMde�nition.Fromtheimage

2

1

1

3

3

3

1

0

2

2

3

3

1

1

2

0

2

1

0

1

0

0

1

0

0

wecancomputeqfora=0andd=p

2as

NGLDM

0

1

2

3

4

5

6

7

8

0

1

0

1

0

0

0

0

0

0

1

0

0

0

4

0

0

0

0

0

2

1

0

1

0

0

0

0

0

0

3

0

0

1

0

0

0

0

0

0

whereNd=9.

Thefeatures,thatSunandWeesuggestcomputedfromtheNGLDM,are

listedbelowwiththemodi�cationthatthefeaturecomputationhereis

2.3


31

basedonthenormalizedNGLDM.Thismeansthatthefeaturesareinde-

pendentofNd .

1.SmallNumberEmphasisN

1=

G�1

Xk=0

SXs

=0

Qks

1+s2


ofthecountsandthevalue1i�onlyonecellisnonzero.

2.LargeNumberEmphasisN

2=

G�1

Xk=0

SXs

=0s2Q

ks

3.NumberNonuniformity

N3=

SXs

=0 [ G

�1

Xk=0

Qks ] 2

4.SecondMoment

N4=

G�1

Xk=0

SXs

=0Q2k

s

5.Entropy

N5=�

G�1

Xk=0

SXs

=0QkslogQks

32


2.4

Statisticsforbinaryimages

Forbinaryimagesthenumberofgraylevels,G,equals2andthe�rst-order

statisticsaredeterminedbythefractionof1-pixels,p1=n1 =N.Wehave:

�=p1

�2=p1 (1�p1 )

AGLCMhastheform:

GLCM

0

1

sum

0

n00

n01

n0:

1

n10

n11

n1:

sum

n:0

n:1

Nh

Thenormalizedversionis:G

LCM

0

1

sum

0

p00

p01

p0:

1

p10

p11

p1:

sum

p:0

p:1

1

Forstationaryimageswehavep:0 �p0: �p0 ,p:1 �p1: �p1and

p01 �p10 �p1 �p11

p00 �p0 �p1+p11

Wesee,that,givenp1 ,allsecond-orderstatisticscanbeexpressedasa

functionofe.g.p11 ,i.e.thereisonly1degreeoffreedominaGLCMgiven

the�rst-orderstatisticp1 .

2.4

Statisticsforbinaryimages

33

ItismoreinstructivetoseethenormalizedGLCM

expressedintermsof

the�rst-orderstatisticsandthecorrelation�=p11�p1p1

p0p1

.

GLCM

0

1

sum

0

p0 p0 (1+p1

p0

�)

p0 p1 (1��)

p0

1

p1 p0 (1��)

p1 p1 (1+p0

p1

�)

p1

sum

p0

p1

1

Thusallsecond-orderstatisticscanbeexpressedintermsofthe�rst-order

statisticsandthecorrelation�.Weshallshowthisforthe�2measureand

the�measure.

The�2measureisforbinarytextures

�2=Nh[ (p00 �p0 p0 )2

p0 p0

+2(p01 �p0 p1 )2

p0 p1

+(p11 �p1 p1 )2

p1 p1

]

=Nh[p20 �2+2p0 p1 �2+p21 �2]=Nh[�2(p

0+p1 )2]=Nh�2

andthe�measureis

�=p00+p11 �p0 p0 �p1 p1

1�p0 p0 �p1 p1

=2p0 p1 �

2p0 p1

=�

Higher-orderstatisticsareelegantlyexpressedintermsofmathematical

morphologyasinSerra(1982).

34


2.5

Fourierfeatures

ThediscreteFouriertransform(DFT),F,anditsinverse,F�1,arede�ned

fortheimage,ff(m;n);m=0;::;M

�1;n=0;::;N�1g,as

F(f)=F(u;v)=

1MN

M

�1

Xm=0

N�1

Xn=0f(m;n)e�j2�(m

uM

+nv

N

)

and

F�1(F)=f(m;n)=

M

�1

Xu=0

N�1

Xv=0F(u;v)ej2�(m

uM

+nv

N

)

TheFourierpowerspectrumisjFj 2

=FF�

(2.2)

whereF�denotesthecomplexconjugateofF.Thepowerspectrum

usu-

allyvariesoverseveralordersofmagnitude,whichmakesitinterestingto

considerthelog-powerspectrumlog �1

+jFj 2 �

(2.3)

StandardlibraryFFTroutinesusuallyhavetherestrictionthattheheight

andthewidthoftheimagehastobeapowerof2.

ThepowerspectrumistheFouriertransformoftheautocorrelation,i.e.it

onlycontainsinformationaboutthesecond-orderstatistics.Itisrecognized

thatthephasespectrumcontainsmuchrelevantinformation,butitisvery

hardtomakeituseful.

FromtheFouriertransformoftheimage,thepowerspectrumandthelog-

powerspectrum

wecancomputeanumberoffeatures.Averagesofthe

powerspectrum

overring-shapedandwedge-shapedregionsarecommon

2.5

Fourierfeatures

35

features(seee.g.Weszkaetal.(1976)).Liu&Jernigan(1990)extracts28

featuresfromthepowerandphasespectrum.

36


2.6

Measurementofenzymatictreatmentef-

fectontextile

Thee�ectofcellulaseenzymatictreatmentontextileshasbeeninvestigated

usingstandardtexturealgorithms.AnextensivestudyinboththeFourier

domainandthespatialdomainhasrevealedthenatureofthechangesand

resultedinonesinglefeaturethatmeasuresthesechangesinafastand

robustway.

2.6.1

Background

ThisprojectstartedwhentheR&Dgroupinthedetergentenzymedivision

ofNovoNordisk(aworld-leadingmanufacturerofdetergentenzymes)ex-

pressedthewishtoquantifythee�ectsofenzymatictreatmentoftextiles

usingdigitalimageanalysis.Untilnowthisquanti�cationhasbeendone

qualitativelyusingmicroscopicinspectionandquantitativelyusingpanel

testsandlightmeasurements(Huntercoordinates).Therewasaneedfora

newobjective,robust,fastandrelativelyinexpensivemethod.

2.6.2

Imageacquisition

Theimageacquisitioniscarriedoutasfollows.Thetextileisplacedin

homogeneousandplentifullighting.Acameraispositionedsuchthatit's

opticalaxisisperpendiculartothetextileplaneandtherectangularvisual

2.6


37

areacoversasmuchofthetextileaspossiblewithoutincludingnon-textile

areas.Thesizeofthetextilesinthisstudyis15x10cm.

WeusedanRGBhigh-resolutionslow-scancamera.Thecameraoutputis

digitizedbyaframegrabberthatgeneratesframesof978by768pixelsinthe

red,greenandblueband.Theseframesarecutto969by711toeliminate

acquisitionartifacts.Subsequentlywewillonlyshowresultsderivedfrom

thegreenbandsincethetextilesusedinthisexperimentareblackandgray

andthuscontainsverylittleornocolorinformation.

Thisstudyregardstheenzymatictreatmente�ectforasingletypeofcel-

lulase.Wewanttoassessthee�ectatdi�erentpHvaluesandfordi�erent

doses.Toassesstheday-to-dayvariationthetextileswerewashedondif-

ferentdaysforeachpH-level.Thuswehavethreefactorsthatwewantto

investigate.

�pH:3levels,123

�dose:8levels,01025405075100200

�day(pH):3levels,123(forpHvalues7.0,8.0and9.0)

Wehavetworepetitionsforeachcombination,thusweendupwith144

images.In�gure2.2wesee8textilesrepresentingthe8dosesforpH1,day

1andrepetition1.

38


Figure2.2.8textilesrepresentingthe8dosesforpH1,day1.

2.6


39

2.6.3

Descriptionofvisualproperties

Theobjectofthedigitalimageanalysisistocomputeonefeaturethatquan-

ti�esagivenvisualpropertyfromtheimagearray.Inthiscasethevisual

propertyisthehumanperceptionofwear.Thefeaturehastocorrelatewell

withpaneltests.Forcellulaseenzymatictreatmentwithknowne�ectsthis

meansthatthefeaturehastoshowimprovementasafunctionofdoseand

showbestresultsforpHvaluesclosetothepHwithhighestenzymeactivity

(between7.0and8.0inourcase).

Obviouslymanydi�erentfeaturescanbecomputedfromtheimage.Asim-

plefeatureistheaverageintensity,lightness.Thishasastrongresemblance

towhatismeasuredbytheHuntercoordinates.Probablythislightness

featurealsohasastrongin uenceonapaneltest.Figure2.3showsthe

averageintensityasafunctionofdoseforpHlevel1.Weseethatlightness

onlyhasdiscriminativecapabilityforsmalldoses.Inthecontextofimage

analysislightnessisanon-robustfeatureinthesensethatitdependsheavily

onlightingconditionsandcamerasensitivity.

Anotheraspectofenzymatice�ectonthetextilesestimatedbythepanel

testisthedistinctnessoftheregulartextilepattern.Thisdistinctness

shouldincreaseasaresultofthecellulaseenzymatictreatment.Theregular

patternintheinvestigatedtextilesresemblesarectangulargridstructure.

Thewellde�nedperiodofthisgridmakesitappropriatetolookatthe

textilesintheFourierdomain.Thisisdoneinthenextsection.

40


100

102

104

106

108

110

112

0

50

100

150

200

mean

dose

day13

33

33

33

33

33

33

33

33

day2+

++++

+ +

+++ +

++

++

++

day32

22

22

22

2 2

22

22

22

2 2

Figure2.3.AverageintensityasafunctionofdoseforpHlevel1.

2.6


41

2.6.4

AnalysisintheFourierdomain

Frequencybasedmethods.

Theclassicalwayofobtaininganestimateofthepowerspectrum

isby

equation2.2.Thelog-powerspectrumisgiveninequation2.3.

Theperiodogramisanon-consistentestimateofthepowerspectrum.Welch's

methodisonewaytodealwiththis.Theimageissplitupinanumberof

non-overlappingsubimages.Theperiodogramiscalculatedineachsubim-

age,followedbyanaveragingoverthesubspectra.

Figure2.4showsthefullresolutionpowerspectraofthetextilesin�gure

2.2.Theconcentriccirclesareisolinesforthespatialfrequency.Several

high-intensityspotsinthepowerspectrumisshowingtheperiodicityofthe

weaves.Thespotsoflowerintensityinthehigh-frequencyareasarehigher

harmonics.Weseethattheintensityinthelow-frequencyareas(nearthe

centerofthepowerspectrum)isfadingforhigherdosesofenzyme.To

illustratethise�ectwecomputedtheaverageofthepowerspectruminthe

ringsbetweentheconcentriccirclesandplotteditversustheradiusofthe

rings.Theseaveragesarecomputedforeachofthepowerspectrain�gure

2.4,andtheaveragecorrespondingtodose0subtractedfromtheaverages

ofeachoftheotherdoses.Theplotisshownin�gure2.5,anditisobvious

thattheaveragesinthelow-frequencyareasaredecreasingforhigherdoses.

Wealsonotethatallthecurveshasapproximatelythesameintersection

atafrequencycorrespondingtothefrequencyoftheweaves.Thushaving

establishedthatthepowerspectrumactuallycontainsrelevantinformation

aboutthetextilewear,wewilltrytoquantifythisinasingleFourierfeature.

42


Figure2.4.Powerspectraofthetextilesin�gure2.2.

2.6


43

-6e-07

-5e-07

-4e-07

-3e-07

-2e-07

-1e-07 0

1e-07

2e-07

0

2

4

6

8

10

12

14

Energy

Ringnumber

dose=0

dose=103

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

dose=25+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

dose=402

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

dose=50�

��

��

��

�

dose=754

444

44

444

444444

4

dose=100

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

dose=200

Figure2.5.Averageofpowerspectraringsrelativetodose0forthespectra

in�gure2.4.

44


Spectraltexturefeatures.

Texturefeaturesderivedinthespatialfrequencydomainhavebeeninvesti-

gatede.g.inWeszkaetal.(1976)andLiu&Jernigan(1990).Thefeatures

testedinthepresentcontextarelistedbelow.

1.Rings

2.Wedges

3.Inertia

4.Entropy

5.Anisotropy

Allfeatureshasbeencomputedonboththefullresolutionpowerspectrum

andthepowerspectrumestimatedusingWelch'smethod.TheWelchspec-

tralestimateperformedsigni�cantlybetterthanthefullresolutionpower

spectrum.

Thefeatureswerecomputedonboththepowerspectrum

andthelog-

powerspectrum.Itturnedoutthatthefeaturescalculatedonthelog-power

spectrumperformedsigni�cantlybetterthanthepowerspectralfeatures.

Furthermorewefoundthatinertiaandentropyfeaturesperformedbetter

thantheotherfeatures.Theinertiafeatureperformedgenerallyalittle

betterthantheentropyfeature,anditseemstobeamorenaturalway

summarizethephenomenaobservedin�gure2.5.

2.6


45

TheinertiafeatureIandlog-powerinertiaLIiscomputedas

I= X(u

;v) (u

2+v2)jF(u;v)j 2

LI= X(u

;v) (u

2+v2)log(1+jF(u;v)j 2)

wherewearesummingoverallfrequencies.Thenormalizedinertiaisthe

inertiadividedbytheinertiaofthecorrespondingtextilewithdose=0.

In�gure2.6weshowthenormalizedlog-powerinertiavs.log(dose)forall

threevaluesofpH.Thusthemeasureisaveragedoverdaysandrepetitions.

Itcanbeseenthatthereisacleardistinctionbetweentheperformance

oftheenzymesatthethreepHvalues.Inadditionthereseemstobean

approximatelylinearrelationbetweentheinertiaandlog(dose).

Discussionofresults

Thespectralapproachhasprovideduswithausefulfeatureandalotof

insightregardingthenatureofthisproblem.TheuseoftheFFTalgorithm

howeverintroducessome,somewhattechnical,limitationsregardingthesize

oftheimageandcomputationalspeed.Itisalsoless exibleinremoving

textileirregularitiesfromtheanalysis.

2.6.5

Spatialdomainfeatures

Thedistinctnesspropertyandothertexturalpropertiescanalsobemea-

suredbytexturalfeaturesinthespatialdomain.Siewetal.(1988)used

featuresbasedondi�erenttexturematricesforcarpetwearassessment.The

46


1.05

1.1

1.15

1.2

1.25

1.3

1.35

2

2.5

3

3.5

4

4.5

5

5.5

normalizedinertia

log(dose)

pH=13

3

3

3

3

3

3

3

pH=2+

+

+

+

+

+

+

+

pH=32

2

2

2

2

2

2

2

Figure2.6.Normalizedlog-powerinertiaversuslog(dose).Weseethatthe

measurere ectstheexpectedranking.

2.6


47

conclusionofthepaperwas,thatfeaturesbasedontexturematrices(e.g.

GLCM)canbeusedtocharacterizetheappearanceofcarpetsandchanges,

theyundergoduringwear.Theproblemofcarpetwearassessmentissimi-

lartomeasuringe�ectsofenzymatictreatment,andthereforeweincluded

GLCM

featuresinourstudy.

Spatialfeatures

Thespatialdomainfeaturesincludedinthisstudywereallthe�rst-order

statisticsofsection2.1andthefollowing15GLCM

features.

1.Energy

2.Entropy

3.Maximumprobability

4.Correlation

5.Diagonalcorrelation

6.Kappa

7.Di�erenceenergy

8.Di�erenceentropy

9.Inertia

10.Localhomogeneity

11.Sumenergy

48


12.Sumentropy

13.Sumvariance

14.Clustershade

15.Clusterprominence

Thefeatureswerecomputedforseveralnumbersofgraylevelsandatseveral

resolutions.Attemptstomakethefeaturesrobusthaveincludedcorrection

forinhomogeneouslightingandautomaticremovaloftextileirregularities.

Theoperationalfeature

Manyofthetestedfeaturesperformedwellonsubsetsoftheimages,but

onlyafewfeaturesgaveanoverallgoodandrobustmeasurement.

Itwaspossibleto�ndarelativelysimplefeaturewithanoverallgoodand

robustperformance.Thisfeatureiscomputedasfollows.Theimageis

transformedtoaresolutionwheretheregulartextilepatternhasjustdis-

appeared(inourcasetheimageswerelowpass-�lteredandsubsampledto

1/16size).Thenthevarianceofthisimageiscomputed.Thevariances

arenormalized(divided)bythevarianceofthecorrespondingtextilewith

dose=0.Theaverageoverdaysandrepetitionsofthisfeatureisshownin

�gure2.7inalog-logplot.Itranksthetextilesjustasexpectedandit

seemsthatalinear�tisappropriateforeachpHlevel.Thisfeatureshallbe

calledthecoarse-scalenormalizedvariance(csnv)feature.Thecsnvfeature

canbecomparedtothetheFourierinertiafeatureintheFourierdomain.

Thelowpass�lterweusedcorrespondapproximatelytoamultiplication

2.6


49

-1.4

-1.2 -1

-0.8

-0.6

-0.4

-0.2

2

2.5

3

3.5

4

4.5

5

5.5

log(csnv)

log(dose)

pH=13

3

3

3

3

3

3

3

pH=2+

+

+

+

+

+

+

+

pH=32

2

2

2

2

2

2

2

Figure2.7.Plotof(log)coarse-scalenormalizedvarianceversuslog(dose).

Weseethatthemeasurere ectstheexpectedranking.

50


withaGaussianweightingfunctioncenteredat(0,0)intheFourierdomain.

FortheFourierinertiafeaturetheweightingfunctionis(u2+v2).Thus

thecsnvfeaturemeasurestheenergyinthelowfrequenciesandtheinertia

featuremeasurestheenergyinthehighfrequencies.Sincethemeasuresare

normalizedtheywillactuallymeasuresimilarproperties,butasthetextile

wearseemstobebestdescribedinthelowfrequencies,theinertiafeature

isnotasrobustasthecsnvfeature.

FittingagenerallinearmodelwiththeSASGLM-procedure:

procglm;

classphday;

modellogvar=logdosephday(ph)ph*logdoselogdose*day(ph);

lsmeansph;

randomday(ph);

givestheresults:

2.6


51

DependentVariable:LOGVAR

Source

DF

SumofSquares

MeanSquare

FValue

Model

17

10.32849194

0.60755835

138.29

Error

108

0.47448146

0.00439335

Corr.Total

125

10.80297340

R-Square

Pr>F

RootMSE

LOGVARMean

C.V.

0.956079

0.0001

0.066282

-.92821182

-7.140862

Source

DF

TypeISS

MeanSquare

FValue

Pr>F

LD

1

8.54219513

8.54219513

1944.35

0.0001

PH

2

1.60099025

0.80049512

182.21

0.0001

D(PH)

6

0.12027099

0.02004517

4.56

0.0004

LD*PH

2

0.01070209

0.00535104

1.22

0.2999

LD*D(PH)

6

0.05433348

0.00905558

2.06

0.0637

Source

DF

TypeIIISS

MeanSquare

FValue

Pr>F

LD

1

8.54219513

8.54219513

1944.35

0.0001

PH

2

0.10533370

0.05266685

11.99

0.0001

D(PH)

6

0.05593527

0.00932254

2.12

0.0565

LD*PH

2

0.01070209

0.00535104

1.22

0.2999

LD*D(PH)

6

0.05433348

0.00905558

2.06

0.0637

whereLD=LOGDOSEandD(PH)=DAY(PH).

52


LeastSquaresMeans

PH

LSMEAN

1

-1.04568416

2

-0.96280088

3

-0.77615044

ItfollowsthattheamountofvariabilityexplainedbypHanddoseareorders

ofmagnitudegreaterthantheremaininge�ects,inclusivetheday-to-day

variability.ThustheconclusivemodelwillonlyincludethepHanddose

e�ects.TheleastsquaremeansforthethreepHlevelsshowtheexpected

ranking.

2.6.6

Conclusion

Wehaveobtainedasinglefeaturefromdigitalimageanalysistodescribethe

e�ectofcellulaseenzymatictreatmentoftextiles.Thisfeatureisalsofast

tocomputeandseemstoberobust.Otherfeaturesmeasuringthevariation

inthetextilethatiscoarserthantheregulartextilepatterncanpossibly

describethesametextileproperties,butthecoarse-scalenormalizedvari-

anceseemstobethefeaturethathastheoverallbestperformanceofthe

featuresconsidered.Thefeaturemayalsobeusefuline.g.carpetwear

assessment.

2.7

GLCM

featureperformance

53

2.7

GLCM

featureperformance

Theperformanceof15GLCMfeaturesistestedinCARTclassi�cationof15

Brodatztextures.Wetherebyinvestigatehowmuchtexturalinformationis

containedinthesimultaneousdistributionof(horizontal)neighborpixels.

Thecooccurrencematricesarecomputedontherawtextures,onthetex-

turesafterahistogramequalization,andonthetexturesafteraGaussian

histogrammatch.

2.7.1

Imagematerial

15Brodatztextureswereselectedonthebasisthattheyshouldhavea

�ne-grainedandhomogeneoustexture.Apartofeachofthesetexturesare

shownrawin�gure2.8,afterahistogramequalizationin�gure2.9,and

afteraGaussianhistogrammatchin�gure2.10.Thenamesoftheselected

texturesareshownin�gure2.11.ThetexturesD16,D21,D53,D77and

D84willsubsequentlybecalleddeterministicduetotheirrelativelystrict

ordering.Therestwillbecalledstochastic.Thisgroupingwillbehelpful

intheinterpretationoftheclassi�cationresults.

Thetextureswerescannedfromthepaperwithan8-bit,300dpiscanner.

Theoutputfromthescannerisa2400x1800image,whichisthenreducedby

twostepsinaGaussianpyramid(Burt,1981).TheapproximatelyGaussian

operatorisaseparable,symmetric�lterwithvalues

0:05;0:25;0:40;0:25;0:05

Theresultisa600x450 oatingpointimage,wherealmostnopixelshave

identicalvalues.Threebyteversionsofeachimageisnowgenerated.

54


Figure2.8.15Brodatztextures(nohistogrammatch).

2.7

GLCM

featureperformance

55

Figure2.9.15Brodatztexturesafterahistogramequalization.

56


Figure2.10.15BrodatztexturesafteraGaussianhistogrammatch.

2.7

GLCM

featureperformance

57

Pressedcork

Grasslawn

Woolencloth

(D4)

(D9)

(D19)

Herringbone

Frenchcanvas

Calfleather

weave(D16)

(D21)

(D24)

Beachsand

Pressedcork

Woodgrain

(D29)

(D32)

(D68)

Orientalstraw

Handmadepaper

Pigskin

cloth(D53)

(D57)

(D92)

Cottoncanvas

RaÆa

Calffur

(D77)

(D84)

(D93)

Figure2.11.Namesofthe15Brodatztexturesin�gure2.10.

58


�The oatingpointimagescaledlinearly.

�Ahistogramequalizedversion.

�AGaussianmatchedversion(mean=127.5,sdev=40.0).

Thehistogram

equalizationandGaussianmatchareperformedbysort-

ingallpixels,whiletheimageisin oatingpointformat,andthenassign

bytevaluesaccordingtothedesiredhistogram.Thusweobtainaperfect

histogram

match.Histogram

equalizationhasbeenusedfrequently(e.g.

Haralicketal.(1973)andLaws(1980))byresearchersstudyingtheperfor-

manceoftexturefeatures.Theequalizationhasinthesecasesbeenmade

usingalessaccuratebytetobytematch.

Fortheselected,�ne-grainedtextureswecorrectedforbackgroundvaria-

tionsbysubtractinga25x25median�lteredversionofeachtexturefrom

itself.

2.7.2

GLCM

Wecomputedtheright-neighborGLCM(h=(0,1))forthethreeversionsof

all15textures.In�gures2.12,2.13and2.14areshownplotsofthecooc-

currencematricesforrespectivelytherawversions,thehistogramequalized

versionsandtheGaussianmatchedversions.

Eachimagewaspartitionedin108disjoint50x50subimages.From

the

right-neighborGLCMofthesesubimageswecomputedthefollowingGLCM

features:

2.7

GLCM

featureperformance

59

Figure2.12.Cooccurrencematricesofrawtextures.

60


Figure2.13.Cooccurrencematricesofhistogramequalizedtextures.

2.7

GLCM

featureperformance

61

Figure2.14.CooccurrencematricesofGaussianmatchedtextures.

62


1.Energy(Enrg)

2.Entropy(Entr)

3.Maximumprobability(Maxp)

4.Correlation(Corr)

5.Diagonalmoment(Diag)

6.Kappa(Kapp)

7.Di�erenceenergy(Derg)

8.Di�erenceentropy(Dent)

9.Inertia(Iner)

10.Inversedi�erencemoment(IDM)

11.Sumenergy(Serg)

12.Sumentropy(Sent)

13.Sumvariance(Svar)

14.Clustershade(Shad)

15.Clusterprominence(Prom)

2.7.3

CARTclassi�cation

Classi�cationandregressiontreesisanonparametricalternativetoclassical

discriminantanalysis.Abinarydecisiontreeisconstructedandaclassi�ca-

tionismadebyrunningdownthetreeandchoosetheclasscorrespondingto

2.7

GLCM

featureperformance

63

theterminalnode.TheCARTprogramfromCaliforniaStatisticalSoftware,

Inc.wasused.ThereaderisreferredtoBreiman,Friedman,Olshen,&

Stone(1984)andtheCARTdocumentationfordetailedinformationabout

CART.

Onlysplitsbasedonsinglefeatureswereallowed.10-foldcross-validation

wasusedforestimatingtheprobabilityofcorrectclassi�cation.

WemadeaCARTclassi�cationonsevensubsetsofthe15texturesinall

threeversions.Thesevensubsetsare:

1.The�vetexturesintheleftcolumn.

2.The�vetexturesinthemiddlecolumn.

3.The�vetexturesintherightcolumn.

4.The�vedeterministictextures:D16,D21,D53,D77andD84.

5.The�vestochastictextures:D4,D9,D29,D32andD57.

6.Thetentexturesintheleftandmiddlecolumns.

7.All15textures.

Linearlyscaledversions

In�gure2.15weseetheclassi�cationtreesuggestedbyCARTforset2.A

textureisclassi�edbystartingatthetopnodeandthenrundownthetree

untilaterminalnodeisreached.Everyterminalnodeisassociatedwith

atextureclass,andthisclassisassignedtothetexturethatwewishto

64


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................................................................................................

......................................................................................................

......................................................................................................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................................................................................

...........................................................................................................................................................................................

D32

...........................................................................................................................................................................................

D9

Enrg

D21

D84

D57

Shad

Diag

Iner

Figure2.15.Classi�cationtreeforset2withnohistogrammatch.

classify.Ateverynonterminalnodeadecisionismadebasedonthevalue

ofonefeature.Ifthevalueofthefeatureislowerthanthesplitvaluefor

thatnodewegoleftinthetree,otherwisewegoright.Thisclassi�cation

isapartitioningoffeaturespaceintoboxes.Thecross-validationestimate

wasonemisclassi�edtextureoutof540.CARTalsoshowedthatmany

alternativetreeswouldhaveasimilarperformance.Figure2.16showsa

scatterplotoftheinertiaversusthediagonalmomentforthetexturesin

set2.Weseethatthetextureclassesareeasilydiscriminated.

Whennohistogrammatchisperformedthe�rst-orderstatisticswillin u-

encethecooccurrencefeatures.Asthe�rst-orderstatisticsoftheBrodatz

texturesdi�ersigni�cantly,thesetof15GLCMfeatureswillbeabletodis-

criminatebetweenanysubsetofthe15textures(actuallyevenanysubset

ofallthe112Brodatztextures)withcloseto0%errorrate.Henceweshall

concentrateonthehistogrammatchedversions.

2.7

GLCM

featureperformance

65

200400

600800

-600 -400 -200 0

11

11

11

11

11

1

11

1

1

111

111

11

11 1

1

11

1 11

111

1

11

1

1

11

1

1

111

11

11

11

1

11

11

1

1 11

11

11

1

11 1

1

1

11

11 1

1 11

1

11

1

11 11

1

1

1

11

1

1 1

1 1 1

1

1

1

1 111

11

2

2 22

22

2

2

22

2

2

2

22

2

22

2

2

22

2

22 2

22

22

22

22

22

22

2

2

2

2

22

2

22

22

22

2

2

2

22

222

2

22

2

2

2

22

2

22

22

22

22

222

2 2

2

2

22

22

22

2

22

22

2

22222

2 22 2

2

22 2

33 3

3

3333

3

33

3

3

33 3

3

3

3

3

33 33

3

3 3

3

3 33

3

33

3

3 3

33 333

33

3

3

3 333 333

3 33 3

3

3

3

33

3

3333

33

33 3

33

3

3

333 3 3

3

33

33

3

3 3

3

3

3

3

33

33 3

3

33

33

3

3

33

34

44

4

4

4444

44

44

4

4

44

4 4 44

4

4

444 4 4

44

444

444

4

4 444

4 44

4

4

4 4 44

444 4

4 44 4

44

44

4 444

44

44

44

4 44 44

44

44

4

44

444

44

4

444

4 44 4

444

444

44

444

55

55 5

55

55

5

5 5

5 55 55 5

5 55

55

55 55

5

5 5

55

5

555

55

55 5

55

55

55

555

555

55

5

5

5

5

5

555

5

5 55

5

55

555 5

55

5 55

5 555

5

55

55

5

5

55 55 5

55

55

5

55 55

55 5

5

Inertia

Diag

200400

600800

-600 -400 -200 0

Figure2.16.Scatterplotofthediagonalmomentversustheinertiaforthe

texturesinset2.1=D9,2=D21,3=D32,4=D57,5=D84.

66


Histogram

equalizedversions

Asummaryoftheclassi�cationresultsforthehistogramequalizedtextures

islistedinthefollowingtable.

Setno.

Classes

Terminal

%correctly

Mostimportant

nodes

classi�ed

feature

1

5

10

95.9

Iner

2

5

10

96.5

Derg

3

5

6

80.9

Svar

4

5

6

98.5

Corr

5

5

13

81.3

Corr

6

10

27

89.3

Corr

7

15

54

74.3

Derg

Theresultsshowthatthesetswithseveraldeterministictextureshashigher

percentageofcorrectlyclassi�edtextures.i.e.thedeterministictextures

inthisstudyarerelativelyeasytodiscriminate.Thecorrelationfeature

anduniformityfeaturesbasedonenergyandentropyareimportantforthe

classi�cation.

Weshallnowstudytheclassi�cationresultsofset2inmoredetail.The

classi�cationtreeisshownin�gure2.17.The�rstsplitisbasedonthe

correlationfeature,anditdiscriminatesthetexturesD57andD84fromthe

otherthree.Thisisagoodsplit(highdiscriminatorypower)andsoare

thetwosplitsonthesecondlevel.Howeverontherightbranchofthesplit

basedonthediagonalmomentweseearelativelycomplexsubtreetrying

todiscriminatebetweenthetexturesD9andD32.Thefeaturesusedfor

thispurposearedi�erenceenergy,energyandentropy.Figure2.18showsa

2.7

GLCM

featureperformance

67

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................................................................................................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................................................................................................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................................................................................................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................................................................................................

......................................................................................................

......................................................................................................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................................................................................

......................................................................................................

......................................................................................................

...........................................................................................................................................................................................

...........................................................................................................................................................................................

D32

D9

D9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................................................................................................... D32

Corr

D32

D9

D32

D21

Enrg

Entr

Derg

Enrg

Derg

Enrg

Derg

D84

D57

Diag

Figure2.17.Classi�cationtreeforset2afterhistogramequalization.

68


0.600.65

0.700.75

0.800.85

0.90

-2000 -1500 -1000 -500 0 500 1000

1

11

1

11

1

11

1

1

11

1

11

1

1

11

1

1

11

1

11

11

1

11 1

1

1

1

1

1

1 1

1

1

1

1

11

11

1

1

1

1

1

1

1

11

11

1

11

11

11

1 1

1

1

1

1

11

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

11

1

1

1

11

11

1

1

1

11

1 1

1

1

22

2

22

2

22

2

2

222

2

22

22

2

22

22

22 2

2

2

2

22

22

22

2

2

22

2

2

2

22 2

2 22

2

2

22

2

2

22

22 2

2

22

2

2

2 22

2

2

22

2

22

22

2

22

2

2

22

2

2

2

2

22

22

22

2

2

22

2

2

2

2

22

2

22

22

33

3

3

3

33

3

3

33

3

33

3 3

3 3

3

3

33

3

3

3

3

3

3 3

3

3

3

3

3

33

33

3

3

33

3

3

3

3

3

33

3

3

3

3

3 3

33 33

333

3

33

3

3

33

33

3

3

3

33

33

3

3

33

3

33

3

3

3

3

3

33

33

33

33

33

3

3

33

3

3

3

3

4

4 4

444

44

4

4

4

44

4

4

44

44

4

4

444

4

4

44

44 4

4

4 4

44

44

44

44

4 4

4

44

4

44

4 4

4

44

4

4

44

44

4

4

44

44

4

4

4

44

44

44

44

4

44

44

44

4

4

44

4

444

44

4

4

4444 4

4

444

4

45

55

5 555

55

5

55

5

55

5

5 5

5 5

5

5

55

5

5

5

5

5

5

55

5

5 55

5

5

55

55

5

55

55 55

55

5

5

555 5

5

5

5

5

5 55

55

5

5

5

5

5

55

5

5

5

5 555

555

5

5

555

5

5

5

5

55

5

55

55

5

5

5 5

55 5

5

5

Corr

Diag

0.600.65

0.700.75

0.800.85

0.90

-2000 -1500 -1000 -500 0 500 1000

Figure2.18.Scatterplotofthediagonalmomentversusthecorrelationfor

thetexturesinset2afterhistogramequalization.1=D9,2=D21,3=D32,

4=D57,5=D84.

scatterplotofthediagonalmomentversusthecorrelation.Weseethatthe

majordiscriminatoryde�ciencyinthesetwofeaturesisthemixtureofthe

classesD9andD32.Figure2.19showsascatterplotoftheenergyversus

thedi�erenceenergy.Itisobviousthatthereisnoeasywayoutofthe

discriminatoryproblem.

Gaussianmatchedversions

Asummaryoftheclassi�cationresultsfortheGaussianmatchedtextures

islistedinthefollowingtable.

2.7

GLCM

featureperformance

69

0.0100.015

0.0200.025

0.002 0.004 0.006 0.008

11

1

11

11

11 1 1

1 111

1 1

1

1

1

11

1

1

11

1

11

11

111 1

111

11

11

1

11

1

11

11

1

1 1

11

11

111

11

11

1

11

1

1

1

1

1

11

11

1

11

11

1

11

1

1 11 1

1

1

1

11

1 1

1

1

1

1 1

1

1

1

1

1

1

1

22

22

22

22

222

222 2

22

22

22

2222

22

2222

222

2 22

2 22

22

22

22

22

22

22 2

22

222

22

22

22

22

22

2 22

222 2

22

22

222

222

22

22

2 2

22

22

222

2

22 2

22

222

23

3

3

3

3

3

3

3

3

3 3

3

3

3

3

3

3

3

33

3

3

3

3

3

3

3

33 33

33

3

3

3

33333 33 3

333

3 33

33

3

3

33

333 3

33

33

333

333

33

33

3

3

3

333

3

3333

33

33

33

33 3

33

33

3

33

33

33

3

3

34

4

44

4

4

4

4

44

44

4

4

44

4

44

4

4

4

4

44

4

4

44

4

4

4

4

4

44

4

44

4

4

44

44

44 4

4

4

4

44

4

4 444

444 4

44

4 4

444

44

444

4 444

44

44

44

444

44

4

44

44 4

4

44 4

444

44

44

44

55

55

5

5

5

5

5

5

5

5

5

5

5

55

55

5

5

5

5

5

5

5

55

5

5

5

5

5

5

5

55

5

55

5

5

5

5

55

55

55 5

55

55

5

5

5

5

5

5

5

5

55

55

5

5

5

5

55

5 5

55

5

55

555

55

5

5 55

55

55

5

555

5

5

55

5

5 5

5

5

55

Difference energy

Energy

Figure2.19.Scatterplotoftheenergyversusthedi�erenceenergyfor

thetexturesinset2afterhistogramequalization.1=D9,2=D21,3=D32,

4=D57,5=D84.

70


Setno.

Classes

Terminal

%correctly

Mostimportant

nodes

classi�ed

feature

1

5

7

93.5

Corr

2

5

6

97.6

Diag

3

5

8

85.9

Diag

4

5

5

97.2

Corr

5

5

6

84.3

Corr

6

10

18

89.7

Corr

7

15

40

80.9

Corr

Againweseethatthedeterministictexturesarerelativelyeasytodiscrimi-

nate.Allsetsexceptthesetswithamajorityofdeterministictextures(set

1andset4)wereclassi�edmorecorrectlywiththesefeaturesthanwiththe

featuresbasedonhistogramequalization.Generallythediagonalmoment

wasanimportantfeature,andfortwosetseventhemostimportant.It

canalsobeseenthatingeneralthetreeshasfewerterminalnodesthan

treesbasedonthehistogramequalization,thuswegetsimplertrees.The

energyandtheentropyfeatureswerefoundtobehighlycorrelatedforall

15texturesaswerethedi�erenceentropyandtheinertia.

Theclassi�cationtreeforset2isshownin�gure2.20.Thetreeissim-

plerthanthetreebasedonthehistogram

equalization.Onlythethree

featurescorrelation,diagonalmomentandinversedi�erencemomentare

used.Figure2.21showsascatterplotofthediagonalmomentversusthe

correlation.ThereishardlyanyconfusionbetweentheclassesD9andD32.

2.7

GLCM

featureperformance

71

...................................................................................................... ......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

...........................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................

D84

Corr

D57

D9

D32

D21

D21

IDM

Diag

Diag

IDM

Figure2.20.Classi�cationtreeforset2afterGaussianhistogrammatch.

2.7.4

Classi�cationsummary

Theresultsoftheclassi�cationsaresummarizedasfollows:

�ItiseasytodiscriminatetheBrodatztexturesifnohistogrammatch

isperformed.

�FeaturesbasedonaGaussianmatchperformedbetterthanfeatures

basedonhistogramequalizationforthestochastictextures.

�Featuresbasedonhistogram

equalizationperformedalittlebetter

thanfeaturesbasedonaGaussianmatchforthedeterministictex-

tures.

72


0.600.65

0.700.75

0.800.85

0.90

-600 -400 -200 0 200 400

1

1

11

1

1 1

11

1

1

1

11

1

11

1

11

11

1

1

11

11

11 11

11

11

11

1 11

1

1

1

1

1

11

1

111

1

1

11

1

11

11

11

11

1

1 11

11

11

1 1

1

1

1

11

1

11

1

11

1

11

11

1

1 1

1

1

1

11

11

1

1

111

1

1

2

2 22

22

22

22

22

2

22

2 22

2

2

22

22

2 2

2

2

22

22

2

2

22

22

2

22

2

2

22

222

2

2

2

2

22

22

22

22

22

2

2

2

22

22

2

22

2

2

22

2

22

22

2

22

2

22

2 2

22

2

22

2 22

22

22

22

2

2

22

2

3

33

3

3

3

3

3

3

33

33

3

33

3

3

3

33

3

3

3

33

3

3 33

33

3

333

33 3

3

3

3333

333

33

3

33

3

3

3

3

33

33

3

3

3

3

3

3

3

3

3

33

3

33

3

3

33

33

3

3

3

33

33

3

3

3

3

3

3

33 3

3

3

3

33

3

3

33

3

3

4

44

44

44

4 4

44

4

4

4

4

4

44

44

44

4

4

4

4 44

44

4

4

444

44

4

4

44

4

44

4 4

4

4

444

44

4

4 4 44

444 4

4

44

4

4

44

44

444

44

4

44

4

444

4

44

44 4

444

4

4

44

444

44 4

44

44

4

4

5

55 5 5

5

5

5

5

5

55

5

5

5 55

5

5 5

5

5

5

5

5

5 55

5

5

5

55

5

555

5

5

5

5

555 55

5

5 5

5

55

5

5

5

5 5

5

55

55

55

5

55

5

55

55

5

5

55

5 55

5

5

5

555 5

5

5 5

5

55

5

5

5

5

5

5

5

5

5

5

5

55

5

55

Corr

Diag

0.600.65

0.700.75

0.800.85

0.90

-600 -400 -200 0 200 400

Figure2.21.Scatterplotofthediagonalmomentversusthecorrelation

forthetexturesinset2afterGaussianhistogrammatch.1=D9,2=D21,

3=D32,4=D57,5=D84.

2.7

GLCM

featureperformance

73

�Thedeterministictextureswereeasiertodiscriminatethanthestochas-

tictextures.

�Featuresbasedonhistogramequalizationgenerallyproducetreeswith

morenodesthanfeaturesbasedonaGaussianmatch.

�Generallycorrelationwasthemostimportantfeature.

�Thediagonalmomentwasaveryimportantfeature.Manysplitswere

basedonthediagonalmoment.

�Theuniformityfeaturesenergy,entropy,di�erenceenergy,di�erence

entropy,sumenergyandsumentropyseemstobemoreimportantfor

histogramequalizedtextures.

�Themaximumprobabilityfeaturewasgenerallyunimportant.

�Theenergyandtheentropyfeatureswerehighlycorrelatedaswere

thedi�erenceentropyandtheinertia.

2.7.5

Conclusion

Theperformanceof15right-neighborGLCM

featuresinCARTclassi�ca-

tionof15Brodatztextureshasbeeninvestigated.

Thisstudyhasshownthathistogrammatchingoftextureshasasigni�cant

e�ectonthediscriminatoryperformanceofGLCMfeaturescomputedfrom

thetextures.Especiallyitseemsthathistogramequalizationistoocrude

forstochastictextures.ForsuchtexturesaGaussianmatchwillgivebetter

performanceandasimplerandmoreinterpretableclassi�er.TheBrodatz

texturesareeasilydiscriminatedifnohistogrammatchismade.

74


Thediagonalmomentisanimportantfeature.Asthisfeaturecannotbe

computedfromthegrayleveldi�erencehistogram(GLDH)andthegray

levelsumhistogram(GLSH),thereisalossofrelevantinformationwhen

replacingtheGLCM

withthesetwohistograms.

Generalizationoftheconclusionsofthisstudyshouldbedonewithgreat

caution.Theselectionof15texturesthatweusedrepresentaninsigni�cant

fractionofreal-worldtextures,andonlythehorizontalneighborrelation

hasbeeninvestigated.

Chapter3

Markovrandom

�elds

Thischapterdealswithparametricdescriptionoftexturebasedonaclass

ofmodelscalledMarkovrandom

�elds.ThetheoryofMarkovRandom

�eldsisreviewedtogetherwiththetheoryoftheassociatedGibbsrandom

�elds.ThetheoryofGibbsrandom�eldswerefoundedinstatisticalphysics

(Ising,1925)andsomerelevantresultsfromthisareaispresentedinanew

statisticalsetting.AvarietyofMarkovrandom�eldsisreviewedwithan

emphasisondiscretemodels.Furtherweintroduceasetofmorphological

Markovrandom�elds,thatextendsthestandardsetofmodelsbyusingthe

operatorsofmathematicalmorphology(Serra,1982).

75

76

Chapter3.Markovrandom

�elds

. . . . . . . . . . . . . . . . . . . . . . . . . .

.......................... ..........................

..................................................... . . . .. . . . . . . . . .. . . . . . . . . ..

........................... . . . . . . . . . . . . . . . . . . . . . . . . .

..................................................... . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . ...................................................... . . . .. . . . . . . . . .. . . . . . . . . ..

..................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . ...........................

. . . . . . . . . . . . . . . . . . . . . . . . . ...........................

. . . . . . . . . . . . . . . . . . . . . . . . . ...........................

........................... . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

..........................

........................... . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

...................................................

........................................

...........

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

...................................................

........................................

............ . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . ......................

................................ . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .......................

............................... . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

..........................

Figure3.1.Regular2Dtessellations.Rectangular,triangularandhexago-

nal.

3.1

Random

�elds

Oneofthemaintasksinstatisticalimageprocessingistoconstructstochas-

ticmodelsforobservedimagesandespeciallyfortextures.Thepixelvalues

fxi ;i=0;1;::;n�1garerepresentedasrealizationsofrandomvariables

fXi ;i=0;1;::;n�1g,andtheprobabilitymeasurerepresentingthejoint

distributionofallpixelvaluesonanimagegridiscalledarandom

�eld.

P(x)istheprobabilityofaparticularimageorcon�gurationx2,where

isthesetofallpossiblecon�gurationsonthegivengrid.

3.1.1

2Dgrids

Thereexiststhreewaysofpartitioningthetwo-dimensionalplaneindis-

junct,regularpolygonsofequalsize.Suchapartitioningiscalledaregular

tessellation.Thethreeregulartessellationsaretheregularsquaretessella-

tion,theregulartriangulartessellationandtheregularhexagonaltessella-

tionasshownin�gure3.1.

3.1

Random

�elds

77

..................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . ...................................................... . . . .. . . . . . . . . .. . . . . . . . . ..

..................................................... . . . . . . . . . . . . . . . . . . . . . . . . ............................ . . . . . . . . . . . . . . . . . . . . . . . . .

..................................................... . . . .. . . . . . . . . .. . . . . . . . . ..

..........................

. . . . . . . . . . . . . . . . . . . . . . . . . ...........................

..........................

. . . . . . . . . . . . . . . . . . . . . . . . . .

..........................

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

........................................

...........

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

...................................................

........................................

............ . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . ......................

................................ . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . ......................

................................ . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . ..

...................................................

........................... . . . . . . . . . . . . . . . . . . . . . . . . .

........................... . . . . . . . . . . . . . . . . . . . . . . . . .

........................... . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . ...........................

. . . . . . . . . . . . . . . . . . . . . . . . . ...........................

Figure3.2.2Dpixelgrids.Rectangular,honeycombandhexagonal.

Letthepolygonsofatessellationcorrespondtopixels,thenthegraphcor-

respondingtothepixelgridwillbedualtothegraphofpolygonborders,

i.e.�Asquaretessellationcorrespondtoasquarepixelgrid

�Atriangulartessellationcorrespondtoahoneycombpixelgrid

�Ahexagonaltessellationcorrespondtoatriangular(hexagonal)pixel

grid

Thehoneycombgridisusedinstatisticalmechanicsbutveryrarely(ifat

all)usedinimageanalysis.Sincetheneighborhoodofapixelinatriangular

gridishexagonal,andthepixelsarehexagonal,thisgridisoftencalledthe

hexagonalgrid,eventhoughthisterm�tsjustaswellforthehoneycomb

grid.Herewewillfollowthecommonpracticeinimageanalysis,i.e.subse-

quentlyahexagonalgridhashexagonalpixels.Thehexagonalgridisquite

popularinmathematicalmorphology(Serra,1982)duetotheattractive

neighborhoodstructure.Thesquaregridisusedinthevastmajorityof

situations,andwherenothingelseismentionedthiswillbesynonymousto

grid.Thegridscorrespondingtothetessellationsof�gure3.1areshownin

�gure3.2.Pixelsarelocatedatthelineintersections.

78


�elds

3.2

Gibbsrandom

�elds

3.2.1

Historicalperspective

In1877Boltzmanninvestigatedthedistributionofenergystatesinmolecules

ofanidealgas.AccordingtotheBoltzmanndistributiontheprobabilityof

amoleculebeinginastatewithenergy"is:

P(")=1z

e�

1kT

"

wherezisanormalizationconstant,thatmakestheprobabilitiessumtoone.

Tistheabsolutetemperature,andk,Boltzmann'sconstant,isaconstantof

nature,thatrelatestemperaturetoenergy.Inallsubsequentformulasthe

temperaturewillbeassumedmeasuredinenergyunits,hencekTwillbe

replacedbyT.

Gibbsusedasimilardistributionin1901toexpresstheprobabilityofa

wholesystemwithmanydegreesoffreedombeinginastatewithacertain

energy.LetxdenoteastateinstatespaceandU:7!Rbetheenergy

function.Then

P(X=x)=

1Ze�1T

U(x)

(3.1)

where

Z= Xx

2

e�1T

U(x):

Ziscalledthepartitionfunction.Tcontrolsthedegreeofpeakinginthe

probabilitydensityfunction.AsT!1thedistributionwilltendtoa

uniformdistributionamongallpossiblestates.AsT!0thedistribution

willtendtoauniformdistributionamongtheminimumenergystates.The

distribution3.1iscalledtheGibbsdistributionorcanonicaldistribution.

Subsequentlytheformertermwillbeusedexclusively.

3.2

Gibbsrandom

�elds

79

Ising(1925)usedtheGibbsdistributiontodescribethebehaviorofferro-

magneticmaterials.Anysiteorpixelinsuchamaterialisthoughtofasa

smalldipole,whichcanbeinstate"spinup"or"spindown"corresponding

tovalues1and-1.

TheIsingmodelonasquaregridisde�nedthroughtheenergyfunction

U(x)=�J Xi�

jxi xj �mH X

i

xi

wherei�jmeansthatpixeliandpixeljareeitherhorizontalorvertical

nearestneighbors.Jisapropertyofthematerialthatdeterminesthe

interactionbetweenneighboringspins.IfJ>

0neighboringspinstend

tobeequal.IfJ<

0neighboringspinstendtobeopposite.J=

0

meansnointeraction.Theconstantm

>0isapropertyofthematerial

thatdeterminesthesensitivityofthespinstoanexternalmagnetic�eld

ofintensityH.H

>0willfavoraspinup,whereasH

<0willfavora

spindown.TheIsingmodelhasbeensuccessfulinexplainingferromagnetic

phenomena,buthasalsofoundedaninterestinthemoregeneralGibbs

random�elds.

Brush(1967)reviewsthehistoryoftheIsingmodel.

3.2.2

Generalproperties

Gibbsrandom�eldsarerandom�eldsde�nedthroughequation3.1.This

meansthatforeveryenergyfunctiononthereexistsacorresponding

Gibbsrandom

�eld.NotalloftheseGibbsrandom

�eldsareusefulfor

ourpurposesandinthenextsectionweshalllimitourattentiontoavery

interestingsubclass.

80


�elds

TheGibbsmeasurehasaninterestingpropertywithrespecttoentropy.

TheentropySisfrequentlyusedasauniformitymeasureofarandom�eld

P,andisde�nedas

S(P)=� Xx

2

P(x)logP(x):

Ofallprobabilitymeasuresde�nedthroughanenergyfunctiontheGibbs

measure(3.1)isthemeasurewhichmaximizesentropyamongallmeasures

withthesameexpectedenergy(Jaynes,1957).

3.3

Markovrandom

�elds

Hassner&Sklansky(1980)introducedMarkovrandom�eldstoimageanal-

ysisandthroughthelastdecadeMarkovrandom

�eldshavebeenused

extensivelyasrepresentationsofvisualphenomena.Inthisthesisthereis

putastrongemphasisonMarkovrandom�eldswithdiscretepixelvalues

i.e.discreteMarkovrandom�elds,butmostoftheresultsareeasilyex-

tendedtocontinuousMarkovrandom�elds.Formorethoroughexpositions

onMarkovrandom�eldsthereaderisreferredtoGeman(1990),Dubes&

Jain(1989),andRipley(1988).

Intherestofthissectionweshallrestatesomede�nitionsregardingMarkov

random�eldsandatheorem

thatshowsanequivalencebetweenMarkov

random�eldsandGibbsrandom�elds.

De�nition1.LetS=fs0 ;s1 ;:::;sn�1 gbeasetofsites.Aneighbor-

hoodsystem

N=fNs ;s2SgisacollectionofsubsetsofSforwhich

3.3

Markovrandom

�elds

81

1.s62Ns

2.r2Ns ,s2Nr

Nsaretheneighborsofs.

Whensitesiandjareneighborswewritei�j.Thesetofallpossible

con�gurationsonSiscalled.

De�nition2.AcliqueCisasubsetofSforwhicheverypairofsites

areneighbors.

Singlepixelsarealsoconsideredcliques.Thesetofallcliquesonagridis

calledC.

De�nition3.Arandom

�eldX

isaMarkovrandom

�eld(MRF)

withrespecttotheneighborhoodsystemN=fNs ;s2Sgi�

1.P(X=x)>0forallx2

2.P(Xs=xs jXr=xr ;r6=s)=P(Xs=xs jXr=xr ;r2Ns )

foralls2Sandx2

ThestructureoftheneighborhoodsystemdeterminestheorderoftheMRF.

Fora�rstorderMRFtheneighborhoodofapixelconsistsofitsfournearest

neighbors.InasecondorderMRFtheneighborhoodconsistsoftheeight

nearestneighbors.Thecliquestructuresareillustratedin�gure3.3and

82


�elds

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .. ....................................... . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ......................................... . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .. ....................................... . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .. ....................................... . . . . . . . . . . . . . . .

Figure3.3.Cliquesfora�rst-orderneighborhood.

. . . . . . . . . . . . . . . ......................................... . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ......................................... . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ......................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....................................... .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....................................... .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .

....................................... .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . ......................................... . . . . . . . . . . . . . . .

.......................................

........................................

........................................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........................................

........................................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure3.4.Additionalcliquesforasecond-orderneighborhood.

5

4

3

4

5

4

2

1

2

4

3

1

.

1

3

4

2

1

2

4

5

4

3

4

5

Figure3.5.Ordercodingofneighborhoodstructure.Then-orderneighbor-

hoodofthecenterpixel(.)containsthepixelswithnumberslessthanor

equalton.

3.3

Markovrandom

�elds

83

�gure3.4fora�rst-orderMRFandasecond-orderMRF.Theordercoding

oftheneighborhooduptoorder�veisshownin�gure3.5.

De�nition4.X

isaGibbsrandom

�eld(GRF)withrespecttothe

neighborhoodsystemN=fNs ;s2Sgi�

P(X=x)=

1Zexp(�U(x)=T)

whereZisanormalizingconstantcalledthepartitionfunction,Tisa

controlparametercalledtemperatureandUistheenergyfunctionof

theform

U(x)= XC

2C

VC(x)

whereVC

iscalledapotentialandisafunctiondependingonlyonxs ;s2

C,

Theorem

1.(Hammersley-Cli�ord).Arandom�eldXisaGibbsrandom

�eldwithrespecttotheneighborhoodsystemNi�X

isaMarkovrandom

�eldwithrespecttoN.

AsimpleproofmaybefoundinGeman(1990).Usingthisequivalence

wehavebothalocalandaglobaldescriptionofthedistribution.Inthe

presentcontextweusetheterm

Markovrandom

�eldtoemphasizethe

Markovproperty.

84


�elds

3.4

BinaryMarkovrandom

�elds

3.4.1

Isingmodelrevisited

ThebestknownandmostinvestigatedMarkovrandom

�eldistheIsing

model.Thismodelhasbeenstudiedinstatisticalphysicssinceitsintro-

ductioninIsing(1925),whereasstatisticiansjoinedthee�ortsinthe1960's.

WeshallgiveathoroughdescriptionoftheIsingmodelusingstatisticalter-

minology.Thuswiththenotationintroducedintheprevioussectionwe

willtalkaboutthe�rst-orderbinaryMarkovrandom�eld.Thereaderis

referredtoKinderman&Snell(1980)forbackgroundmaterialonthisissue.

Inournotationeverysitecantakethevalues0or1.Theneighborhood

ofapixelisthefournearestneighbors.Thecorrespondingthreecliques

aresinglepixels,horizontalneighborsandverticalneighbors.Singlepixels

withvalueonehavethepotential��.Horizontalneighborcliqueshavethe

potential��1ifbothpixelsareone.Thecorrespondingverticalneighbor

cliquepotentialis��2 .Ifanypixelinacliqueis0thecliquepotentialis0.

Thisgivesustheenergyfunction

U(x)=�� X

i

xi ��1 Xi$

jxi xj ��2 Xil

jxi xj

(3.2)

andthejointdistribution

P(X=x)=

1

Z(�;�1 ;�2 )exp(� X

i

xi+�1 Xi$

jxi xj+�2 Xil

jxi xj )

(3.3)

wherei$jmeansthatiandjarehorizontalneighbors,andiljmeans

thatiandjareverticalneighbors.If�1=�2thecon�gurationswillshow

3.4

BinaryMarkovrandom

�elds

85

nodirectionalityandwecallthisanisotropicmodel.Themoregeneral

formulationin(3.3)representstheanisotropicmodel.Thejointdistribution

fortheisotropicmodelis

P(X=x)=

1

Z(�;�)exp(� X

i

xi+� Xi�

jxi xj )

wherei�jmeansthatiandjareneighbors.

Theexpectedmeanandvariancecanbeexpressedas

E( X

i

Xi )= X

[ Xi

xi ]P(x)=

1Z

@@�Z=

@@�logZ

V( X

i

Xi )= X

[ Xi

xi ] 2P(x)�1Z

2(@@

�Z)2

=

1Z

@2

@�2Z�1Z

2(@@

�Z)2=

@2

@�2logZ:

Thisresultisvalidforboththeisotropicandanisotropicmodels.

Inthehorizontaldirectionweget

E( Xi$

jXi Xj )= X

[ Xi$jxi xj ]P(x)=

1Z

@@�1Z=

@@�1logZ

V( Xi$

jXi Xj )= X

[ Xi$jxi xj ] 2P(x)�1Z

2(@

@�1Z)2

=

1Z

@2

@�21Z�1Z

2(@

@�1Z)2=

@2

@�21logZ:

Fortheverticaldirectionandfortheisotropiccasetheresultsareanalogous.

Asitcanbeseenfrom

theequationsabovethepartitionfunctionisa

mainkeyinunderstandinganddescribingthebehaviourofthismodel.

86


�elds

Manyattemptshavebeenmadetomakeevaluationofthepartitionfunction

possible.TheonlyexactresultwasfoundbyOnsager(1944)forthezero-

�eldIsingmodelinthelargegridlimit.Zero-�eldmeansthatthemarginal

probabilityof0-pixelsand1-pixelsareequal,i.e.�=��1+�2 .LetNbe

thenumberofpixelsinthegrid.OnsagerfoundthatinthelimitN!1

wecanwrite1N

logZas

log2��1+�2

2

+

12�2 R�0 R�0

log(cosh�1

2

cosh�2

2

�sinh�1

2

cos!1 �sinh�2

2

cos!2 )d!1 d!2 :

Usingthisexpressionwecan�ndthecorrelationbetweenhorizontalneigh-

borsinthelimitN!1as

�1 (�1 ;�2 )=E(4N Xi$

jXi Xj �1)

=

12�2 Z

�0 Z

�0

2sinh�1

2

cosh�2

2

�2cosh�1

2

cos!1

cosh�1

2

cosh�2

2

�sinh�1

2

cos!1 �sinh�2

2

cos!2d!1 d!2 :

(3.4)

Ananalogousexpressionisobtainedfortheverticalneighborcorrelation,

�2 (�1 ;�2 ).Intheisotropiccasewegetthenearestneighborcorrelationin

thelimitN!1as

�(�)=E(2N Xi�

jXi Xj �1)

=

12�2 Z

�0 Z

�0

2sinh�2cosh�2 �cosh�2(cos!1+cos!2 )

cosh2�2 �sinh�2(cos!1+cos!2 )

d!1 d!2 :

(3.5)

Theintegralscanbecomputedbynumericalintegration,e.g.usingGaus-

sianquadratures(Press,Flannery,Teukolsky,&

Vetterling,1988).Fig-

ures3.6and3.7showplotsof�1 (�1 ;�2 )and�(�).

3.4

BinaryMarkovrandom

�elds

87

0

1

2

3

4

0

1

2

3

40

0.5 1

�1

�2

�1

Figure3.6.Nearesthorizontalneighborcorrelationversus�1and�2forthe

anisotropicmodelinthelargegridlimit.Thelinesinthe(�1 ,�2 )planeare

isolinesforthecorrelation.

88


�elds

00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

0

0.5

1

1.5

2

2.5

3

3.5

4

�

�

Figure3.7.Nearestneighborcorrelationversus�fortheisotropicmodelin

thelargegridlimit(Pickard,1987).

3.4

BinaryMarkovrandom

�elds

89

Thelocalpropertiesofthemodelisdeterminedbytheconditionalproba-

bilities

P(Xi=xi jxw;xe ;xn;xs )=

exp(xi [�+�1 (xw

+xe )+�2 (xn+xs )])

1+exp(�+�1 (xw

+xe )+�2 (xn+xs ))

wherexn,xs ,xw

andxearethenorth,southwestandeastneighborsofxi .

Theparametersareeasilyinterpretedinthat�controlsthenumberof1-

pixels,�1controlsthenumberofhorizontal1-1-neighborsand�2controls

thenumberofvertical1-1-neighbors.

Phasetransitions

Aphasetransition(Kinderman&Snell,1980;Pickard,1987)occursina

MRFwhenthelocallyspeci�edinteractionsarehighenoughtodevelopinto

long-rangecorrelations.

Onsager(1944)showedthattheIsingmodelhasaphasetransitionfor

sinh�12sinh�22=1:

Figure3.8showsthecriticalparametersinparameterspace.Fortheisotropic

modelthecriticalvalueis�c

=

sinh�11=

1:7627.Wetalkaboutsu-

percriticalparametersifsinh�1

2

sinh�2

2

>1andsubcriticalparametersif

sinh�1

2

sinh�2

2

<1.Fromthe�gureweseethat1DIsingmodelsdonot

haveaphasetransition.Ifwegotothesupercriticallimitineachofthefour

quadrantswegetthedeterministicpatternsshownin�gure3.9.Thereare

twosuchdeterministicpatternsineachquadrant,onebeingthepixelwise

negationoftheother.Forthe�rstquadrantwehaveablackcon�guration

andawhitecon�guration.Inthethirdquadrantwehavecheckerboard

90


�elds

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

�1

�2

Figure3.8.

Phasetransitionbordersforananisotropiczero-�eldIsing

model.

Figure3.9.Deterministicpatternsforeachofthefourquadrantsinthe

supercriticallimit.

3.4

BinaryMarkovrandom

�elds

91

Figure3.10.Nondeterministicpatternrepresentedineachofthefourquad-

rants.Thefourpatternscanbegeneratedfromeachotherinaverysimple

way.

andnegatedcheckerboard.Wecanusethisknowledgeofthedeterministic

patternstounderstandtherelationbetweennondeterministicpatternsin

di�erentquadrants.Thevalueofeverypixelinanondeterministicpattern

willcorrespondtothevalueofthesamepixelinoneofthetwodeterminis-

ticpatterns,i.e.wecanpartitiontheimagebasedondeterministicpattern

membership.Ifwethenreplacepixelsbelongingtoeachdeterministicpat-

ternwiththevaluesofthecorrespondingdeterministicpatternsinanother

quadrant,theresultisatransformationofthenondeterministicpatternto

theotherquadrant.Figure3.10showsanondeterministicpatternrepre-

sentedinallofthefourquadrants.Thevisualsymmetrythusobtained

elegantlymatchesthealgebraicsymmetryofparameterspace.

92


�elds

00.2

0.4

0.6

0.8 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

�

�

Figure3.11.Theexpectedfractionof1-pixelsasafunctionof�foran

isotropiczero-�eldIsingmodel.

Thebifurcationpointoccursfor�=

2sinh�1(1)=1:7627.

Anexactexpressionfortheexpectedfractionof1-pixels,�,hasbeenob-

tainedforthezero-�eldisotropicIsingmodelinthelargegridlimit.

�(�)= 8>><>>:

12+12(1�

1

(sinh�2

)4

)18

if�>�c ,whitecon�gurations

12 �12(1�

1

(sinh�2

)4

)18

if�>�c ,blackcon�gurations

12

if��c

ThisresultoriginatedintheworkofOnsager(1944)andYang(1952).In

�gure3.11wesee�plottedversus�.Thebifurcationoccurringat�cmeans

zero-�eldcon�gurationsdoesnothave50%1-pixels,but50%ofthecon�g-

urationshavealmost100%1-pixelsandtheother50%havealmost100%

0-pixels.Theareabetweenthetobranchesforsupercritical�represents

con�gurationswithverylowprobabilityforallvaluesof�.In�gure3.12is

3.4

BinaryMarkovrandom

�elds

93

Figure3.12.Simulationsofisotropic�rst-orderIsingmodelsfor�-values

0.00,0.50,1.00,1.50,1.70,1.76,1.80,2.00and3.00.

94


�elds

shownsimulationsofisotropic�rst-orderIsingmodelsforincreasing�.The

simulationsareconditionalon50%ofeachphase.Theyareperformedusing

10000iterationsoftheMetropolisspin- ipalgorithmdescribedinsection

5.3.Weseethatlong-rangecorrelationsoccuraroundthecritical�.

3.4.2

Morphologicalextension

Insection3.4.1onlycliqueswithoneortwopixelswereconcerned.Markov

random�eldswiththisrestrictionarecalledpairwiseinteractionmodelsor

auto-models(Besag,1974).Theparametersofapairwiseinteractionmodel

willbeabletocontroltwoveryimportantsetsofdescriptivefeatures:�rst-

orderstatisticsandsecond-orderstatistics.However,thesefeaturesdonot

describealltherelevantaspectsofatexture.Forbinarypairwiseinteraction

modelsweknowthatwewillalwayshavethesamestructurefortheblack

phaseandthewhitephase,andthisdoesnotseemlikeanaturalassumption

formanypracticalpurposes.Di�erencesbetweenthetwophasescanonly

becontrolledusingcliqueswithanoddnumberofpixels.Ripley(1988)

summarizedbinaryimagesthroughmorphologicaloperations.Thestudy

showedthataseriesofopeningsandclosingsmadeitpossibletodiscriminate

betweenimages,wheretheautocorrelationhadlittlediscriminatorypower.

GeneralsurveysonmorphologicaloperationscanbefoundinSerra(1982,

1988),Haralick,Sternberg,&Zhuang(1987).Weadoptthenotationof

Haralicketal.(1987)insubsequentmorphologicalexpressions.

Ifweconsidertheanisotropic�rst-ordermodelwithenergyfunction(3.2),

andifweletC1 (x)bethecircumferenceofoneofthephasesmeasuredby

thetotalnumberof0-1-transitionsintheimage,thenwecanexpressthe

3.4

BinaryMarkovrandom

�elds

95

energyfunctionas

U(x)=�(�+2�)A(x)+12

�C1 (x);

whereA(x)= Pn�1

i=0xi .Thustheenergyisproportionaltothecircumfer-

ence.Thisshowsthatareformulationofamodelcanprovidenewinsight.

Theenergyfunction(3.2)canalsobeexpressedusingthemorphological

operatorerosion(),as

U(x)=��A(x)��1 A(xB1 )��2 A(xB2 )

where

B1=

.

,B2=

.

andxBmeanserosionofthe1-phaseofxwithstructuringelementB.

WewillnowreformulatethebinaryMarkovrandom�eldsonthebasisof

mathematicalmorphology.Ingeneraltheenergyfunctionwillhavethe

form:

U(x)=��A(x)�

fXi=

1�i A(xBi )

(3.6)

wherethestructuringelementsfBi ;i=1;::;fgcanbechosenarbitrarily.

Weshallthenturntotheformulationoftheconditionalprobabilities.Let

xi;k=(x0 ;x1 ;::;xi�1 ;k;xi+1 ;::;xn�1 )then

P(Xi=1jrest)

P(Xi=0jrest)=P(xi;1 )

P(xi;0 )=exp(�U(xi;1 )+U(xi;0 ))

=exp(�+

kXj

=1�j [A(xi;1 Bj )�A(xi;0 Bj )])

96


�elds

.

.

Figure3.13.Isotropicandanisotropicstructuringelements.

=exp(�+

kXj

=1�j ni;1 (Bj )):

Thuswhencomputingtheconditionalprobabilitiesweconsiderthepixels

overlappedbyBjplacedatpixeli.ni;1 (Bj )isde�nedasthenumberofthese

pixelsthataremembersofxi;1 Bj .Thecomputationoftheconditional

probabilitiesislocal,andthisisaveryimportantpropertyforthemodel

tobecomputationallyfeasible.

Twointerestingstructuringelementsareshownin�gure3.13.Theisotropic

elementcanbeusedtomodelisotropicdi�erencesbetweenthetwophases,

andtheanisotropiccanbeusedtomodelanisotropicdi�erencesbetweenthe

twophases.IfweletC2 (x)bethecircumferenceofthe1-phasemeasured

bythetotalnumberof1-pixelswithaneighboring0-pixel,thentheMRF

de�nedthroughthismeasureisequivalenttoamodelwiththeisotropic

structuringelement.

The�gures3.14to3.27showsomeexamplesofsimulatedsamplesfrom

MRFswiththestructuringelementsof�gure3.13.Thesimulationswere

runona128�128toroidalgrid.Allsampleshaveapproximately50%

blackand50%whitepixels.Theparameters�1and�2correspondtothe

structuringelementsof�gure3.13.Alltheparametersets,excepttheone

3.4

BinaryMarkovrandom

�elds

97

Figure3.14.MorphologicalMRF.�1=2:0,�2=0:0.

usedin�gure3.27,aresupercritical.Thesupercriticalsamplesshownare

thusintermediatestepstowardssomerelativelyuninterestingsteady-state

pattern.50iterations(fullsweeps)ofthealgorithm

wereusedtocreate

these�gures.Inalltheexamplesweseeastructuraldi�erencebetween

thetwophases.Wehavewhitedotsintheblackphasebutnoblackdots

inthewhitephase.Thestructuraldi�erenceisalsore ectedinthelarger

structures.Insomeoftheimagesthereisvisuallynodoubtthatitiswhite

objectsenclosedinablackphase.Suchadi�erencebetweenthetwophases

issimplynotpossiblewithbinarypairwiseinteractionmodels.

98


�elds



3.4

BinaryMarkovrandom

�elds

99



100


�elds



3.4

BinaryMarkovrandom

�elds

101



102


�elds

Figure3.23.MorphologicalMRF.�1=2:0,�2=�1:0.


3.4

BinaryMarkovrandom

�elds

103

Figure3.25.MorphologicalMRF.�1=4:0,�2=�1:0.


104


�elds


WhyformulatemorphologicalMRFs?

Theenergyfunction(3.6)isonlyareformulationoftheenergyfunctionde-

�nedthroughcliques.Toeverystructuringelementthereisacorresponding

cliquewiththesameshape.Weproposethisreformulationbecauseitbrings

coherencebetweenthestatisticalmodelsanddescriptiveimageanalysis.It

makesitmoreobviouswhentousemulti-spincliquesandwhichitshould

be.Itprovidesuswiththeeverincreasingtoolboxofmorphologicalim-

ageanalysisasmodellingtools.Otherenergyfunctionsthan(3.6)witha

moreintricaterelationbetweenstructuringelementsandandcliquesmay

beformulatedinsimplemorphologicalterms.

3.5

Pottsmodels

105

3.5

Pottsmodels

ThePottsmodelisageneralizationoftheIsingmodeltomorethantwo

unorderedstates(phases).Ithasbeenstudiedinstatisticalphysicssince

itwasintroducedinPotts(1952).Atutorialreviewoftheresultsofthis

researchcanbefoundinWu(1982).AreviewofthePottsmodelsina

statisticalsettingcanbefoundinBesag(1986).Weshallnowreviewthree

examplesofPottsmodels.Theyareofincreasingcomplexity.

Letqbethenumberofstatesandf1;2;::;qgthecorrespondingpixelvalues.

Furtherlet

Æ(x1 ;x2 ;::;xk )= (1

ifx1=x2=:::=xk

0

otherwise

thenthestandardnearest-neighborPottsmodelischaracterizedbythejoint

distribution

P(x)=

1Zexp(� Xi�

jÆ(xi ;xj ))

(3.7)

where

Z= X

exp(� Xi�

jÆ(xi xj )):

Thiscorrespondstonearest-neighborcliqueshavingthepotential��ifthe

twopixelsbelongtothesamestateandzerootherwise.Fortheconditional

probabilitiesweget

P(Xi=kjxj ;j2Ni )=

exp(�ui (k))

Pl exp(�ui (l))

whereui (k)isthenumberofneighborsofpixeliwithvaluek.IfthisPotts

modelhastwostatesitisequivalenttoanisotropiczero-�eldIsingmodel,

when�fromthePottsmodelismultipliedbytwo.

106


�elds

ThePottsmodelabovecanbeextendedbyallowingeachstatetohavea

speci�cstructureandfrequencyofoccurrence.Thisiseasilydonebyin-

troducingstate-dependentparametersforneighbor-pairs,f�k ;k=1;::;qg,

andforsinglepixelsf�k ;k=1;::;qg,thusobtainingtheconditionalprob-

abilities

P(Xi=kjxj ;j2Ni )=

exp(�k+�k ui (k))

Pm

exp(�m

+�mui (m)):

(3.8)

Someorderingbetweenthestatescanbeobtainedbylettingtheparameters

bespeci�cforthecolorsofbothneighbors,giving

P(Xi=kjxj ;j2Ni )=

exp(�k � Pl6=k�kl ui (l))

Pm

exp(�m

� Pl6=m

�ml ui (l)):

(3.9)

3.5.1

Phasetransitions

Fortheq-statePottsmodelwehavephasetransitionsasthosedescribed

fortheIsingmodel(Potts,1952).Thecriticalvalueof�,�c ,forthemodel

(3.7)is

�c=log(1+p

q)

andforthe2-statePottsmodelthisgives

�c=log(1+p

2),

sinh�c=1:

Thus�c=0:8814.

3.5

Pottsmodels

107

3.5.2

Morphologicalextension

Itispossibletoincludemulti-spincliquestoincorporatemorphological

propertiesinthemodels.Wegeneralizethenotationfrom

thelastsec-

tionby�rstde�ningaseriesofbinaryimages,fx(k);k=1;::;qg,fromthe

q-stateimage,x,i.e.

x(k)= (1

ifxi=k

0

otherwise

WecannowintroduceamorphologicalPottsmodelas

U(x)=

qXk

=1 [�

�k A(x(k))�

fXi=

1�ikA(x(k)Bik )]:

(3.10)

Examplesofthismodelanditsapplicationwillbeshowninsection5.4.

3.5.3

Otherextensions

Theliteratureofstatisticalphysics(Wu,1982)providesuswithsomeother

extensionsofthePottsmodel.

�Site-dilutedPottsmodel

Thismodelincludesvacanciesonthegrid.Thesevacanciescanbe

chosenatrandomorinadeterministicway.Examplesofasite-diluted

Pottsmodelmodelanditsapplicationwillbeshowninsection5.4.

�Bond-dilutedPottsmodel

Inthismodelweallowneighborswithnointeraction(orbond).The

missingbondscanbechosenatrandomorinadeterministicway.

108


�elds

�Random-bondPottsmodel

Inthismodelthepotentialofeachbondischosenindependentlyfrom

someprobabilitydistribution.

�"Spin-glass"Pottsmodel

Anextensionofthebinaryspin-glassmodel.Thepotentialsofthe

bondsisanotherrandom�eld(usuallyGaussian).

3.6

GaussianMarkovrandom

�elds

TheGaussianMarkovrandom

�eldmodelisfrequentlyusedtodescribe

continuousphenomena.Theconditionaldensityisgivenbytheexpression

P(xi jxj ;j2Ni )=

1

p2��2expf�1

2�2[xi �� Xj2

Ni �

j (xj ��)] 2g:(3.11)

Thismodelisalsocalledaconditionalautoregressive(CAR)model.More

detaileddescriptionsofthismodelcanbefoundinBesag(1974),Ripley

(1981)andChellappa(1985).

TospecifythejointdistributionoftheCARmodelletBbean�nmatrix

withunitdiagonalentriesando�-diagonalelementsf��ij ;i6=jg,where

�ij=0unlessiandjareneighbors.Wheniandjareneighbors,�ijequals

the�thatcorrespondstotherelativepositionsofthesetwopixels.Thusif

themodelisde�nedonatoroidalgrid,thenBwillbeblockcirculantwith

circulantblocks;seee.g.Chellappa(1985)orDubes&Jain(1989).Ob-

viouslyBissymmetric.Thejointdistributionisthenmultivariatenormal

withmeanvector�,dispersionmatrix�2B�1anddensity

f(x)=

1

p2��2n pjBjexpf�1

2�2(x��)TB(x��)g:

(3.12)

3.6


�elds

109

ForthismodeltobevalidwehavetorequirethatBispositivede�nite.

TheCARmodelsarerelatedtothesimultaneousautoregressive(SAR)mod-

els(Besag,1974;Ripley,1981;Kashyap&Chellappa,1983).SARmodels

areextensionsoftheautoregressivemodelsoftimeseriesanalysistotwo

dimensions.

3.6.1

Alternativegrayleveldistributions

Thejointdensityinequation3.12correspondstotheenergyfunction

U(x)=� Xi�

j�ij(xi ��)(xj ��)

�2

:

(3.13)

Besag(1989)presentsanalternativeclassofjointdistributions,wherethe

energyfunctioninvolvespairwisedi�erencesonly.Theyarede�nedby

U(x)= Xi�

j�(xi �xj )

(3.14)

where�isafunctionthatsatis�es

�(z)=�(�z);�(z)increasingwithjzj:

Jointdistributionsde�nedbyequation3.14areimproperinthattheycan

notbenormalized(Besag,1989).Theydohoweverhaveaperfectlyproper

conditionaldensityp

(xi jxj ;j6=i)/expf� Xj2

Ni �

(xi �xj )g:

Itispossibletoformulatemorphologicalalternativestotheenergyfunc-

tion(3.13)usingtheoperatorsofgraylevelmorphology(Sternberg,1986;

Haralicketal.,1987).Suchmodelsmayturnouttobefeasibleanduseful.

110


�elds

Chapter4

Markovrandom

�eld

parameterestimation

FormostpracticalapplicationsofMarkovrandom�eldsitisessentialthat

wehaveaccurateandfeasiblealgorithmsforparameterestimation.This

chapterreviewsaselectionofestimationmethods.Someofthesemethods

areappliedinchapter5.Anextensionoftheasymptoticmaximumlikeli-

hoodestimator(Pickard,1987)totheanisotropiccaseisproposedinsection

4.4.2.

111

112



4.1

Introduction

MaximumlikelihoodestimationoftheMRFparametersisingeneralcom-

putationallyintractableduetothelikewiseintractablepartitionfunctionin

thejointprobabilitydensity.Therearehowever,asweshallsee,exceptions

tothisrule.But�rstwewilldescribesomealternativestoML-estimation.

4.2

Codingestimation

Besag(1974)introducedcodingestimationasanalternativetoML-estimation.

Thegridispartitionedintoanumberofdisjointsetofpixels,calledcoding

patterns.Thecodingsarechosensuchthatthedistributionofthepixelval-

ueswithinonecodingpattern,conditionalonthepixelvaluesoftheother

codingpatterns,areindependent.Thissimplymeansthatapixelandits

neighborcannotbemembersofthesamecodingpattern.Thenumberof

codingpatternsiskeptaslowaspossibletoobtainthemosteÆcientes-

timator.Thuswegettwocodingpatternsfora�rst-orderMRFandfour

codingpatternsforasecond-orderMRF.Thesecodingpatternsareshown

in�gure4.1and�gure4.2respectively.Sincethevariablesassociatedwith

pixelsfromonecodingpatternareconditionallyindependent,giventheob-

servedvaluesofallotherpixels,wecanexpresstheconditionallikelihood

as

Lk= Yi2

Ck

P(xi jxj ;j2Ni )

whereCk

isthesetofpixelsbelongingtocodingpatternk.Wegetone

setofestimatesforeachcodingpatternbymaximizingthecorresponding

4.2

Codingestimation

113

1

2

1

2

1

2

2

1

2

1

2

1

1

2

1

2

1

2

2

1

2

1

2

1

1

2

1

2

1

2

2

1

2

1

2

1

Figure4.1.Codingpatternsfora�rst-orderMRF.Pixelswiththesame

numberbelongtothesamecodingpattern.

1

2

1

2

1

2

3

4

3

4

3

4

1

2

1

2

1

2

3

4

3

4

3

4

1

2

1

2

1

2

3

4

3

4

3

4

Figure4.2.Codingpatternsforasecond-orderMRF.Pixelswiththesame

numberbelongtothesamecodingpattern.

114



likelihoodfunction.Thesesetsmaythenbecombinedappropriately,e.g.

bycomputingthearithmeticorharmonicmean.

4.3

Pseudolikelihoodestimation

Besag(1975)suggestedusingtheproductofconditionalprobabilitiesforall

pixelsasapseudolikelihoodfunction,i.e.parameterestimateswerefound

bymaximizing

PL= Y

i

P(xi jxj ;j2Ni ):

Thisisobviouslynotareallikelihoodfunctionbecausetheconditionalprob-

abilitiesarenotindependent.Geman&GraÆgne(1987)showedhowever

thatthismethodproducedconsistentestimatesinthelargegraphlimit

undermildconditions.Thereasonforusingmaximumpseudolikelihoodes-

timationinsteadofcodingestimationistoincreasetheeÆciency.Maximum

pseudolikelihoodestimatescomparefavorablytocodingestimatesinBesag

(1977),whereGaussianMRFsareconsidered.Besag(1977)alsonotedthat

forthe�rst-orderGaussianMarkovrandom�eldonasquaregridthemax-

imumpseudolikelihoodestimatorisequivalenttotheharmonicmeanofthe

twoalternativecodingestimator.Inthesubsequenttechnicaldescription

ofestimatorsforspeci�cmodelsthecodingmethodwillgiveresultssimilar

tothepseudolikelihoodmethod.

4.4

BinaryMRF

115

4.4

BinaryMRF

4.4.1

Maximum

pseudolikelihood

InthissectionwepresenttheresultsforthebinomialMRFbecausethese

areimmediateextensionsoftheresultsforthebinaryMRF.

Let�bethevectorofMRFparametersandsibethevectorofthecorre-

spondingneighborsumsforpixeli,i.e.foraanisotropic�rst-orderMRF

wehave

�= 0BB@

��1

�2 1CCA

;

si= 0BB@

1

xW

+xE

xN

+xS 1CCA

:

Furtherlet

Ti=�Tsi

and

pi=

exp(Ti )

1+exp(Ti ):

ThenwecanexpresstheconditionaldistributionsofabinomialMRFas

(xi jxj ;j6=i)2B(n;pi ):

ForabinaryMRFnwillbeequaltoone.

ForthebinomialMRFtheconditionalprobabilityofanobservedpixelvalue

giventherestoftheobservedimageis

P(xi jxj ;j2Ni )= nx

i !pxi

i

(1�pi )n�xi:

116



Theresultingpseudolikelihoodis

PL= Y

i nx

i !pxi

i

(1�pi )n�xi:

Thuswemaximize

logPL= X

i

[log nx

i !+xi Ti �nlog(1+exp(Ti ))]:

ThebinomialcoeÆcientdoesnotdependon�thusthemaximumpseudo-

likelihoodestimateof�isfoundbymaximizing

f(�)= X

i

[xi Ti �nlog(1+exp(Ti ))]

(4.1)

withrespectto�.Forthisfunctionwecan�ndthegradientvectorrf(�)

andHessianmatrixasr2f(�)

rf(�)= X

i

[xi �npi ]si

r2f(�)=�n X

i

exp(Ti )

(1+exp(Ti ))2si si T:

TheHessianmatrixisnegativesemi-de�nite,andthemaximizationproblem

isnoweasilysolvedbystandardoptimizationprocedures.

Dubes&Jain(1989)expressestheconcernthatwhenmaximizingthefunc-

tionfin(4.1)wemayrunintoalocalmaximum.Thisrequiresthatthe

optimizationisrepeatedforseveralinitialguesses.However,wehaveexpe-

riencedthatweobtainthesamesolutionfromseveralinitialguesses,and

thatforsimulatedtexturesthissolutioncorrespondtotheparametersused

inthesimulation.Thefunctionfseemstobewell-behavedevenforreal

textures.In�gure4.3weshowf(�;�)foranisotropic�rst-orderMRF

estimatedonabinarygrasslawntexture(BrodatztextureD9).

4.4

BinaryMRF

117

-2

-1.5-1

-0.50

0.5

0

0.5

1 -1.2-1.1 -1

-0.9-0.8-0.7

�

�

f(�;�)

Figure4.3.Pseudolikelihoodsurfaceforbinarygrass.Maximumisreached

for�=�1:27and�=0:64.Thelinesinthe(�,�)planeareisolinesforf.

118



4.4.2

Asymptoticmaximum

likelihood

Inthecaseofazero-�eld�rst-orderbinaryMRFwecanusetheresults

ofOnsager(1944)forthelarge-gridlimittoestimate�.Themethodwas

introducedbyPickard(1987)fortheisotropiccase.Heusedequation3.5

andappliedittoa�nitegrid.Thusinournotationhegottheequation

Corr(xi ;xj ji�j)=

2N Xi�jxi xj �1=�(�):

Theequationcanbesolvednumericallyusinge.g.Brent'smethod(See

e.g.Pressetal.(1988)).Forgridslargerthan100�100Pickardshowed

thatthe�nite-gridgamma-functionsarenearlyidentical.Theresultscan

beextendedtotheanisotropiccase,usingequation3.4andthevertical

analogue.Wegettheequations

Corr(xi ;xj ji$j)=

4N Xi$jxi xj �1=�1 (�1 ;�2 )

Corr(xi ;xj jilj)=

4N Xilj

xi xj �1=�2 (�1 ;�2 )

Solvingthesetwoequationswillprovideuswithestimatesof�1and�2 .

ThesolutioncanbefoundusingaNewton-Raphsonmethod(Seee.g.Press

etal.(1988)).

4.4.3

Otherestimationmethods

Derin&Elliot(1987)introducedanalternativeestimationmethodthat

involvesthesolutionofanoverdeterminedsystemoflinearequations.This

andotheradhocmethodsarereviewedinDubes&Jain(1989).

4.5

Pottsmodel

119

4.5

Pottsmodel

Inthissectionweconsidertheq-statePottsmodelwithconditionalproba-

bilityde�nedin(3.8).

4.5.1

Maximum

pseudolikelihood

Theconditionalprobabilitiesaregivenby

pi (k)=

exp(�k+�k ui (k))

Pm

exp(�m

+�mui (m))

where,asbefore,ui (k)isthenumberofneighborsofpixeliwithvaluek.

Let�bethevectorofMarkovparametersandsi (k)bethevectorofthe

correspondingneighborfunctionsforpixeliandcolork,i.e.

�= 0BBBBBBB@

�1

�1...�

q�q 1CCCCCCCA

;

si (k)= 1

ui (k) !:

ThepseudolikelihoodfunctionP

L= Y

i

pi (xi )

isthenmaximizedbymaximizing

f(�)= X

i

[�xi

+�k ui (xi )�log X

m

exp(�m

+�mui (m))]:

120



Thegradientvectoriseasilyobtainedas

rf(�)= 0BBBBB@ P

i [1xi =1 �pi (1)]si (1)

Pi [1xi =2 �pi (2)]si (2)

...

Pi [1xi =q �pi (q)]si (q) 1CCCCCA

However,ifaconstantisaddedtoevery�k

wegetexactlythesamemodel.

Thusone�kcanbechosenarbitrarilyandwethenremovethecorresponding

equationabove.

4.6

GaussianMRF

ThissectiondescribestowaysofestimatingparametersoftheGaussian

Markovrandom�eldmodelde�nedinsection3.6.

4.6.1

Maximum

pseudolikelihood

Fromtheconditionaldistributiongivenbyequation(3.11)we�ndthatthe

thepseudolikelihoodfunctionisgivenby

PL= Y

i

1

p2��2expf�1�

2(xi �� Xj2

Ni �

j xj )2g:

4.6

GaussianMRF

121

Let�bethevectorofparametersandsibethevectorofthecorresponding

neighborsumsforpixeli,i.e.

�= 0BBBBBBB@

��1

�2...�

r 1CCCCCCCA;

si= 0BBBBBBB@

1

xW

+xE

xN

+xS

...

xU

+xV 1CCCCCCCA

:

Bysettingthepartialderivativesofthelog-likelihoodequaltozerowe

obtain

^�=[ X

i

si si T] �1 X

i

si xi

and

^�2=

1N Xi

(xi �^�Tsi )2

=

1N( X(i;j

)x2i;j �^�T X(i;j

) sxi ):

Thusthesolutionoftheestimationproblemisgiveninclosedform.

4.6.2

Maximum

likelihood

Thejointdistributiongivenbyequation(3.12)providesuswiththelikeli-

hoodfunctionL

=

1

p2��2n pjBjexpf�1

2�2(x��)TB(x��)g:

(4.2)

Letusassume(Besag,1974)that�=0andthatwehaveanestimateof

B,^B.ThentheML-estimateof�2willbe

^�2=1n

xT^Bx

122



andbysubstitutingthisintoequation4.2andtakingthelogarithmleads

usto�ndingtheML-estimatebymaximizing

logjBj�nlogxTBx:

Wearenowleftwiththenumericalproblemofevaluatingthisfunctionand

especiallythedeterminantjBj.Thishasbeentriedine.g.Besag&Moran

(1975)andKashyap&Chellappa(1983).

Chapter5

Markovrandom

�eld

simulation

InthischapterwereviewasetofiterativesimulationschemesforMarkov

random�eldsimulation.Wethenpresentafastnewparallelalgorithmfor

simulatingMarkovrandom�eldsconditionalongiven�rst-orderstatistics.

FinallyweinvestigatetheuseofthisalgorithmandamorphologicalPotts

modelinthesimulationofgeologicalstructures.

123

124


�eldsimulation

5.1

Introduction

TheproblemofgeneratingsamplesfromaMRFdistributionisimportant

foranumberofreasons.Obviouslyinimageanalysisweareconcernedwith

thevisualpropertiesofthesamples.Instatisticalphysicsitismoreinter-

estingtousethesamplesforcomputingexpectedvaluesofthermodynamic

quantities.

Ifwedisregardthespatialnatureofimagedataandconsiderthepixelvalues

asidenticallyandindependentlydistributedthenthepixelvaluehistogram

willbeasuÆcientstatisticforourrandom�eld.Simulatinganimagefrom

the�rst-orderstatisticswouldonlyrequiresamplingfromaunivariatedis-

tributionwhichisrelativelyeasybutratheruninteresting.Simulatinga

moregeneralrandom�eldcorrespondstosamplingamultivariatedistribu-

tionofveryhighdimension,andaselectionofiterativesimulationschemes

hasbeendeveloped(Seee.g.Dubes&Jain(1989)).

5.2

Iterativesimulation

TheiterativeprocessofMRFsimulationhasfruitfullybeenthoughtofas

adiscrete,�nite-stateMarkovchain.Thestate-spaceofthisMarkovchain

isthesetofallpossiblecon�gurationsandthelimitingdistributionwe

wantistheMRFdistribution.

Fromthetheoryofdiscrete,�nite-stateMarkovchainswegetthefollowing

de�nitionsandresults.LetP=fpij ;i;j2gbethematrixoftransition

5.2

Iterativesimulation

125

probabilities,wherepij (t)denotestheprobabilityofatransitionfromstate

itostatejintsteps.

De�nition5.AMarkovchainisirreducibleornon-decomposable

i�

8i;j29t:pij (t)>0

De�nition6AMarkovchainisaperiodici�

9t0 8t>t0 8i;j2:pij (t)>0

Lemma1.AnirreducibleMarkovchainisaperiodicif

9i2:pii>0

Proof.SeeAarts&Korst(1989).

De�nition7.Aprobabilitydistribution�isinvariantorstationaryfor

aMarkovchainwithtransitionprobabilitiesfpij gi�theglobalbalance

equationsaresatis�ed,thatis

8j2:�j= X

i

�i pij

Theorem

2.ForanirreducibleandaperiodicMarkovchainthereexists

auniqueinvariantdistribution.

Proof.Seee.g.Feller(1968).

126


�eldsimulation

De�nition8.AMarkovchainisreversibleorself-adjointi�thede-

tailedbalanceequationsaresatis�ed,thatis

8i;j2:�i pij=�j pji

Lemma2.ForanirreducibleandaperiodicMarkovchain�istheunique

invariantdistributionifitsatis�esthedetailedbalanceequations.

Proof.Seee.g.Aarts&Korst(1989).

5.2.1

TheMetropolisalgorithm

Metropolis,Rosenbluth,Rosenbluth,Teller,&Teller(1953)describedan

algorithmforcomputersimulationofGibbsdistributedsystems.Thisalgo-

rithmisnowknownastheMetropolisalgorithm.

Algorithm

1.Metropolisalgorithm.LetQbeasymmetricirreducible

transitionmatrixwithstatespace.

1.Startwithcon�gurationx2

2.Chooseanewcon�gurationyfromthedistributionintherowcorre-

spondingtoxinQ

3.Replacexbyywithprobability

p=min(1;P(X=y)=P(X=x))

4.ifnotstopthengoto2

5.2

Iterativesimulation

127

NoticethatwhiletheMetropolisalgorithmwillalwaysmakeachangetoa

newcon�gurationwithhigherprobabilityitwillalsowithsomeprobability

makeachangetoanewcon�gurationwithlowerprobability.

ItistrivialtoshowthattheMetropolisalgorithmde�nesanirreducibleand

aperiodicMarkovchain.Thedetailedbalanceequationsgivefori6=j

qij pi min(1;pj

pi )=qji pjmin(1;pi

pj)

whichforbothpi �pjandpi<pjleadsto

qij=qji

ThisexplainsthesymmetryconditiononQ.

5.2.2

Spin- ipalgorithms

Inspin- ipalgorithmssinglepixelsarevisitedsuccessivelyandtheirvalues

arechangedaccordingtosomecriteria.Kirkland(1989)considered ipping

2x2and3x3blocksofpixelsbuttheresultswerenotencouraging.Thetwo

mostpopular ippingcriteriaprovidesthefollowingalgorithms.

Algorithm

2.Metropolisspin- ipalgorithm.LetQbeasymmetric

irreducibletransitionmatrixwithstatespacef0;::;G�1g,whereGisthe

numberofgraylevels.

1.Startwithcon�gurationx

2.Chooseapixelsandapixelvaluegfromthedistributionintherow

correspondingtoxsinQ.

128


�eldsimulation

3.Setcon�gurationyequaltoxwithpixelssettog




Algorithm

3.Gibbssamplerorheatbathalgorithm.


2.Chooseapixels

3.ReplacexsbyavaluesampledfromtheconditionaldistributionofXs

giventhevaluesoftheneighborsofs.


Inbothalgorithmswehavetochoose(visit)apixelforeachiteration.One

waycouldbetochoosearandompixeleverytime,butmakingasystematic

sweepovertheimageismoreeÆcientbothintermsofrateofconvergence

andintermsoftimepersweep.Ifwemakesurethatwecontinuetovisit

everypixelthentheorderinwhichwesweepthroughtheimagedoesnot

matter.Usingasimplerastersweepdoesensureconvergencebutimposes

anarti�cialanisotropyontheintermediateresultsasseen(unintentionally

?)intheisotropicsimulationsof�gure4inDerin&Elliot(1987).Toavoid

thearti�cialanisotropyandtoenablesimultaneousupdatingofmanypixels

wedividetheimageincodingpatternsasdescribedinsection4.2.Pixels

fromeachcodingpatterndonotinteractwithotherpixelsfromthesame

codingpattern.Wethensweepthroughthecodingpatterns,oneatatime.

5.2

Iterativesimulation

129

5.2.3

TheMetropolisspin-exchangealgorithm

Thespin-exchangealgorithmwasintroducedinimageanalysisbyCross&

Jain(1983).

Algorithm

4.Metropolisspin-exchangealgorithm.


2.Choosetwopixelsrandsatrandom

3.Ifxr=xsthengoto2

4.Setyequaltoxwithpixelsrandsswitched




Insteadof ippingsinglepixelsthisalgorithmexchangesthevaluesoftwo

randomlychosenpixels.Thestepwillmaintainthepixelvaluehistogram

andthusthe�rst-orderstatistics.Asastop-criteriaCrossandJainchecked

ifthenumberofsuccessfulswitchingsdroppedbelow1%ortheestimated

parametersmatchedtheinputparameterswithin5%.Thisresultedina

varietyofinterestingtextures.

ToelaborateonthisalgorithmfortheisotropicIsingmodelweconsiderthe

casexr=1andxs=0,andthusyr=0andys=1.Thecon�gurationsx

130


�eldsimulation

andyareidenticalexceptatpixelsrands.TheratioRisthencomputed

as

R=P(Y=y)

P(X=x)=

1

Z(�;�)exp(� Pi yi+� Pi�jyi yj )

1

Z(�;�)exp(� Pixi+� Pi�jxi xj )

=exp(� Xi�

j [yi yj �xi xj ])=exp(�[Ws (y)�Wr (x)])

whereWk (z)isthenumberof1-neighborsofkincon�gurationz.Ripley

(1987)discussesaproblemintheexpositionofCrossandJainforthecase

whererandsareneighbors.Thisproblemdoesnotoccurusingthepresent

exposition.

Thefactthat Pixi= Pi yimeansthat�isaredundantparameter.This

seemsquitenaturalsince�istheparameterthatcontrolstherelativenum-

berof1-pixels,andthisnumberiskeptconstantbythespin-exchange

algorithm.

5.2.4

Swendsen-Wangalgorithm

Arelativelynewtypeofsimulationalgorithminvolves ippingclusters.A

clusterisaconnectedsetofpixelswithidenticalvalues.Swendsen&Wang

(1987)describedaclusteralgorithmforthebasicPottsmodel3.7.

Algorithm

5.Swendsen-Wangalgorithm.

1.Startwithpixelcon�gurationx

5.2

Iterativesimulation

131

2.Createabondcon�gurationbyintroducingabondbetweenneighboring

pixelswiththesamecolorwithprobability

p=1�exp(��)

3.Findtheclustersjoinedbybonds

4.Independentlyassignarandomcolortoeachcluster


Clusteralgorithmsisanactiveresearchareaandextendedandnewversions

haveoccurred(e.g.(Wol�,1989)).

Figure5.1shows24iterationsoftheSwendsen-Wangalgorithmona5-state

Pottsmodelwith�=2:0.Convergenceseemstobeveryfast.Figure5.2

isaplotofthemaximumpseudolikelihoodestimate�asafunctionofthe

iterationnumber.

132


�eldsimulation

Figure5.1.24iterationsoftheSwendsen-Wangalgorithmona5-statePotts

modelwith�=2:0.

5.2

Iterativesimulation

133

00.2

0.4

0.6

0.8 1

1.2

1.4

1.6

1.8 2

0

5

10

15

20

25

30

35

�

iterations

Figure5.2.Maximum

pseudolikelihoodestimate�asafunctionofthe

iterationnumber.�=2:0forthesimulation.

134


�eldsimulation

5.3

The�

-controlledspin- ipalgorithm

TheMetropolisspin-exchangealgorithmisthemostwidelyusedalgorithm

forsimulatingMarkovrandom�eldsconditionalonthe�rst-orderstatistics.

Inthissectionweproposetwospin- ipalternativesbasedontheGibbssam-

plerandtheMetropolisalgorithmandincludesasanewfeatureafeedback

looptoachievetheconditioning.Therateofconvergenceforlargeattrac-

tionparameters�iscomparedtotherateofconvergenceoftheMetropolis

spin-exchangealgorithm.Thespin- ipalgorithmsturnouttobefasternot

onlyintimepersweepbutalsoinrateofconvergence.Furtherthespin- ip

algorithmsareeasyto0,andthisisdoneusingaSIMDmassivelyparallel

computer.

5.3.1

Introduction

SimulatingMarkovrandom�eldsconditionalontheir�rst-orderstatistics

hasbeenverypopular,sincethiscanprovideinterestingtexturesforlarge

�seeminglyavoidingthephasetransition.Thespin-exchangealgorithm

ishoweververyslowforlarge�andthisispartlyduetothefactthat

theintermediatecon�gurationshave�xed�rst-orderstatisticsandthusthe

numberofpossiblepathsbetweentwocon�gurationsareverylimited.

Inthespin- ipalternativespresentedherewedonotstrictlymaintainthe

�rst-orderstatistics,butstabilizesthesearoundapresetvaluethrougha

feedbackloop.

5.3


135

5.3.2

Thefeedbackloop

Theideaistoconstructaspin- ipalgorithmwithalmostconstant�rst-order

statistics.ForanIsingmodelthe�rst-orderstatisticsisfullydescribed

bythemean�.Supposewewant�tohavethevalue�0 .Thismaybe

accomplishedbyafeedbackloopsuchthat�isadjustedaftereachiteration

tokeep�(t)near�0 .Incontroltheory(e.g.�Astr�om&Wittenmark(1984))

thestandardtextbookPID-controllercanbewrittenas

�(t)=Kp e(t)+Ki

1

1�q�1e(t)+Kd (1�q�1)e(t)

wheree(t)=�0 ��(t)istheerrorfunctionandq�1isthebackshift-operator,

i.e.q�1e(t)=e(t�1).Ifwemultiplywith(1�q�1)onbothsidesweget

(1�q�1)�(t)=Kp (1�q�1)e(t)+Ki e(t)+Kd (1�q�1)2e(t)

(5.1)

andthisistheformactuallyusedhere.Thename,PID-controller,comes

fromthethreecontrollingactionsinexpression5.1.

�P-proportionalaction.Thebasicideaistohaveacontrolaction

proportionaltotheerror.

�I-integralaction.Thisactionisusedtoeliminateastationaryerror

inthemeanvalue.

�D-derivativeaction.Thisactionisusedtoincreasethespeedofthe

controlsystem.

TheP-,I-andD-actionsareadjustedthroughKp ,KiandKdrespectively.

ThejointPID-actionandthedynamicofthespin- ipsystemdeterminesif

thecontrolsystemisstable.

136


�eldsimulation

. . . . .. . . . .. . . . ... . . . .. . . . .. . . . ..

. . . . . . . . . . . . . . ................

..............................

. . . . . . .. . . . . . ... . . . . . .. .. . . . ..

. . . . .. . . . .. . . . ... . . . .. . . . .. . . . ..

. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . ... .. ....................................................................................................................... .... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .. . . . . . ... . . . . . .. .. . . . ..

Spin- ip

�(t)(1�q�1)

e(t)Kp (1�q�1)

-1

x(t)

�(t)

�0

Mean

Figure5.3.P-controller.Multiplicationisperformedistheboxeslabelled

Kp (1�q�1)and�1.

AsimpleP-controller,withKi

=

Kd

=

0,isshownin�gure5.3,and

thiscontrollerwasinvestigatedinthepresentwork.Here�isdirectly

proportionaltotheerrorfunction,e(t).ExtensiontoaPID-controllerwith

integralandderivativeactionsispossible.Thiscanbedonewithoutmuch

computationale�ortbutonehastobemorecarefulinchoosingtheright

constants.Toincludeknowledgeaboutx(t�1)incomputing�(t)would

becomputationallyharderand,asweshallsee,completelyunnecessary.

Thetimestepusedinthecontrolloopisselectedasatrade-o�between

computationalcostandperformance.Thetimestepusedherecorresponds

toafullsweepofthespin- ipalgorithm.

Thiscontrolloopapproachisgenerallyapplicabletoiterativesimulation

schemesandisnotcon�nedtocontrolofthemeanvalue.Otherproperties

ofthecon�gurationx,e.g.second-orderstatistics,canbemeasuredand

usedtocontrolthesimulationparameters.

5.3


137

5.3.3

Relationtoimportancesampling

Green(1986)suggestedthatglobalproperties(e.g.�rst-orderstatistics)

couldbeintegratedinaMarkovrandom�eldP(x)byconsideringthemod-

i�ed�eld

P�(x)/e��D(x)P(x)

whereD(x)isanon-negativerandomvariablemeasuringthedeviationfrom

theidealproperty,and�isapositiveparameter.Thismodi�ed�eldfocuses

P(x)onrealizationswiththedesiredproperty.Theparameterdetermines

thestrengthofthefocusing.FortheIsingmodelwemightchoose

D(x)=(n(x)�nd )2

wheren(x)istheactualnumberof1-pixelsinx,andnd

isthedesired

numberof1-pixels.Theconditionaldistributionofpixeliinthemodi�ed

�eldisthengivenbyreplacing�with��2�(n(xxi =0 )�nd+12).This

resultcorrespondstoalocalP-controllerwithKp=2�.Weshalladoptthe

termimportancesamplingfromRipley(1992)forthesamplingfromP�(x).

TherearethreeadvantagesinusingthePID-controllerinsteadofimpor-

tancesampling.The�-adjustmentisdoneoncepersweepinsteadofonce

perpixel.Importancesamplingcannotbeparallelizedbecausethecondi-

tionaldistributionisbasedonglobalproperties.Finallywehavetoknow

thevalueof�whendoingimportancesampling,whereasthePID-controller

will(hopefully)convergetothecorrectvaluefromanystartingguess.

138


�eldsimulation

5.3.4

Parallelimplementation

ParallelimplementationsofMRFsimulationschemeshasbeensuggested

andimplementedseveraltimesinthepast.Geman&Geman(1984)dis-

cussedaparallelimplementationoftheGibbssamplerandsuggestedan

asynchronousupdatingschemebasedonanMIMD(multipleinstruction

multipledata)computer.Murray,Kashko,&Buxton(1986)implemented

aparallelversionoftheMetropolisspin- ipalgorithmusingsynchronous

updatingforeverycodingonanSIMD(singleinstructionmultipledata)

computer.

AnapproximationoftheMetropolisspin-exchangealgorithm

wasimple-

mentedonaSIMDcomputerbyMargalit(1989)throughaslave-master

handshakingbetweenthetwochosenpixels.Thisprocedurewasrunboth

withandwithoutusingacodingscheme.Besidesthehandshakingoverhead

onlya40%degreeofparallelismisaccomplished.

The�-controlledspin- ipalgorithmdescribedherewasimplementedona

ConnectionMachineCM-200from

ThinkingMachinesusingtheparallel

C-compiler,C�.Thebasicshapeoftheparallelvariablescorrespondtoa

gridofcodingelements.Acodingelementisagroupofneighboringpixels,

onefromeachcoding.Thecodingelementfora�rst-orderMRFcouldbea

pixelfromcoding1andit'sneighbortotheright.Forasecond-orderMRF

thecodingelementcouldbea2x2squarewithapixelfromcoding1init's

upperleftcorner.Weassociateavirtualprocessorwitheachcodingelement

andthisprocessorperformsthespin- ipoperationsuccessivelyonallthe

pixelsinthatcodingelement.Ifthenumberofcodingelementsisequalto

amultipleofthenumberofphysicalprocessorsthisschemeprovides100%

useoftheparallelcomputer.

5.3


139

5.3.5

Results

Thesimulationswereperformedonatoroidal128by128grid.Oneiteration

correspondstoafullsweepthroughtheimage.

Mean-convergence

Inordertomakethemeanvalueconvergetothedesired�0wehavetochoose

anappropriateconstantKp .Westartedsimulationswithallblackpixels

and�0setto0:5.Di�erentvaluesofKp

werechosenforbothMetropolis

algorithmandtheGibbssamplerandtheresultsareshownin�gure5.4

and�gure5.5.�was3:0forallthecurves.WecanseethatKp=6:0seems

likeanappropriatechoiceforbothalgorithms.Wealsonoticethatringing

e�ectsaremoreapparentfortheMetropolisalgorithm.Thisisduetothe

morefrequent ipping.

�-convergence

Therateofconvergenceofthepseudo-likelihood�-estimateisshownin

�gure5.6.Thevalueof�is3.0andwecanseethatthespin-exchange

algorithmconvergesslowerthanit'sspin- ipalternatives.Toillustratethe

visualconvergenceofthethreealgorithmsweshowtypicalcon�gurations

after50,100,200,400,800,2000,4000and8000iterationsfor�=3:0.

In�gure5.7weseethattheMetropolisspin- iphasconvergedtoastable

patternalreadyafter200iterations.TheconvergenceoftheGibbssampler

isshownin�gure5.8andastablepatternisreachedafter2000iterations.

140


�eldsimulation

0 10

20

30

40

50

60

0

20

40

60

80

100

120

140

160

180

200

%

iterations

1.0

2.0

6.0

Figure5.4.Mean-convergenceforMetropolisspin- ip.Percentageof1-

pixelsversusthenumberofiterationsfor�=3:0.

5.3


141

0 10

20

30

40

50

60

70

0

20

40

60

80

100

120

140

160

180

200

%

iterations

1.0

2.0

3.0

4.0

6.0

9.0

Figure5.5.Mean-convergenceforGibbssampler.Percentageof1-pixels

versusthenumberofiterationsfor�=3:0.

142


�eldsimulation

00.5 1

1.5 2

2.5 3

3.5

0

20

40

60

80

100

^�

iterations

Spin-exchange

Metropolis

Gibbssampler

Figure5.6.Convergenceinpseudo-likelihood�estimatesfortheGibbssam-

pler,Metropolisspin- ipandMetropolisspin-exchangealgorithms.

Forthespin-exchange,shownin�gure5.9,wehavetowait30000iterations

beforethepatternstabilizes.

Duringthesesimulationswenoticedthattherewerethreedi�erenttypesof

steady-statepatterns.Theseareshownin�gure5.10.The�rstcorresponds

tothesemi-steadystatereportedinRipley&Kirkland(1990)fortheuncon-

ditionalsimulation.Thesecondpatternconsistsofonephaseencapsulated

intheotherphase.Thethirdpatternshowsdiagonalstripingandthisrelies

onthetoroidalstructureofthegrid.Simulating300samplesusing20000

iterationsoftheMetropolisspin- ipresultedin56%

ofthe�rstpattern,

41%ofthesecondpatternand6%ofthethirdpattern.Wethentriedto

simulatethesamemodelwithfreeboundaryconditions,i.e.thepixelson

theboundariessimplyhavelessneighborsthannonboundarypixels.This

5.3


143

Figure5.7.ConvergenceofMetropolisspin- ipalgorithm.Con�gurations

after50,100,200,400,800,2000,4000and8000iterationsfor�=3:0.

Figure5.8.ConvergenceofGibbssampler.Con�gurationsafter50,100,

200,400,800,2000,4000and8000iterationsfor�=3:0.

144


�eldsimulation

Figure5.9.ConvergenceofMetropolisspin-exchangealgorithm.Con�gu-

rationsafter50,100,200,400,800,2000,4000and8000iterationsfor

�=3:0.

resultedinthetwotypesofsteady-statepatternsshownin�gure5.11.At

upto100000iterationsthedistributionwasmaintainedat80%oftheleft

patternand20%oftherightpattern.

TheseresultsalsoshowthatCross&Jain(1983)neversimulatedtosteady-

stateforsupercritical�,andthatthestop-criteriaweremoreimportantin

Figure5.10.Natureofsteady-statepatternssimulatedonatoroidalgrid.

5.3


145

Figure5.11.Natureofsteady-statepatternssimulatedwithfreeboundary

conditions.

determiningthevisualpropertiesofthesimulatedtexturesthanthemodel

parameters..

Timing

TheparallelimplementationofthisalgorithmontheCM-200isinaverage

40timesfasterthanasequentialimplementationonanHPApollo9000/750,

whichismarketedasthefastestworkstationintheworldatthemoment.

ThetimeontheCM-200wasonan8kprocessorsystem

withexclusive

access.

5.3.6

Conclusion

TheiterativesimulationofMarkovrandom�eldsconditionalonthe�rst-

orderstatisticshasbeenstudied.Untilnowsuchsimulationshasbeendone

usingtheMetropolisspin-exchangealgorithm,whichwasmadepopularby

Cross&Jain(1983).Presentedherearetwospin- ipalternativesthathave

146


�eldsimulation

severaladvantages.Theyarefasterpersweep.Therateofconvergenceis

higher,severalordersofmagnitudeforsupercritical�,andtheyareeasyto

parallelize.Theessentialpartfortheconditioningisasimplefeedbackloop.

Itisstraightforwardtoextendtheuseofsuchafeedbackloopiniterative

simulationschemestoconditioningonotherimagefeatures.

UsingimplementationsofthesealgorithmsonanSIMDmassivelyparallel

computerwehaveshownthatCrossandJaindidnotsimulatetosteady-

stateforlarge�andthattheirrealizationsforlarge�dependsheavilyonthe

stop-criteriaused.Statisticshasbeenmadeforthenatureofsteady-state

con�gurationsforbothsimulationsonatoroidalgridandforsimulations

withfreeboundaryconditions.

5.4

Simulationofgeologicalstructures

147

5.4


WeusethemorphologicalPottsmodelsde�nedinequation(3.10)andapply

the�-controlledspin- ipalgorithmfromsection5.3forthesimulationof

geologicalstructuresinanoil�eld.

5.4.1

Introduction

Inthebusinessofpetroleumexplorationandproductionitisofgreatimpor-

tancetoassessthepropertiesofanoil�eld.Computersimulationstudies

isapowerfultoolinthisassessment.Theyareperformedbysimulating

owinsimulatedstochasticreservoirs.Thusthewordsimulationisusedin

twosensesinthe�eldofpetroleumtechnology.Itisusedforthestochastic

simulationofthespatialdistributionofsedimentaryfaciesandpetrophys-

icalpropertiesaswellasforthenumericalsimulationof owinamedia.

Thiscasestudyisconcernedwithsimulationsinthe�rstsense.When ow

simulationsismeant,thisshallbestatedexplicitly.

Thesimulationisbasedonareservoirmodel.Inthismodelwehaveto

incorporategeologicalknowledgefromsimilarstructuresaswellasthege-

ologicalknowledgeobtainedfromwelldata.Theinformationusedinthe

designofareservoirmodelisoftenreferredtoassoftdata.Whensimu-

latingthereservoirmodelthewelldatashallbe�xedatthecorresponding

locationthushonoringwhatiscalledharddata.Asourceofinformation

thatseemssomewhathardertoincorporateisdatafromseismicstudies.

148


�eldsimulation

Whenmodellingthedistributionofrocktypesweuseadiscretecodingof

thelithology,whereasmodelsforpetrophysicalpropertieslikeporosityand

permeabilitymaybemorenaturallybasedoncontinuousvariables.

Forliteratureonthesubjectofthiscasestudythereaderisreferredto

Ripley(1992),Dubrule(1989)andHaldorsen,Brand,&Macdonald(1988).

5.4.2

Modeltypes

Thetwomaingroupsofstochasticmodelsusedinreservoirsimulationare

objectmodels(orBooleanmodels)andvoxelmodels(orblockmodels).We

areprimarilyconcernedwithvoxelmodels.

Voxelmodels

Thesemodelsarebasedonaregulargridandthedistributionofvoxelvalues

ischosentosatisfye.g.acertainvariogram(correlogram)oraconditional

probabilitydistribution.Thevariogram

andtheconditionalprobability

distributionmaybeinferredfromharddata.

Threerecentpublicationswithdi�erentapproachesare:

�Adler,Jacquin,&Quiblier(1990)simulatedporousmediabasedthe

measuredporosityandvariogram.

�Farmer(1989)generatedgraylevelnumericalrocksby�rstcomput-

ingthehistogram,cooccurrencematricesandautocorrelationofarock

5.4


149

sample.Thenapatternwiththesamehistogramisgenerated,and

thispatternisusedasastartingcon�gurationforaspin-exchange

simulatedannealingprocedure.Deviationfromthesamplecooccur-

rencematricesandautocorrelationisusedaspenaltyintheenergy

function.

�Ripley(1992)simulatesthedistributionofrocktypesusinga3DPotts

modelconditionalontheharddatapoints.

TheapproachinthisstudyissimilartotheapproachinRipley(1992).We

includemorphologicalpropertiesinthemodelbyusingthemorphological

Pottsmodelsde�nedinequation(3.10),andweapplythe�-controlledspin-

ipalgorithmfromsection5.3forthesimulation.

Objectmodels

Anobjectmodeldescribesthedistributionofrockbodiesofrandomshape

atrandomlocations.Thetheoryofpointprocessesandrandomsetmodels

canbeaveryusefultoolforspecifyingandsimulatingobjectmodels.From

theviewpointoftextureanalysisobjectmodelscorrespondstothestructural

approachwithprimitivesandplacementrules.

5.4.3

AMarkovrandom

�eldreservoirmodel

WeshalltrytoapplythemorphologicalPottsmodel3.10forreservoirsimu-

lationonthegigascopicscale(Haldorsenetal.,1988).Themodelisintended

forthedescriptionofthedistributionofbothrocktypesanddiscretized

150


�eldsimulation

petrophysicalproperties.Thegoalisthatthereservoirsimulationscheme

shallincorporate:

�Fixed�rst-orderstatistics.

�Anisotropyinthedi�erentfacies.

�Spatialtrends(instationarity).

�Harddata.

�Planardiscontinuities(faults)asharddata.

Thesimulationisperformedusingthefeedbackloopdescribedinsection5.3

tokeepthe�rst-orderstatistics�xed.Theconstantsforthecontrolactions

hastobeselectedforeachstate.API-controllerwasusedinthesimulations

below.

Anisotropyinthedi�erentfaciesisimplementedthroughtheuseofthe

morphologicalPottsmodels.

Instationaritycanbeimplementedbylettingthemodelparametersvary

acrossthe�eld.

Harddatapointsarehonoredbysimplynotvisitingthemduringthesim-

ulation,i.e.theywillneverchangetheirvalue.

Discontinuities(faults)areintroducedasharddata.Thisissimplydone

byconsideringdiscontinuitiesasanewphase,thediscontinuityphase(The

discontinuityphasecanalsobeconsideredasvacanciesinasite-diluted

Pottsmodel).Thediscontinuityphaseisnotconsideredinthe ipping

5.4


151

process.Tobe100%

e�ectivethediscontinuityphasehastobeaswide

asthelongestdistancebetweentwoneighborsintheMRF.Toavoidthe

wrap-aroundthatisduetothetoroidalgridwecanapplythediscontinuity

phasetothesidesofthegrid.Horizontalwrap-aroundmaybedesirablein

manycases,whereasthisisrarelytrueforverticalwrap-around.Wewould

inthiscaseapplythediscontinuityphasetothetopand/orbottomlinesof

thegrid.Analternativetothediscontinuityphaseistouseabond-diluted

Pottsmodel.Insuchamodelwehavenobonds(nointeractions)acrossthe

discontinuityzone.

5.4.4

Simulationresults

Areservoirsimulationprogram,rocksample,hasbeenimplemented(See

appendixA).Weshallnowshowafewexamplesofsimulationsin2Dbased

onthesemodels.Thesimulationsweremadeona128x128grid,wherethe

pixelsarerectangleswithheight1andlength4.Themodelusedisafour-

statemorphologicalPottsmodelwiththetwostructuringelementsshown

in�gure3.13.In�gures5.12to5.15weshowfoursimulationexamples.

50iterationsoftheMetropolisspin- ipalgorithmwereused.Simulations

likethesecon�rmedthat�rst-orderstatisticsandanisotropyofthedi�erent

faciescanbecontrolled.

Figures5.16and5.17illustratestheconditioningonharddataandfaultsas

harddata.Thesimulationsareconditionalonharddatainverticalcolumns

onbothsidesofthefault.Figure5.16showstwoindependentsimulations,

onewithafaultandonewithout.In�gure5.17thefaultisintroducedin

theresultofthesimulationwithoutthefaultandthensimulationisdone

again.Inthiscasethesimulationslookmoresimilar.

152


�eldsimulation

Figure5.12.Simulationresultoffour-statemorphologicalPottsmodel.



5.4


153


Instationarityhasnotbeenimplementedinthesimulationprogram.

Alltheparametersusedinthesimulationsweresupercritical,i.e.

the

steady-stateresultwouldhaveonlyonecolorifwedidnotconditionon

the�rst-orderstatistics.Thusforthesesimulationstobeusefulinprac-

ticewehaveto�ndasuitablestop-criterion.Thestop-criterioncouldbea

globalstructuralstatistic,e.g.averageclustersizeorlength.

5.4.5

Conclusion

TheusefulnessofMarkovrandom�eldsinreservoirsimulationisdependent

onaneÆcientimplementationofthesimulationscheme.Thisisparticularly

truewhensimulating3Dstructures.The�-controlledspin- ipalgorithm

implementedonamassivelyparallelcomputerhasprovedveryeÆcientand

wouldbeanappropriatechoice.Theexamplespresentedinthissectionhas

beensimulatedonaserialworkstationin2D,buttheyareeasilyextended

to3D.

154


�eldsimulation

Figure5.16.Resultoftwosimulationswithidenticalparameters.Thelower

imagehasafaultasharddata.Thesimulationsareconditionalonhard

datainverticalcolumnsonbothsidesofthefault.

5.4


155

Figure5.17.Resultoftwosimulationswithidenticalparameters.Thelower

imageismadebyintroducingafaultintheupperimageandthensimulating

again.Thesimulationsareconditionalonharddatainverticalcolumnson

bothsidesofthefault.

156


�eldsimulation

SimulationsbasedonthemorphologicalPottsmodelsuggestthatitispos-

sibletosatisfyasetofcriteriathatarerelevanttoreservoirgeologists.

Thenumberofsimulationsthatwehavecomputedsofarisverylimited,

andmuchmoreresearchisneededtoevaluatethepotentialofmorhological

Markovrandom�elds(andMarkovrandom�eldsingeneral)inreservoir

simulation.

Chapter6

Bayesianparadigm

TheBayesianparadigmisaframeworkforincorporatingstochasticmodels

ofvisualphenomenaintoaverygeneralsetoftasksfromimageprocessing

andimageanalysis.SincetheseminalpaperofGeman&Geman(1984)

therehasbeenanincreasinginterestinthissubject(Besag,1986;Marro-

quin,Miter,&Poggio,1987;Geman&McClure,1987;Ripley,1988;Besag,

1989;Geman,Geman,GraÆgne,&Dong,1990).Wegiveashortreview

ofBayesianimageanalysisandpresentanapplicationthatmakessuccess-

fuluseofMarkovrandom�elds,theMetropolisalgorithmandsimulated

annealinginaBayesianframework.1

57

158

Chapter6.Bayesianparadigm

6.1

Introduction

TheBayesianparadigminimageanalysiscanbedescribedasfollows:

1.WeconstructapriorprobabilitydistributionP(x)forthevisualphe-

nomenaX,thatwewanttomakeinferencesabout.

2.WethenformulateanobservationmodelP(yjx).Thisisthedistri-

butionofobservedimagesYgivenanyparticularrealizationxofthe

priordistribution.

3.Thepriordistributionandtheobservationmodelarecombinedtothe

posteriordistributionP(xjy)byBayestheorem

P(xjy)/P(yjx)P(x):

P(xjy)isthedistributionofthevisualphenomenaXgiventheimage

ythatwehaveobserved.

4.Finallywemakeinferencesaboutthevisualphenomenabasedonthe

posteriordistributionP(xjy).

6.2

Priordistribution

ThegeneralityofBayesianimageanalysisliesinthevarietyofvisualphe-

nomena,thatwecanmodel.

Inimagerestorationwewanttomakeinferencesaboutthetrueundegraded

imagerepresentedbyXfromanoisyobservedimagey.AGaussianMRF

6.3

Observationmodel

159

couldthebeanappropriatepriordistributionforX.Priorsthatmodelthe

jointgrayleveldistributionarecalledpixelpriors.

Thegoalofimageclassi�cationistoassignaclassorlabeltoeachpixel

inanimagey.E.g.inremotesensingwecanassignland-useclasseslike

forest,lake,roadetc.topixelsinsatelliteimages.Thejointassignment

oflabelstoallpixelsisalabellingx.PriorsP(x)forthelabellingcould

bediscreteMarkovrandom�eldssuchasbinaryMRFsandPottsmodels.

Priorsthatmodelalabellingarecalledlabelpriors.

Ifwewanttomakeinferencesaboutgeometricalshapes,representedbyX,

inanimagey,weareintheareaoftemplatematching.Templatepriorsare

modelsofgeometricalrelationsinobjectsorbetweenobjectsinanimage.

TheapplicationofaMRFtemplatepriorisillustratedinthecasestudyof

section6.6.

FortheBayesianapproachtobesuccessfulitisimportantthattheprior

densityre ectsourknowledgeofthevisualphenomenabehindtheobserved

images.

6.3

Observationmodel

TheobservationmodelP(yjx)isthedistributionofobservedimages

Y

givenanyparticularrealizationxofthepriordistribution,i.e.ittells

ushowthevisualphenomena,thatwewanttomakeinferencesabout,is

actuallyobserved.Inimagerestorationxistypicallyconsideredobserved

afterconvolutionwithablurringfunctionhandadditionofanoiseimage

160


�,as

Y=h�x+�:

Inimageclassi�cationtheobservationmodelcouldbeatextureand/ornoise

modelforeachclass,e.g.aforesttextureonforestlabels,alaketextureon

lakelabelsetc.

Afterhavingspeci�edthepriormodelandtheobservationmodelweare

readytoextractinformationfromtheposteriordistribution.

6.4

Maximum

aposteriori(MAP)estimates

TheMAPestimate^xofxgivenanobservedimageyisde�nedby

^x=argmaxP(xjy):

ThusMAPestimationinvolvesmaximizationofahighdimensionaljointdis-

tribution,andthisisusuallyconnectedwithaconsiderablecomputational

cost.

6.4.1

Simulatedannealing

Asimulatedannealingschemeisasuccessivesamplingfromthedensity

PT(xjy)/[P(yjx)P(x)]

1T

(6.1)

wherethetemperaturestartsataninitialvalueT0>0andthenfallsto-

wards0.Ifthetemperatureisloweredslowenough,then(6.1)willassign

6.5

Marginalposteriormodes(MPM)

161

unitprobabilitytotheMAPimageinthelimit(Geman&Geman,1984).

Thesamplingalgorithmforsimulatedannealingcanbee.g.theGibbssam-

plerortheMetropolisspin- ipalgorithm.Asimulatedannealingscheme

willbeusedinsection6.6.ThereaderisreferredtoAarts&Korst(1989)

fordetailsonthesimulatedannealingalgorithm.

6.4.2

Iteratedconditionalmodes(ICM)

TheICM

algorithmconsistsofanumberofsweepsovertheimage,where

eachpixelisvisitedandsettothemodeoftheconditionalprobability,i.e.

^xi=argmax

t

P(Yi jyi )P(Xi=tjxj ;j6=i):

TheICM

algorithm

usuallyconvergesintheorderof10sweeps,which

isgenerallymuchlessthanwouldberequiredforasimulatedannealing

scheme.AnICMschemeisontheotherhandmorelikelytogettrappedin

alocalmaximumoftheposteriordensity.

6.5

Marginalposteriormodes(MPM)

Marroquinetal.(1987)generatedaseriesofsamplesfromthe(discrete)

posteriordistributionand,foreachpixel,chosethemodeofthemarginal

posteriordistribution,i.e.x

�i=argmaxP(xi jy)

Thesamplingalgorithm

forMPM

canbee.g.theGibbssamplerorthe

Metropolisspin- ipalgorithm.AMPMschemeusedwithalabelpriorwill

162


minimizetheexpectedmisclassi�cationerrorundertheposteriordistribu-

tion.

6.6

Hybridization�lteranalysis

163

6.6


Analgorithm

forautomaticlocalizationandclassi�cationofspotsona

hybridization�lterhasbeendevelopedandimplemented.Thealgorithm

representsasuccessfulapplicationofaMarkovtemplateprior,theMetropo-

lisalgorithmandasimulatedannealingscheme.

6.6.1

Background

ThegenomeanalysislabatImperialCancerResearchFund(ICRF)in

Londonisworkingonthehumangenomeproject.Thisprojectinvolves

amassiveamountofhybridizationexperiments.Theintentionofthework

presentedhereistoanalyzehybridization�ltersautomaticallyforthemap-

pingofthehumangenome.

The�lterisasquaresheetofnylonwithasidelengthof23.2cm.Arobot

placesa96x96gridofspotsonthis�lter,whereeachspotisaspeci�ccosmid

clone.AcosmidcloneisastretchofDNA,about40000baseslong.When

aradioactiveDNAprobeisappliedtothe�ltertheprobewillonlybind

(hybridize)tothosecosmidclonesthatcontainthesameDNAsequenceas

theprobeitself.Theunboundprobesarewashedo�,andspotscontaining

cloneshybridizedtotheprobeappeardarkerthantheotherspots,whenan

autoradiographistakenofthe�lter.Whenaphosphorimageistakenthe

spotscontaininghybridizedcloneswillappearlighterthantheotherspots.

164


6.6.2

Robotdynamics

Thecosmidclonesareplacedonthe�lterbyarobot.Theyarekepton

microtiterdisheswithan8x12gridofwells,thustherobotarm

consists

ofanarrayof8x12pins.Whentherobotarmisdippedintoamicrotiter

dishasmallquantityofeachcosmidcloneadherestoitscorrespondingpin.

Thearmisthenmovedtothe�lterwhereitappliesthecosmidclonesas

anarrayofspots.Afterthistherobotsterilizesthepinsandmovesonto

thenextdish.Thisisdone96timesforeach�lterproducingthe96x96

gridofspots.Thisgridismadeupof6almostindependent32x48subgrids

asshownin�gure6.1.Eachsubgridcontains4x4interleaved8x12grids

correspondingtothemicrotiterdishgrid.Thespacingbetweenthewellsof

themicrotiterdishesis8mm,thusthespacingbetweenthespotsis2mm.

6.6.3

Imageanalysisproblem

Theproblemtobesolvedthroughtheuseofimageanalysisistoautomati-

callydetectwhichcosmidcloneshybridizestotheprobe.Thisinvolvesthe

correctassignmentofeachspotonthe�ltertoacorrespondingregioninthe

imageandclassi�cationofeachspotastothedegreetowhichhybridiza-

tionhasoccurred.Severalcircumstancescomplicatethesolutionofthese

problems.Forthespotlocalizationproblemwehavethat

�Therobotmovementsareimprecise.

�Themembranemayphysicallywarp.

�Somepinsoftherobotarmmaybebent.

6.6


165

. . . . . . . . . . . . . . . . . . ........................................ . . . . . . . . . . . . . . . . . .

......................................

.. . . . . .. . . . . .. . . . . ... . . . . .. .. . . .. . . . . .

96

48

96

32

32

32

48

Figure6.1.Arrangementofthe6subgrids.Thefullgridisa96x96spot

array.

166


�Somespotsaremissing.

�Somespotsmayhavemerged.

Forthespotclassi�cationproblemwehavethat

�Thebackgroundradiationlevelvariesacrossthe�lter.

�Somespotsmayhavemerged.

�Somespotsmayhavebeenmisplaced.

Weattempttoprovideane�ectivesetofimageanalysistoolsthatarerobust

underthesecircumstances.Thespotlocalizationproblemisconsideredto

bethemostdiÆcultandwillbeourmainconcern.

6.6.4

Digitization

Theautoradiographisdigitizedbyacameraandaframegrabber.Forthe

setupusednowtheresultisan8bit512x512image.Thegray-scaleofthese

imagesisinvertedtogetwhitespots.

Thephosphorimagerscanswith88�mperpixel.Itiscapableofscanning

anareaofupto35�43cm

with16-bitgray-scaleresolution.The�lter

anditsimmediatesurroundingsarescannedandtheresultingimageissub-

sampledtoan8-bit1024x1024image.Thisimageisthestartingpointof

theprocessing.Thespacingbetweenthespotsisabout8.5pixelsandthis

seemsreasonableforourpurpose.Theexamplesshowninthisthesisare

phosphorimages.In�gure6.2weshowanexampleofarawimage.

6.6


167

Figure6.2.Rawimage.Thisisagoodqualityphosphorimageshowingthe

full96x96spotarray.

168


6.6.5

Preprocessing

Thepreprocessingservesfourpurposes.

1.Correctionforrotation.

2.Findingtherectangularoutlineofthespotarray.

3.Correctionforbackgroundvariations.

4.Spotequalization.

Thesuccessofthesubsequentspotlocalizationandspotclassi�cationde-

pendshighlyonasuccessfulimplementationofthesepreprocessingsteps.

Toillustratethepreprocessing,thespotlocalization,andthespotclassi�-

cationwewillshowthee�ectofeachstepontheimagein�gure6.3.This

isaphosphorimageofa32x48subgrid.

Correctionforrotation

Thespotarrayisnormallyverywellalignedwiththepixelarrayinphosphor

images,butautoradiographswillingeneralberotatedslightly.Therotation

anglecanbefoundbyusingtheHoughtransform(Seee.g.Duda&Hart

(1972))andsearchfortheanglebetweene.g.-5and+5degreeswiththe

highestvarianceoverthepro�leinHoughspace.Theimagecanthenbe

rotatedbackintoalignment.In�gure6.4weseetheimagefrom�gure6.3

afteralignment.

6.6


169

Figure6.3.Phosphorimageofa32x48subgrid.

170


Figure6.4.Alignedversionoftheimagein�gure6.3.Therectangular

outlineisshown.

6.6


171

Inparticularlyhardcasesthefourcornersofthespotarraycanbepointed

outmanually.Inthiswaywecanaligntheimageandobtaintherectangular

outlineofthespotarray.

Findingtherectangularoutlineofthespotarray

Wecannowassumestrictlyhorizontalandverticalbordersonthespot

array.Thesebordersarefoundby�rstcomputingthesum

ofeachrow

(column),fsi ;i=

0;::;1023g.Thenwecomputethedi�erenceoflag8

obtainingfdi ;i=8;::;1023g,wheredi=si �si�8 .Lag8ischosenbe-

causeitisclosetothedistancebetweenspotrows(columns).Finallywe

�ndthestartingrow(column)andendingrow(column)asargmaxi di �4

andargmini di+4,whereargmaxidiistherow(column)numberwith

themaximumdi�erence,andargmini diistherow(column)numberwith

theminimum

di�erence.Figure6.4showstherectangularoutline,where

threeofthesideswerefoundbythismethod.Theendingrowhadtobe

repositioned.

Correctionforbackgroundvariations

Thebackgroundvariesovertheimagesandthiswillcauseproblemsinthe

localizationandclassi�cationprocess.Thestandardwayofcorrectingfor

varyingbackgroundistosubtractalowpass�lteredimagefromtheoriginal.

Asalowpass�lterwewillchooseagray-scaleopening.Usinga at9�9

gray-scaleopeningwillremoveallthespotsandleavethebackground,which

isthensubtractedfromtheimage.Thisoperationcanbewrittenas

R=I�O(I)

172


whereIistheinputimage,O(I)istheopeningoftheinputimage,andR

istheresultingimage.

Spotequalization

Tomakethelocalizationprocesseasierweequalizetheintensityofthe

spots,thusweightingthespotsequally.Thisisdoneusingamorphological

equalization,

R=

I

D(I)�E(I)

whereIistheinputimage,Disthedilatedinputimage,Eistheeroded

inputimage,andRistheresultingimage.Againa at9�9structuring

elementisused.Basicallythemorphologicalequalizationmakesthelocal

graylevelrangeconstantovertheimage.Figure6.5showsthee�ectof

backgroundcorrectionandspotequalizationoftheimagein�gure6.4.

6.6.6

Spotlocalization

Thespotlocalizationinvolvesmatchinga96�96gridonthespotsin

theimage.Thisgridshouldadaptgloballytoprovidetheabsolutespot

coordinatesandlocallytotakeintoaccountallthesmalldistortionsinthe

grid.

6.6


173

Figure6.5.Backgroundcorrectedandspotequalizedversionoftheimage

in�gure6.4.

174


Initialassignment

Ifwecanobtainagoodinitialguessonthespotlocationsthenthesub-

sequentprocessingwillbefaster.Astheinitialguesswecovertheoutline

rectanglewitharegular96�96grid.

Simulatedannealingscheme

AsabasisforimprovingthespotlocationsweuseaMarkovrandom�eld

asatemplatepriorfortheregulargridstructureofthespotarray.The

variablesinthismodeldoesnotrepresentapixelvaluebutthe(x;y)image

positionofaspot.Thisvariableisnotde�nedonthepixelgridbutonthe

spotgrid.Thepriorisde�nedas

P(g)/exp(��0 Xi�

jd20 (i;j)��1 Xi�

j (d1 (i;j)�D)2)

whereiandjrepresentsspotsandg=f(xi ;yi );i=1;::;ns gcontainsthe

locations(x;y)ofallthespots.nsisthenumberofspotsinthespotarray.

Theneighborhoodisthefournearestneighbors.d0 (i;j)isthedeviationin

alignmentofthespotsiandj,andd1 (i;j)�Disthedeviationfromthe�xed

griddistance,D,betweenneighbors.Figure6.6illustratesthemeaningof

d0andd1 .

Giventhespotlocationsgwethenspecifyanobservationmodelforthe

observedimageyas

P(yjg)/exp(� X

i

�(i))

wherethesummationisoverallspotsi,and�(i)isthesum

ofthegray

levelsina5x5neighborhoodaroundspoti.

6.6


175

. . . . . . . . . . . . . . . ......................................... . . . . . . . . . . . . . . .. .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .

......................................

. . .. . . . . .. . . . . .. . . .. . .. . . . . .. . . . . .. . ..

. . . . . . . . . . . . . . . .. ....................................... . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . ........................................ . . . . . . . . . . . . . . . . . .

D

......................................

. . .. . . . . .. . . . . .. . . .. . .. . . . . .. . . . . .. . ..

d0

d1

Figure6.6.De�nitionofthedistancemeasuresd0andd1forahorizontal

neighbor-pair.Disthedistancebetweenneighborsonaperfectgrid.

Wecanregardthissetupasastructuraltexturemodel.Thepriormodel

representstheplacementrulesandtheobservationmodelrepresentsthe

primitives.

Theposteriordistribution

P(gjy)/P(yjg)P(g)

isobviouslyanewMarkovrandom�eld,andre ectsatrade-o�betweenthe

regularityofthegridandthetrustintheimagedata.Theenergyfunction

oftheposteriordistributionisgivenas

U=�0 Xi�

jd20 (i;j)+�1 Xi�

j (d1 (i;j)�D)2�� X

i

�(i):

Inthisenergyfunctionwecancontrolthepropertiesofthe�ttedgrid.The

faithinthedataiscontrolledby�,sincethisparameteristheweightofthe

intensityofthespots.Theregularityofthegridiscontrolledby�0and�1 .

�0determinesthedegreeoflinearityofthegridand�1controlsdeviations

fromthe�xedgriddistancebetweenneighboringspots.

176


Thismodelhasaproblemforspotsontheedges.Ifnothingisdonethe

threeorfourouterrowsandcolumnswillbedraggedtowardsthecenter

ofthespotarraybecauseofthelackofspotspullingtheotherway.To

eliminatethise�ectwede�nearti�cialspotsaroundtheedgesofthespot

array.Thearti�cialspotsareinitiallypositionedjustoutsidetheoutline

ofthespotarray,andtheonlyrestrictionintheirmovementsisthatthey

cannotcrossthisoutline.

WecannowapplyasimulatedannealingschemeandtheMetropolisalgo-

rithmusingthisMarkovrandom�eld.Everyspotisvisitedandanattempt

ismadetochangeitspositiontoarandomlyselectednearestneighbor.

In�gure6.7thelocationofeveryspotin�gure6.5ismarkedbyadot.

Figure6.8showthespotlocationsinaclose-upofthelowerrightcornerof

�gure6.7.

Robotgridcontrol

Thesame8�12gridofrobotpinsisused96timesoneach�lter.This

gridcanberegardedas�xedandwecanusethatinformationtodetect

andcorrectmisplacedspots.We�rstcomputethemeanoftherelative

positionsofneighborsinthe8�12grid.Thenforeachspotwecompute

thedeviationsfromthismeanforallfourneighbors.Thetrimmedmean

(minandmaxtrimmedo�)ofthesefourdeviationswillgiveagriddeviation

numberforeachspot.Ifthegriddeviationnumberexceedsaspeci�cvalue

thespotwillbeconsideredasmisplaced.Themisplacedspotscanthenbe

relocatedusingtherelativepositiontotheneighborinthe8�12gridwith

thelowestgriddeviation.

6.6


177

Figure6.7.Spotlocations.Thelocatedspotsoftheimagein�gure6.5are

markedwithadot.

178


Figure6.8.Spotlocationsinaclose-upofthelowerrightcornerofthe

imagein�gure6.7.

6.6


179

Therearenomisplacedspotsin�gure6.7,butanexampleoftherobotgrid

controlprocedurewillbeshowninsection6.6.8.

6.6.7

Spotclassi�cation

Thespotclassi�cationisbasedonthemeangraylevelintheneighborhood

ofthespotlocationfromthebackgroundcorrectedimage.Thresholdsare

selectedtoclassifyeachspotinoneofthreeclasses:positive(+),negative

(-)ormissing(x).Spotclassi�cationsofthelocatedspotsin�gure6.8are

shownin�gure6.9.

Ifthereisanydoubtwhetheraspothasbeencorrectlylocateditwillbe

classi�edasmissing.

6.6.8

Results

Figures6.10and6.11showclose-upsofthelocalizationresultoftheimage

in�gure6.2.Theyillustratetherobustnessofthealgorithm.In�gure6.10

thereisaverticalgapdownthemiddleoftheimage.Thisgapdoesnotcause

anyproblemsinthelocalizations.Inthecenterof�gure6.11weseethat

tworowsofspotsmergeandsplitupagain.Thisisalsointerpretedcorrectly

bythealgorithm.Inboth�guresweseethatmissingspotsarelocatedina

satisfactoryway.Toobtaintheseresultsweusedtheparameters:

�Priormodel:�0=�1=5:0

�Observationmodel:�=0:2

180


Figure6.9.Spotclassi�cationsofthelocatedspotsin�gure6.8.Theclasses

are:positive(+),negative(-)ormissing(x).

6.6


181

Figure6.10.Close-upsofthelocalizationresultoftheimagein�gure6.2.

Thereisaverticalgapdownthemiddleoftheimage.

�Startingtemperature:T0=4:0

�Temperaturescheme:Tn

=Tn�1log(n+2)

log(n+3)

�Numberofiterations:100

Untilnowwehaveonlyshowngoodqualityphosphorimages.In�gure6.12

weseeanoisyphosphorimage,wheretheregularspotpatternishardly

noticeableinlargeareasofthespotarray.Inthesimulatedannealingscheme

weusedthesameparametersasbeforeexceptthatweset�=0:1toputless

trustinthedataandmoretrustinthegridstructure.Thelocationsfound

onthefullgridaremarkedwithdotsandshownin�gure6.13.Aclose-up

ofthis�gureisshownin�gure6.14.Wecanseefromthe�gures,thatthe

182


Figure6.11.Close-upsofthelocalizationresultoftheimagein�gure6.2.

Inthecenterweseethattworowsofspotsmergeandsplitupagain.

6.6


183

Figure6.12.Noisyphosphorimage.Theregularspotpatternishardly

noticeableinlargeareasofthespotarray.

184


Figure6.13.Spotlocationsfromtheimagein�gure6.12markedwithdots.

clearlyvisiblespotsarelocatedcorrectly.Evenforareaswherespotsare

hardlynoticeableweseethatalgorithmmakesareasonablechoice.

Figures6.15and6.16showthee�ectoftherobotgridcontrol.Agroupof

4x3spotshasbeenshiftedtotheleftin�gure6.15.Inthiscasetheshift

wasduetoafastcooling.In�gure6.16weseethemisplacedspotspointed

outbytherobotgridcontrolalgorithm.Wecannowrelocatethemisplaced

spotsandrunthelocalizationalgorithmagain.

6.6


185

Figure6.14.Close-upofthespotlocationsshownin�gure6.13.

186


Figure6.15.Errorsinthelocalization.Agroupof4x3spotshasbeen

shiftedtotheleft.

6.6


187

Figure6.16.Misplacedspotspointedoutbytherobotgridcontrolalgo-

rithm.

188


6.6.9

Conclusion

Wehavepresentedanalgorithmforautomaticlocalizationandclassi�ca-

tionofspotsonahybridization�lter.ThealgorithmisbasedonaMarkov

templatepriorforthespotarray,andthelocalizationisobtainedasatrade-

o�betweenthismodelandtheobserveddata.Thecomputationisbased

onasimulatedannealingscheme.Asetofoperationswasusedtoprepro-

cesstheimages.Thesepreprocessingstepshelpedsigni�cantlyinmaking

thesimulatedannealingschemesuccessfulandcomputationallyfeasible.A

postprocessingstepthatimplementsacheckonthelocalizationhasbeen

implemented.

Thealgorithm

hasbeensuccessfullyappliedtomanyhybridization�lter

images.Itseemstobebothe�ectiveandrobustcomparedtopreviously

testedautomaticmethods(unpublished).Itseemstobeabletoclassify

spotsmuchfasterandinmanycasesmoreaccuratelythanamanualoper-

ator.

Chapter7

Conclusion

Textureisanimportantcharacteristicofvisualphenomena,andmanyat-

temptshavebeenmadetocapturetherelevanttexturalpropertiesinaset

oftexturefeaturesorasatexturemodel.Wehavecontributedtothese

attemptsbygoingthroughselectedtheoryandpracticalapplications.

7.1

Summary

Fortexturedescriptionwehavebasedourstudiesonthe�rst-andsecond-

orderstatistics.Wehaveshownthat�rst-orderstatisticscanprovidevalu-

abletexturalinformationiftheyarecomputedatseveralscales(resolutions).

Wefoundthatacoarse-scale�rst-orderstatisticrobustlymeasuredenzy-

matictreatmente�ectsontextile.Thisshowsthatitmaybefruitfulto

189

190

Chapter7.Conclusion

considerthesimplestfeatures�rst,whensolvingatexturedescriptionprob-

lem.

Wehavesurveyedfeaturesbasedongraylevelcooccurrencematrices.The

e�ectofmatchingthegraylevelhistogramtoaspeci�cdistributionbefore

computingthecooccurrencefeatureshasbeenstudied.Classi�cationre-

sultssuggestthatthefrequentlyusedhistogramequalizationreducesthe

discriminatorypowerofthefeaturessigni�cantlyforstochastictextures.A

relativelyneglectedfeature,thediagonalmoment,turnedouttobeveryim-

portantfordiscriminatingtexturesafteraGaussianhistogrammatch.This

suggeststhatingeneralwelooseimportantinformationwhenreplacing

thecooccurrencematrixwiththegrayleveldi�erenceandgraylevelsum

histograms.ThecombinationofGaussianmatchedtexturesandCART

classi�cationresultedinsimple,easilyinterpretableandrelativelyaccurate

classi�ers.

Markovrandom�eldshavebeensurveyedastexturemodels.Manyimpor-

tantresultsaboutthesemodelsfromthe�eldofstatisticalphysicsarestill

fairlyunknowninthe�eldofimageanalysis.Wehaverestatedsomeofthe

resultsinastatisticalsetting.Theseresultsleadsustoanextensiontothe

asymptoticmaximumlikelihoodestimatorofPickard(1987).

StandardMarkovrandom�eldsarebasedonpairwiseinteractionbetween

pixelsthusfailingtoincorporatemorphologicalproperties.Wesuggesta

reformulationofthediscretemodels,inwhichtheoperatorsofmathematical

morphologyreplacetheconceptofcliques.Theadvantagesofmorphologi-

calMarkovrandom�eldsare,thatmorphologicalpropertiesbecomemore

apparentandthatweobtainacoherencebetweentexturedescriptionand

7.2

A

comment

191

texturemodels.IllustrativesimulationsofmorphologicalMarkovrandom

�eldsshowthatinterestingvisualphenomenacanbecreated.

WehavegivenareviewofMarkovrandom�eldparameterestimationand

Markovrandom

�eldsimulation.Anew,fast,parallelalgorithm

forsi-

mulationconditionalonthe�rst-orderstatisticshasbeendevelopedand

implementedonamassivelyparallelcomputer.Theconditioningismain-

tainedbyastandardPID-controller.Longrunsofthisalgorithmhasgiven

usinformationaboutsteady-statepatternsfortheconditionalmodels.The

algorithmhasalsobeenusedforsimulationsofthegeometricalstructureof

oilreservoirsbasedonamorphologicalMarkovrandom�eldmodel.

Markovrandom�eldshavebeenusedsuccessfullyinaBayesiansettingto

analyzehybridization�ltersautomaticallyforthehumangenomeproject.A

�rst-orderMarkovrandom�eldisusedtomodelthegeometricalstructure

ofaspotarray,andthismodelisthenusedaspriorknowledgeforthe

accuratelocalizationofthesinglespots.Thelocalizationisdoneusinga

simulatedannealingscheme.

Anextensivecollectionofsoftwarehasbeendevelopedduringthecourseof

thiswork.ThemainsoftwaredevelopmentsarelistedinappendixA.

7.2

Acomment

Textureanalysishasbeenstudiedextensivelybymanyresearchersoverthe

lasttwodecades.Thestandardreferenceformostofthesestudieshasbeen

theBrodatztextures.Althoughthesetexturescancontinuetoprovide

192

Chapter7.Conclusion

insightabouttexturefeatures,therearetwopointsofcriticismtosuchan

approach.TheBrodatztexturesonlyrepresentanin�nitesimalfraction

ofrealworldtextures,andtheBrodatztexturesareverydi�erent.Even

thoughtheliteratureontextureanalysis,basedonBrodatztextures,is

fullofsuccesses,therearestillplentyofchallengesfortextureresearchers

in�eldslikeindustrialinspection,biologicalandmedicalimaging,remote

sensing,geologyetc.

AppendixA

Developedsoftware

Anextensiveselectionofsoftwarehasbeendevelopedduringthecourseof

thiswork.Theserial(nonparallel)programsweredevelopedinConHP

workstationsrunningHP-UX.TheparallelprogramsweredevelopedinC�

onaConnectionMachineCM-200withaSun-4frontend.Serialprograms

havethesuÆx.candparallelprogramshavethesuÆx.cs.

Standardnumericalalgorithmsweretakenfrom

Pressetal.(1988).On

theConnectionMachineweusedthesuppliedCMSSLlibrary.Forrandom

numbergenerationunderHP-UXweusedthewell-knownlinearcongruen-

tialalgorithmwith48-bitintegerarithmetic(drand48).OntheConnection

Machineweusedalagged-Fibonaccialgorithm(Knuth,1973)implemented

intheCMSSLlibrary.

193

194

AppendixA.DevelopedSoftware

Bothserialandparallelprogramshavebeenmadetoworkasmodulesof

thepipe-orientedHIPSandHIPS-2imageprocessingsoftware.

Thefollowinglistcontainsthemainsoftwaredevelopments.

�Texturestatistics

1.histinfo

Histinfocomputes�rst-orderstatisticsfromtheinputimage.

2.glcm

Glcmcomputesthegraylevelcooccurrencematrixand15fea-

turesfromthismatrix.

3.fhist

Fhisttakesa oatingpointinputimage,sortsallthepixels,and

outputsbyteimagewithaspeci�edhistogram.Thehistogram

canbeuniform(equalization),Gaussian,orabeta-function.

�Markovrandom�eldestimation

1.binest

Binestcomputescodingestimatesandmaximum

pseudolikeli-

hoodestimatesfromabinaryinputimage.Modelsuptoorder

�vecanbeestimated.Isotropy/anisotropycanbecontrolled

foreachneighbor-distance.The�2

teststatisticandthelog-

likelihoodiscomputedforeachcoding.

2.binomest

Binomestestimatesthemaximumpseudolikelihoodestimatesof

abinomialMarkovrandom�eldfromtheinputimage.


195

3.asympest

Asympestcomputestheasymptoticmaximum

likelihoodesti-

mateofa�rst-orderbinaryMarkovrandom�eldfromtheinput

image.

4.pottsest

Pottsestcomputesthemaximumpseudolikelihoodestimateofa

Pottsmodelfromtheinputimage.

5.gaussest

Gaussestcomputesthemaximumpseudolikelihoodestimateofa

GaussianMarkovrandom�eldfromtheinputimage.

�Markovrandom�eldsimulation

1.binsamp

BinsampsimulatesbinaryMarkovrandom�eldsusingtheGibbs

samplerortheMetropolisalgorithm.

2.pottssamp

PottssampsimulatesPottsmodelsusingtheGibbssampleror

theMetropolisalgorithm.

3.morphsamp

MorphsampsimulatesmorphologicalbinaryMarkovrandom�elds

usingtheMetropolisalgorithm.

4.swendsen

SwendsensimulatesPottsmodelsusingtheSwendsen-Wangal-

gorithm.

5.rocksamp

Rocksampsimulatesgeologicalsamplesusingamorphological

Pottsmodelandthe�-controlledspin- ipalgorithm.Model

196


parametersforeachphasecanbespeci�ed.Rocksampisthe

programusedinsection5.4.

6.icrf

Icrfisthepackageofhybridizationanalysissoftwareusedinsec-

tion6.6.

7.bingen

BingenisprograminC�forConnectionMachines.Itsimulates

binaryMarkovrandom

�eldsin2Dand3D.The�-controlled

algorithmisimplemented.Theresultscanbemonitored"real

time"inanX-window.

�Other

1.xshow

XshowisaprogramthatdisplaysHIPSimagesunderX-windows

andletstheuserinteractusingHIPSprograms.

2.frarithmetic

Frarithmeticisaprogramthatcanbeexecutedwithmanynames

(allstartingwith"fr").Itdoesmanykindsofarithmeticopera-

tionsonasetofimages.

3.AHIPSimplementationofthebasicgraylevelmorphological

operations:

{Erosion

{Dilation

{Opening

{Closing

{Morphologicalgradient

{Whitetophat


197

{Blacktophat

{Morphologicalequalization

198


AppendixB

GLCM

forallBrodatz

textures

Thisappendixcontainstheright-neighborGLCM

foralltheBrodatztex-

tures(nohistogram

match).Byobservingthekindofstructuresthese

matricescanhavewemaygetabetterideaofwhichfeaturesgivethebest

summary.

199

200

AppendixB.GLCM


FigureB.1.

Right-neighborGLCM

forBrodatztexturesD1toD10

(byrow).

AppendixB.GLCM


201

FigureB.2.

Right-neighborGLCM


(byrow).

202

AppendixB.GLCM


FigureB.3.

Right-neighborGLCM


(byrow).

AppendixB.GLCM


203

FigureB.4.

Right-neighborGLCM


(byrow).

204

AppendixB.GLCM


FigureB.5.

Right-neighborGLCM


(byrow).

AppendixB.GLCM


205

FigureB.6.

Right-neighborGLCM


(byrow).

206

AppendixB.GLCM


FigureB.7.

Right-neighborGLCM


(byrow).

AppendixB.GLCM


207

FigureB.8.

Right-neighborGLCM


(byrow).

208

AppendixB.GLCM


FigureB.9.

Right-neighborGLCM


(byrow).

AppendixB.GLCM


209

FigureB.10.Right-neighborGLCM


(byrow).

210

AppendixB.GLCM




(byrow).

AppendixB.GLCM


211


forBrodatztexturesD111andD112

(byrow).

212

AppendixB.GLCM


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Index

�-controlledspin- ipalgorithm,

134

asymptoticmaximum

likelihood,

118

autocorrelation,13

backgroundcorrection,170

Bayesianparadigm,157

CARmodel,108

CART,61

classi�cation,61,178

clique,81

codingelement,138

codingestimation,112

ConnectionMachine,138

contingencytable,24

cross-validation,62

diagonalmoment,16

distinctness,38

entropy,80

enzymatictreatment,35

equalization

histogram,13

morphological,171

spot,171

erosion,95

faults,150

Fourierfeatures,33


�elds,

108

Gaussianmatch,52

Gibbsrandom�elds,6,75

GLCM,13,46,52,197

GLDH,19

GLRLM,27

GLSH,20

graylevelcooccurrencematrices,

13

graylevelhistogram,10

grid,76

Hammersley-Cli�ordtheorem,83

223

224

INDEX

Haralickfeatures,23

histogramequalization,13,52

Houghtransform,167

Huntercoordinates,35

hybridization�lter,162

ICM,161

ICRF,162

importancesampling,137

inertia,20

inversedi�erencemoment,20

Isingmodel,83

iso-second-orderconjecture,2

iterativesimulation,124

labelprior,159

log-powerspectrum,34,40

macrotexture,2

MAPestimate,160

Markovchain,124

Markovrandom�elds,6,75

Metropolisalgorithm,126

microtexture,2

morphologicalMarkovrandom�eld,

94,106

MPMestimate,161

multi-resolution,12

neighborhoodsystem,80

NGLDM,29

NovoNordisk,35

objectmodels,149

observationmodel,159

phasetransition,89,106

PID-controller,135

pixelprior,158

placementrule,5

Pottsmodels,104

powerspectrum,34,40

primitive,5

pseudolikelihoodestimation,114

randomnumbergeneration,193

reservoirsimulation,147

rotationcorrection,167

SARmodel,108

simulatedannealing,160,162,173

spin-exchange,129

spin- ip,127

structuringelement,95

Swendsen-Wangalgorithm,130

templateprior,159,162

tessellation,76

texton,5

texture,1

Brodatz,2

deterministic,2

INDEX

225

hierarchical,2

random,2

textureanalysis,5

statistical,5

structural,5

textureelements,5

transitionmatrix,125

voxelmodels,148

description and simulation of visual texture · uleres i statistisk terminologi. s dv anlige mark o...

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