Imag oth

(1)

(2)

(3)

(4)

(5)

Adaptive Filter sfor Edge-Pres er v ingSmoothing of Speckl e Noise in DigitalImages

By

@~larziaRabbiZaman

A thesis submittedtothe School of Graduate Studies in partialfulfillmentof the requirements(or thedegreeof

Masterof Engineering Facultyof EngineeringandApplied Science

MemorialUniversityof Newfoundland December,1992

St.John's Newfoundla nd Canada

(6)

.+.

NalionalUbrary

of Canada Bibhothequenaloooa ic

duCanada Acquisitionsand Direction des~cquisil!01lSet Bibliographicsevces Branch des servces blbhographKjucs 395WellongIOrlSUecl ~.<WWr:looglorl

~~~?"allO ~.rJJ~Q-l!a,.,)

The author has granted an irrevocable non-exclusivelicence allowingthe National Library of Canada t,Q reproduce, loan, distribute or sell copies of his/herthesis by any meansand inan y form orformat,making this thesis available to Interested persons.

The author retainsownershipof the copyright in his/her thesis.

Neither the thesis nor substantial extractsfromit may be printedor otherwise reproduced without his/herpermission .

L'auteura accorde une licence irrevocable at non exclusive permettant it. la Biblioth equ e nalionate du Canada de reproduire,preter,dlstr ibuer ou vendre des copiesdesa these de quelqu e manfere et sous quelqueforme que cesoit pour mettre des exemplaires de cette these

a

^1a disposition des personneslntereesees.

L'auteurconserve la proprlete du droit d'auteur qui protege sa these. Ni la these nldesextraits substantiels de celle-cl ne doivent etre Imprimes ou autrement reproduits sansson autorisation.

ISBN 0-315 -82&17-7

Canada

(7)

Abs tra c t

Speckleisa commonphenomenonin alltypes ofcoherentenergyimagingsysh'llIsuch as Synthetic Apert ureRadar(SAR) , laser,ultras ound , acouatica, sonaretc,Sp""H ' is a multiplie erivc-convcludonalnoise and as such isdifferentfromothcrcom1Jlulll~' found typesof noise suchasAdditiveWhiteGaussianNoise (AW GN).IIl'IIN',lilft'r<'lIl methods ofprocessingarc required to restorespeckledimages. Moreover ,ill mall}' applications , the edgestructureof an image isveryimportant, ana usuallilll'rili/!;

methods arenotwell suitedfor preserving edgesparticularlyin speckledi1llo'W's. III this work,an extensive studyhasbeenmadeto invest igate theapplica hilityuf dif- ferentexist ingnonlinearfilt ering meth odsandalso a!JCWQuadraticVolterraPlhcr (Q VF)based onspeckle-model to solvethe problemofspeckled imagerestorntjuu in termsofnoise smoothingand edgepreservat ion. Edgedetectionitselftillsl', ~· k·

ledimages isamajor problemwhich hasnot beenaddressedbymanyrcs earclu-r s.

This thesis attemp t stoprovide a better approa ch toUte solutionofrest oriugilll;II!,t'.S

corru ptedby specklewhilepreservingtheir edge information.

(8)

Acknowledgements

I would liketo expressmy deepgratitudeandthanks to my superv isor Dr.Cecilia.

M"I'lILcyforher guidance,useful discussions, criticismsandencour agementsformy work throughout the progra mandalsoforthefinancial supportshe hasprovidedme.

IwouldliketothanktheSchoolof Grad uateStudies,allfaculty and staffmembers speciallytheC-CAEstaff membersandalso all fellowsgraduat e studen ts.

Persona lly,J wouldliketo thankDr.J.Sharp ,Associate Dean,Gradua te Studies, Facult y ofEnglneerlng,forhisencouragementsand supportspeciallyncarthe end of IllyM.Ellg. program.ThanksalsogotoDr. T. Chari,theformer Associa teDean, Gradua te studies, Faculty of Engineering, for hiskindconsiderationandassistance.

Iwould alsolike tothank Prof. M.A. Rahman andotherCIDA official srelated ttlMUN/CIDA/B1Tprojectunder whichIiniti ally startedmyMaste rs progra m.

IsincerelyacknowledgeMr.Tony Parsons,a grad uateofElectri cal Eng ineering, Faculty of Engineering,MUN,forthe X-imageprogram he has developedduring his final year projectand also Mr, DipankarBhatt acha rya,aformer graduate student, Facultyof Engineering,MUNforthe usefulmodificati onshemadeonthe original X-imageprogram .

Iwould alsoliketoacknowledgeall examinersofmy thesisfortheirusefulcom- nwnts andsuggestions.

FinallyIthankmy parentsfor allencouragements and helpIgotfrom them during the courseofmystudy.

iii

(9)

"

25 31 36 36

(11)

:JA.::! Fro!!" CFARedge111'II'rlor. :I,.i.:) Bo\,ik':tratioofAn-ragl':' {HOAl :J..I.-I Extend edRatio of,he'ril!;''''CF;\U 3..; Perfcr man re measures

3.5 .2 Edg('preserving n....asutl'S :l.5.3 Othermeasures. 3.6 Conclu d ingr('fl'ark~.

" Methodologyan d Implem e nt a t ion .1.1 ~lethodolog.v

·1.1.1 Choice of speckle mo r!f'l

·1.1.2 Qua,lrAticVolterrafilter .. -1.1.3 Choice offillers.

4.1.4 Types of data .

'1.1.5 Edge de tection .. ,

·1.1.6 Perfor mance measures

·1.2 Imp lementat ion..

-I.:,U Specklesimulatio n _.

-1..2.2 Quadrati cVollerr", filte rimpl...mentation. . 4.2.3 RatioofAveragesedg ede t£'C': tor.. 04.2.4 Edge thinningalgorithm 4..] Concludingremarks.

5 Results 5.1 Int roduc t ion. . .

'I i

:I~l III III II I:!

1:1

,.\ ,.,

11

17

·I~l I~J 1~1

fl '

fjr.

er

(12)

.'i.2 Fillr:rillg results 6'

rj.:U Linear filter n

."1.2.2 ~If'd j a nfiltf'r n

.

'i.2.:1 Ilomcm erphicfiltcr n

'1.2.'! Two-I'oillt Taylorfilter 75

;1.2....1 ~I lt l t i p l ica tiveLet! filfr-r. 7.:'

;1.2.li Sigma filter 78

.'i.2.7 Frost filler. 78

!i.:?8 quadraticVolterra filler 78

;,.2.!I qllHlllitlll ivc m('lISures 'I

;1.:1 Edgf',If't~tionresults 91

;1.:1.1 QlIant,ilativc measure . 93

'1.·1 Criti cal Re view

"

II Conclus io ns 10'

References 109

(13)

List of Figures

3.1 Specklemodel. .1.2 Homomorph icflhoring

.1.1 (a)Origina limage"bands"; (b) One-lookapec kle mrr upl('c!\":r:< i,,";

(e)Four-look speckle corruptedversion.. . .'il

4.2 Scatte rplotofstd/mea nratiovs. meanink' mily(Im.\g..: "ha n,"''' ) . !,,:!

·1.3 Histogram ofII.uniformimagecorruptedbysrwckll:noise (n..·.\o ::.:·Hii !'I:!

4.4 4 one-dim en sionalfiltcnindiffere nt orie nta lio n, . . ..,S -1.5 NeighborhoodpAirorientedin fourdifferentdircdioll~ .. iii

5.1 Original"combine. pic"image

5.2 Speckle corru pted image(a) one-look .(b) rOIIT· look. ill .').3 Linea r averegefilter,window~i1.C("'1 3)( :J .(h) ."1>t:)•••• i'.l 5...1 Median filter.windowsizeCaj .'j;( !i.(hJ 7xi. 7:1 5.5 Homomorphic filter,windowsizt' (a) !))( 5,(b)7x7 71 5.6 Two-pointTaylorfiller, window size (a) .)x,'i,(b)1 x7 7lj 5.1 Multi plicat ive Lee fillc r,windows!z..ta )."jx,"I,(Ii)1"7 77

5.8 Sigma filter,windowsizefa)~x .1,(1))7x7. 7!I

5.9 Frost filter,windowsize(a) 5x.1,(bj7x7 ... •.. 1m

viii

(14)

."dO"f1'lrd"r"ilr1i1~':(a)Original,fb)Speckle corrupted, Linear average

flhcr(c);OJ:/.'i,(<1)iKi. 81

!i.11QVF(vertical],(a)T

=

n,!),(h)T

=

1.1,(c)T

=

L:J 8.1

.'i.I~QVF[ho rizont al ], (a)N

=

i,(b )N

=

5,(c)N=9 85

.""1.1:1QVF (hori7.0ntaland ve rtica l),(a) c "'"0.25,(I.» c

=

0.35,(c) c

=

0.15 86

.'''-1·1qVF(I10 rizo nlaland voet iealerlge adaptive]. 86

.';.l.'iQVF-I,windowsize(a).J)(.1,(h)ix7. 88

Ii.IIiqVF.II,window size (a),J)(.'i,(h)7x7 89

.""di True111gf~mapof the originalimage"combine" 92

!i.I$E,I~emapofthespecklecorrupted image. 95

:'j.I!! E'[gf'map ofthe filteredimage[Median]. 95

:I.:!OEllgemapofthe filteredimage (Mu ltiplicativeLee) 96

.""I.:!lEtlgc mapof thefilteredimage(Frost) 97

,'l.:!'!Etlgc map of thefilt eredimage (QV F) .. 97

ix

(15)

List of Tables

5.1 QVFswith differentparameters(Image:"border"]

5.2 QVPs withdifferent pa rameters(Image: "combine"]

5.3 Quanti tative measuresfor noisesmoothing . 5.4 Qua ntita ti ve measures for edge preserving

87

(16)

List of Abbreviations

pdf ; proh a hilitydens ityfunction

~td :sta ndard deviat ion AweN :Additive WhiteGaussia n Noise CFAR :Const ant False Alarm Ra te COY :Coefficient OfVariation GS], :Gaussian SmoothedLaplacian IIVS :lIuman Visua lSystem

r.xr s :LeastMean Squa re LOG ;Laplacia nOf Gaussian Lil :LikelihoodRa tio r-.IAP :Maximum APosteriori i\IItOA:~Iodifi cdRat io

or

Averages l\ISE :MeanSquaredError 1',1 :ProbabilityOf Detect ion Pfll :Probabil ity of FalseAlar m PSI~ : PointSpread Function QVF :Quadra ti c Volte rra Filter Re OA:lintinand Grad ientOfAverages nOA :Ra tioOfAvera ges SAl{ :SyntheticApert ureRada r SNit : Signalto Noise Ratio

xi

(17)

List of Important Symbols

19 : "Kro necker" product ofmarrlccs

Ir{.) ; tra ceofa matrix

{.)T :tra nsposeofa.matrix

E{.J :expectedvalueoperato r dir((i , j) , (k ,l »,:directionof Iwopixels,(i,j)and(k,/) dist«i ,j),(k,l)) : distancebetweenpixels(i,j)and(k,l)

: setofindepe nd entquadr at iccoefficien tsof theQVF : ind e pendent quadrat iccoeffici ents(orType 0,Ty peI

and Type 2QVF,respectively C :"correct"edgefactor F :"false" edge factor

G : gradie nt magnit ude

G; :gradientmagnitudefortheithsetofpairofthenciglJbuurhuu cls H1JH~ :linea r andquad ratic coefficient matri x

or

theQVF,respectively

:number of independentlooks ,'d :"missed" edge factor

N :window size

Nt :total numberof pixels subjecte d to quadraLicoperation

R :ratio magnitud e

R; :ratio magnitudeforthe ilh se tof pai r ofthenciglihourllol!,h :regionofsupp ortofthe QVF

T T,

: threshold :gradient thres hold

xii

(18)

'I;

w

(l, b

:rat iothreshold : "wrong"edgefacto r

:weightsassigned forAo•Aland A"respectively :Inputmatrix forthe QVF

:Y,0Y2

: stand ard deviationoftheorigi na limage and noise proces s,respectively :powercoefficie nts fortwo point Taylorfilter

:PSFofthe ima ging syst em hu :const anttermofthe QVF

hI!k, :linear and quadratic ope rators, respectively hI,112 :linear and quadrati c coefficients ,respec t ively

:noise process :scalingfactor

r(i ,j) :original or un-corruptedimage atpixelcoordinate(i,j)

i(i,j),i{i,j) :loca l mean andestimated value orz( i,j) ,respect ively y(i,j),y(i,j) :noisyimage atpixellocation(i,i)and itslocalmean ,respe ctively

xiii

(19)

Chapter 1 Introduction

1.1 G enera l

Amajortask indigit alimage processingisrestoringnoisyimages. Image::!oftCIl becomecorrupted with differenttypesofnoises duringtheirformation.Themain aimof imagerestorat ion is to reduce the effectof noiseasmuchaspossill lt~and therebyto producean estimateof animage whatwouldhavebeenproducedhilt!tile imaging system beenideal. There arc manytechnique sfor imagerestoration,hilS l'l l OIl differentcriteria and dependingon differenttypesof noises. Not onlyisitlmpcsaiblc to developa generaltechniquewhichwould be able to restoreanimageexactly,hut itisalso very difficultto esta blish atechnique whichresultsinallimagecstitllillc reasonablyclose to the original.The problemis substantially moredifficultwhenrhc noise issignificant a.nd/ot non-additiveinnature.Therearcseveralrnflllllllhtllat havebeen developed toreducetheeffector multiplica tivenoiseinimages,hul tlll,t!·

is stilla wide scope Ior furtherresearchinthis area.

(20)

1.2 Motivat ion

Tile purpose of thisresearch is to focus on the problem of restoring digitalimages from speckle-degradedversions.Speckleis a noise which occurs dueto thephysics of coher- cnt imaging systemssuchas those based, for example, on laser,ultrasound or synthetic apertureradar (SAR) imaging.Speckle noise may be modeUed as a convolutional- multiplicativeprocess, and isdefinitely non-additive.Linearsignalprocessingis well establishedfor thecase ofadditive noise but the problemarising from multiplicative noise orconvolutionalnoise may bebett er suitedbynonlinear methodswhich has TllIlyet been fullydeveloped.Volterra filters have been used in several nonlinear applicationsin digitalsignalprocessing[8, 21,22, 50].The quadraticVolterrafilter has been usedfor differentimage processing applications such asimagerestoration, enhancement,edge extraction etc.{H , 49, 50\.Images corruptedwithmultiplicative liaisehave been restoredusingfiltersbased on Taylorseriesapproximation[37] which workquite effectively.In terms of its input/outputrelationship, a Volterra filter may be considered as a Taylor series withmemory, the Volterra filter might be ableto givebetterresults Cor thesameproblem.Although,speckle noisehasbeen studied illpreviousdigital images processing research [16, 19,24. 251,the problemof speckle smoothingwhile preservingtheedge struct ure in digitalimage has not been studied . The research of thisthesisis carried outwith an aim to restoring speckled images using thequadratic VolterrafilterwhichwiII tradeoffsmoothingofspecklenoise and preservationor edge structure. The quadraticVolterrafilt er iscompared to other filters whichhavebeenpreviously proposedforthesmoothingofmult iplicative noise.

(21)

1.3 Problem Complexit y

The comp lexity involvedinthe above mentionedproblemarises from twomnjur [nr- tors. Firstofall, nonlinea r signa l processin g isdifficult becausethereisnocompte- hcnsiveand completetheory for nonlinear systemsas these existforlinl'il.lS)"stl'IllS.

Second, the signalto be dealt withis atwodimensionalsignal and hasseveralPl~\I' liarit ics.Someofthe characteris t icsof digit alimages whichmake themdilliculltil treat as signalsare thefollowing :

•Digitalimages are two-dim en sionalsignal.

•They arcnon-st ationary. which maymeanthat locallyadapt i ve urapatinlly- variantprocessingl'~..thodisrequired .

•They are random innature,sothat the precessingofsuch imagesmay needto be basedon the irsta tist ica l properties whichare,ingeneral, not well dclinet!

nor easily estim at ed. Moreover , estima tionatimage statisticsrcquiresalargc numberofdatawhichmaynot beavailab le in a spati ally-varyin g imageorwhen the processingismeantto be locally adaptive.

•Whenimagesareprocessedforhumanviewers, thehuman visu alsyste m(HV:)) becomesan important considerati on ill the developmentof an imagercst or nricn technique .Thereisanumberof different existing models oftheIIVS which have nonlinearcomponents .Hence,standardtechniquessuchesrhc Mean Square Error(MSE)forperforman cemeasurem entarcnotalwa,)'sadequat ein judgillg thevisual quality oftheimages. Asaresult,any general signalestimation techniquesbased onsuchsta ndard performan cemeasures[e.g, LeastMCIlII Square (LMS) adaptivetechnique)mayneedto bemodifiedtousc inimage

(22)

restc re ticnalgorithms.

1.4 Approach to the solution

Theapproach to thesolution of the st atedprobleminvolves the followingthree steps.

• Modellingspeck led images

• Designing speckle-sped fie filters

• Evaluating the perform ance oftheproposedfilte rs

Itis neeeesarytohavea realisticmodelofsp eckleto designamodel-basedfilter.

Are al isticmodelhas been developedformulti-lookSAR imag esbase donprevious work[191 andtheaccuracyoftbemodel hasbeenverified,Synthetic images using themodel havebeen createdforuseintestingthedesigned filter! andother existing filters.Filtersbased on thequadrati cVolterraseries, have been designed wh ich arc madelocally adaptiveaccordingto the statistic al propertiesof speckle. Tode mon-

!lra tctheperformanceofthe filters,evaluation measu r es havebeen usedwhichfocus part ic ularlyalltheirnoise sm oothing and edge preserv at ionqualities.Also,compar- isons withotherwell-knownfiltershavebeenca rried out.

1.5 Organization of the thesis

Thisthesis haa been organized as follows.Ch a pter Two prov ides background infer- mationon four major areasrelated tothe sta te dproblem-i)speckle and especially SARspeckle,ii)nonli nearfiltering me t hodsfo r imageprocessing,iii}the application of Volterrafillersparticularlyin imageprocessin g,lv) edgedetection algorithmspar- licular ly forspeckledimages.ChapterThreeprovidesi)descriptionsofthesp eckle

(23)

model,ii) different filteringmethodsinclud ingbasicbackground on the Qua dratic Volt erra Filter (QVF), allof these areused later in the comparative study,iii)different speckle-specific edge detectorsandfinallyiv)someperformanc e crite riafor evaluatingfilter performance.ChapterFour focusseson the proposedmethod ology fori) speckled SARimagemodeling, ii)thedesignof QVF's forspeckle dimage restore- tion , intended {orboth specklesmoothing andedge preservationandalsoiii)1l1lCW

spec kle-specificedgedetect orwith anedge-m apthinningalgorithm.Resultsobtained from basic experimentswiththe QVFand fromthe comparativ e stud yarc presented in ChapterFive.Finally,conclusionsbasedonthe resultsobtainedantisuggesnons forfutureworkare providedin Chapter Six.

(24)

Chapter 2 Literature Review

2.1 Int roduct ion

This chapterisgrouped intofourmajorsections. The second sectionprovidesa brief survey of the pr..tocework onspeckle-noise modeling, simulationand filtering techniques.Thissectionalso provides a briefbackgroundinformationon SARimage processingmainl yfrom the point of view of the speckl esmoot hing.The thirdsection isanoverview of differentexisting nonline arfiltering methodsandtheirimp ort ance inimage restorat ion.The nextsectiondiscusses the applica tionof quadraticVolterra filtersin variousfields, particularlyinimageprocessingapplicati ons.Abriefsu r veyof differentedgedetectionmethodsispresent edinthefift h section. Since edge det ection for speckledimages is of an importantconsiderat ionin thisstudy, special attention ispaidtospeckle-specific edgedetect ionmethods.

(25)

2.2 Spec kle Noise

Speckle isil.phenomenon which occurs in imagesproducedbyalltypesof coherent orpart ia lly coherentimaging syste msuch assynt heti capert ure radar(SAR),laser, acoustic,ultrasoun d, etc. The effect ofspeckle noise011images isgenerally1101de- sirable becauseitdegrades imagesinsuch a waythatitmayhedifficult toextract usefulinformatio nfromthese.Therearemanyimage restoration and enhancement techniquesusedfor either removing speckle {romtheimages or emphasizingthe In- formativeaspect sofimages.Todesign an optimumimage restorationtechnique ,it isnecessarytoha ve an appropria temathematicalmodel of thespecklenoise based onits statisticalproperties.Speckleprop er tiesarediscussedin severalliter a t ures [5, 63],andinparticular,thestatist icalproperties ofSAR speckleare well described byArsena ultetal.15,6\ .It is wenknown that speckleisa signal-depend entnoise [16, 19. 23, 281.su chthatitsmagnitudedepend s ontheintensityofthe underlyingim- age.Lee'sspecklesmoothing algorithm[28,301for SARimagesisbaaed on employing the sigmaprobabilityofthe Gaussian distri but ion oftheimage intensity.Frostelal.

[161developeda mathematicalmodelforspeckle and designedan adaptivefillerfot multiplicativenoise.Althoughmostof the literaturein this areaassumesthatspeckle is a pointwisemultiplicativenoise,there arecaseswherethis assumptionis not~atis· factory (62).The model derived byKuanelal.[24Jtakesspeckle-correla tioneffeds into accountwhere an adaptivenoise-smoothingfilterand a MaximumaPosteriori (MAP)filter havealso been developed. Raneyhas madean extensivestudyof speckle andreview ed specJclemodelling and simulat iontechniques in the literat ure{511.SAR image restorationhas been attempted byJinelal.1201. wherethe segmentation and classificatio nof SAR imagesareoften verydifficultdue to the presenceofspeckle.

(26)

Theseproblems havebeenaddressedby differentauthors(IS, 19, 25).Hudson and Jcrnig...n {191proposed a modelfor SARimages corrupted by speckle andcompare d theperformance

or

differentfiltersin termsof their abilityin textureclassification

011the filtered images.Themodelthey proposed is basedonthestatistical property ofone-lookspeckleas wellasthe impulse respon se of theradar whichinfactcauses the spatialcorrelation of speckledimages. The model hasbeen verifiedand maybe considered a validmodelforspeckle corrupted SAR images.

2.3 Nonlinear Filtering

Linear systemshal efoundextensive useinsig nal processingbecausethey are rela- tivelysimpleto analyze,designand implement .Because of theirsimplicity,ithas also been possible to establishgeneralizedtheo ries oflinear systems.Butthere are certainsituat ions thatrequirenonlinearprocessingof signals.Because,nonlinearsys- tems arerela t ively difficult to design andimplement,it haabecome usualto assume many practicalsystems to belinearsystems and to applylinear systemstheory. Such approximationsarcsometimes reasonable butthere are cases in which performance qualityissubstantiallyreducedand insuchcases, nonlinear processingmaybecome unavoidable.

2.3.1 Nonlinear systemtheo ry

In1968, Oppenheimetal.(H) made an extensive study witha view of providinga generalized non linear systems theory such as thereexists forlinearsystems. They were successfu lin using the conceptofgeneralized superpositiontodevelop homomorphic filter ing.Moloney[37Jusedthe generalized superpositionprinciplefor the

(27)

problemofImagerestorat ion in thecase of imagesdegradedby pointw isemultiplica- tive ncise,

2.3.2 Application of nonlinear filters

OppenheimeIal.[41)describedthe generalizedsuperpositionprinciplefuruUlIlim';,r filte ringofconvolvedand multiplicativesignalswithtwo examples of itspra rlir;\l applicat ion, namely,i)aud iodynamicrange compressionandexpansion,;llldirlla.~l' enhancem ent; ii}echo removalandSPl'CCh.analysis.

Mathe ws[34Jprovidesquite abroadoverviewon applications. current rt.'l«·ur..1L trends,results,problemareasetc. regardingnonlinea r filters where a latt icest r ucture for nonlin ea radaptivefilters usingtruncziedVolterra_1eriesha~also he...·nc1isrlls~I~1.

2.3,3 Nonlinearfilter s in imageproce ssingapplications

As mentionedabove,nonlinearfilter inghas beenused illimage processing.IP lllim ucne.Oneofthe major reasons for usingno nlinea rprocessing ofirna~e~ i~th atthe human visualsyst emis known tohave nonli nearcom ponents13,581.Thereilrt.'oIlier reasonswhy itbec omes almostessentialto processimagesusing nonlinearlilll'r~.

One suchsit uationoccursin restoring noisyimages.hnagl'll cftengelcorruptedwilh different typesof noises dueto the poor imagingmedia, lion-idealimaging~y~lc'lII~

etc .•and noisemaybe com binedwit hthe image eitherlinearlyor insome ethe rfa.~li·

ion. Buteven ifit islinear,nonlinearfilteri ng isoftenrecom mendedIJI,.'cauM:ufl!IC non-statio narycha racte rist icsof manyimages. To en ha nceimage details,sl,ecific:illJy inpreserv ingedges, nonline ar filteri ngmaybe requ ired.Soitiseasil y1l,rgUl~dthat itis impo r ta nttorestoreimages bynonlinea rmet hodswhenthenoiseit~dris nllt linearly combined with thesignal.

(28)

10

At thispoint,itisclearthatlionlinear6ltcringplays averyimportant role in imagerestoration, AbramaticandSilverman[1]modifiedtheWiener approachto theproble m cf rest oring imagesdegrad ed bywhiteGaussi annoise. Theypaidspecia l attention to preser vingthe edges whichisanotherimportant considerat ion inimag e restorati on.Quantitati veandqua lita ti ve compar isons betweensixwell-knownadap- tive filtersused for imagerestorat ion was made by Mastin [33).Chin andYeh [12J lIIarlca similar kind of stud ywithcommonty used filters.Nagao and Mats uyema[39J

.~Ilggcstcdmeth o dsfor thedifficulttaskoftradingofT betweennoisesmoothing and

l'dgepreser vat ion.

Ll'C'Smulripliceclve filter[27,291.designedfor apointwisemultiplicative noise model,is oneexamp le ofsuch Alters.His algorithm is basedonloca l stat isticsof im- agesand itprod ucesverysatisfac-oryresults. Homomorph ic filterin g[4Jis also an ef- Icctivcmethod of restoringimages corru pt edbypoin twise multiplicative noise.Teklap aud Palcvik[59] alsoinvest igatedseveraltechniqu es forre storing images that are de- gradct.l withpointwisemultiplica ti venoise cause dby sensornonlinearity. Moloney andJernigan138 J developedanew approach tothesolutionof thesame prob lemus- ingtheconcept ofmulti plicativesuperpositi on,The devel oped algorithmis adaptive because itusesthe localstatist icsofthe image wit htheopti mal op erators calculat ed Il~i n gTaylorseri es approximations.

Nonlinear filtersusin g truncat ed Volterraserieshave bee n alsodevelope d andused illdifferentapplicat ionsincludingimageprocessin g whichwillbediscussednext.

(29)

"

2.4 Vo lter r a filters

h h.u almost beena cen turysince Volte rraintroduced his~iC5...hichhas beenf'lIl1ul tobeavery usefultoolfordcscribin~input/outp utrelationshipsorncnlin..ars~·~l.·11I.'"

In recentyea rs,itha.s been extensivelyusedindiffere nt areastohandlen....nli...nr problems. The mainreason foritspopularit yisthatitintrodurcsageneralize..1 theoryfornonl inearsystemsina.relativelysimpleandconceptuallymanageableway.

2.4.1 Volterra seriestheoryan d application

Biglieri171dis cuss edthe nonlinearVolterraproce ssor co vering its general theor y,im- pleme ntationandsom eappl ica tio ns,namely,i)MeanSqua reError(MSE)prctJicl io ll ofdiscrete-tim era ndo mprocessesand H) ide nt i ficatio nofnonlinear5)'11lcmswi t h memory.

System ident ificationbysecondorde rnonlinearVolterraHltera,hft..$ abo11l.'C1I studiedexter.sivelybyKob andPowers(22).Thesolutionfor theopti mumfllter, iterative factorizationto(acilita te theimplemen ta tionofthefilter,LMSadapti ve alsolit hTn!l et c.havebeen coveredin thislite ra t u re.The pra.ctiuJapplica~iOl\they wereinteres tedin,wastoutilize thesecond orde r Volterra filtertomoddandpredict thedynam icbehaviour ofmoored vesselsunder random sea waves. Sicur iUlza and R.a.mponi(55)alsoprc pcaed anDISadapti ve algorithm forsecond orderVoller r....

filtersusing distr ibuti vearithmeti c. Themainconcern

or

theirworkwasto provide anefficientandsimplerealizationorthefilterwhichcou ld be wellsuitedtomodern technologiese.g . VLSItechnology.Chiangelal.1111handled the sameproblem usingadiffere n t approach, withanimplementa ti o nbasedon matrixdecompcsltion.

Byextendingthe Ka lman filteringapproach,afast-response second-orderadaptive

(30)

12

Volterrafilte r was proposedby Davilaetaf.[131. Nonlinearsystemidentificationhas 11I.:cn carriedout using trunca tedVolterraseries modelsbyseveralworkers{22,4.51,but allof them ass umedGaussiannoiseas the system input. Kim and Powers[21] made an attemp t tosolvethe problemofsystemidentificationassuming generalizedrandom iuput and using a truncatedVolterraseries model.Adaptivenoise cancellation [571 for vain !banddata transmission,cancellationof inter-symbolinterfere nceIS]and echocancellation 12Jarc threesimilarapplicationswhere Volterrafiltershavebeen foundt...beeffecti ve.

Inmostsig na l processingapplications ,thesignalsare one-dlrncnsicnal,usuallya function of time. Ilene',workthathasbeen discussed so far,has all been carriedout for1-D signals.Howe ver,rea lizatio n of higher dimensionalquadraticfiltershave been J('al~with in[36, 56],Mcr talosCIal,[36] proposed matrix:decompositiontechniques for rcalil.ing2-Dfilters.

2.4.2 Volterrafilter in digital ima ge processing

Sinceitis very importantin imageprocessingto dealwit h nonlinearity, theThe Volte rrafilter seems to bea possiblechoicefor thenonlinea rfilteringofimages.Non- linear processing itselfis verycomplex andsince an image is atwo-di mensionalsignal, nonlinearprocessingofimagesis morecomplex,However , consideringaVolter ra se- riesonly uptothe quadratic term. itis feasibleto realize2-0 quad raticVolterrafilters.

Workthat has been doneprevious lyusing Volterra filte rsin digitalimage processing applicationshas beenbased on Volterraseriestruncatedup to the first nonlinear term or thequad ratic term and hencethe filtersderived areter med quad ra tic Volterra filte rs,

(31)

2.4.3 Application areas

Sincethis isstilla newfield, there islittl eliteralurc on theapplication of Vlllh'r ra series inimage processing.Oneofthe major st udies inthisarea has Ul'ClLdom-in 1990byRamponi[501. So far,his papercouldbetakenasa frameworkprovidingiI

background of thepropertie sanddesign of quadraticisot ropic 2-D Volterm lillt'rs.

The use of Volterrafilterseems to be quitepromisinginbothcasesofildditi veaud multiplicat ivenoise reduction(501.Designcxamplesinthesame literat ureshowl"]

howdifferentsituat ions couldbedealt with using the Vollerr afilter forimage reslorn- tion.Theenhancementcfim egce isoften ofgreat interest.Asment ionedhyHampcni [4.41, mostoftheimage enhancement techniques have used luminanceSl"itlc modlfi- cation afterprocessing the image th roughlinear ornonlinear filter s.Butthis,[o,'s notimprovethequality ofan image,part icularlyif theimagecontainsIiSigllili.:;lllt amountof noise.Because,inthissituutlcn,norse smoothing isdoneattile'listof losingtheimagedetails,anoptlmi aariou betweennoise smoot hing end preserving image detailsbecomesa veryimportant issue.Arecent approachmadebylliUlll)lJllj

1 44J

uses a luminance independentoper ator. The filterhe usedisaVolletri~scri,~s truncatedupto thefirst nonlinear term.Edgeextractionisone ofthe irnpor tillit tasks in imageprocessingwhich alsoneedsnonlinearprocessing.Harnponi1~!J1also proposed quadratic Volterrafiltersforthispurpose.

2.4.4 Qua dra t ic Volterrafilter design

Themainproblemregardingtl:'ldesign of quadrat icVolt errafillers is the complexity part icularlyfor2·0 signalssuchasimages.A2·0quadraticfilterhas botha setuf linearcoefficients and a set or quadratic coefficients.The numberof filtercoefficients

(32)

14

furthequadraticpartisquil e largeand hencedifficulttodesign (e.g.even(ora3't3 support2-0Volterra filter,the number of linearcoefficientsand quadratic coefficients arc9and 81,respectively].Hut as shownbyRamponi,

1 501,

theselM!.e numbc N may bereducedbyimpo singcertaincon dition s such asisotropy, symmetry,uniform ity in gray leveletc .But undoubtedly,itisdifficult to find ast ra ig bt forward designstrategy such astha texists forfinite suppo rtlinear filters.Becauseofthis inherentcomplexit y of the problem .nosuch designtechniquehas yet been availablewhichisbasedent irely uponthetheoreticalpropert iesof the Volterraseries. Somelitera t ures(46, 47,481 were reportedwhichgive"ro ughidea about thedCllignapproachonecan follow.Themain

~lrat cgyisto definethedesigningproblem,formula te anobj ecti ve function based on tha t,then minimize thefunctionand thereby designthe filtercoefficients .An cprimieeticnapproachhas beensuggested byRamponiandUkovich(47Jwhere three exisl in!no nlinearoptimizationtechniq ues.namelyi)Steepestdescent,ii)Powe l's conjuga~eandiii)Simulal.edannealingareused.

Ramponi in hisliterature{50Jsuggested aslightly differentdesignstralegywhich he referred toas "Bi-impulsc responsedesign"based on thesa me conceptof impulse responseusedin designinglinea r filters. For desi!n ingthequadraticpart of the quadrati cVolterra filtertheresponsetoApAirof impulsesatappropriatelocations is measured.

LinandUnbcha uen (31]intr odu ced an adaptive approachfor designingquadratic Volterrafillers,thatcouldhe used forchannelequalisationandimageresto rat ion.

The techniqu eusedhereis anLMS typewheretheconst ant ,linear,andquadra tic coefficients ofthe quadr atI CVolterrafilters are adjusteddurin giterat io nsunt il a mini mum mean squ a reerrorisachieved.

Allhau ghlitt leworkhas beenreportedin this area,itappearstobe promising

(33)

toapply thequadr aticVolterra filtertorestori ng noisy imagesespeciallywhenthe noiseis notadditive in natur e. Designing thefilteris themainconcern inthisregard.

Though some previous...ork1,16,47,48Jprop osedanoptimiza tion approach,ttu- formulation ofan objective function to obtainthe filtercoefficientsisnol~'l'tclcar, Theadaptive approachdiscussedin [31)dealswitha slightly different issue where theinput/output relationship of the system is assumed tc be known andchannel equa lization is ofmain interest.Since themain objecti veof any image reatcrarion is to smoothout thenoise whi lepreserv ing thefinedetails asmuch aspossihle,iL1lI1 sincean image itself isanon-statio na rysignal. alocallyadapt ive techniquemigluhe abetter approach.

2.5 Edge detection

One ofthe majortasksin image analysisis edgedetection,EdgesarefUlldamcllt illly impor tant primitive features ofanimage becausetheyoftenprovideanindicationof the physical extent ofobj ectswithinthe image.Hence,edgedetectionisveryimpor- tantfor severalimageprocessingtaskssuchaspatt ernrecognitio n, segmentaliolletc.

Anedge is defined asavaria tionor disconti nuityinimage int ensityfC!lulting from changes in some physicalprop ertiesofthesurface, namely, its reflectance,geomet ry and/orincident illumination.Itis interestingto note that whilethehllrll<H1visual syste mperformsedge det ect ion quite easily,itis not very easytoautomatethe process.However,there isno shor t ageofedgedetectionalgorithmswhichwor kalluigitaJ images and provideedge maps asoutput. Classicaledgedetectionoperators developed duringthe1960'sperformquitesatisfactorilyonidea l[un-corrupt ed] images.

But theseedge detector sperform poorlyon noisyimages. Since inreality imagesruay

(34)

16

becorruptedbydifferenttypesof noises,recentatt empts have been carried outwith an aimtofindingedge det ecto rswhichhave been less sensitivetonoise. Techniques whicharcdevelopedparticul arlyfordetectingedges on noisyimages arealso abun- dant.But sofar, attentionhas been paid mainly to images which arecorruptedby acommon typeofnoise, such as additivewhite Gaussian noise (AWGN).Veryfew methods arcfoun d which have been developed (or images corr uptedwith noisesother than AWGN.

2.5 .1 GeneralPod ge detection techniques

Are view paperbyDavis

I J.lJ

discuss esseveral edgedetectionappro acheswhichhave been proposedfor the solution ofthe general edge detectionproblem.Most ofthe previouswork in edge detection uses smalldifferentialoperators appliedto animage followed by detectionoperatorssuch as theRobert . SobelorLapla cian (18).Such metho ds workquite effectivelyon imageswithoutnoise butshowpoorperformance

Oilnolse-dr-geaded images. :\1arrandHildreth(321developeda th eoryCor detecting intensity changesand proposed theLaplacia n of Gaussian operato rforedge detection.

Accordingto them,the Gaussianfilteris theoptimalfilterwhich provides localiza- tion inbothspace andfrequency domains,an imp ortant considerationin accurately locatingedges.Shanm ughanet al. [54] proposedafrequencydoma in filterforedge detect ion whichworksgloballyon theimage.A good edgedetectorsshouldsatisfy severalcrit eria;based onthese criteria.edgedetect ionmeasuresare usua lly defined toquan t ifytilequality ofan edgedetecto r. Pratt 'sfigureoCmerit[43Jisone ofthe sta ndard measuresavailablefor edge detectors.Canny (10] also prop osed someedge detectio nmeasures in his work.Arecenthiera rchicaledgedetection appr oac h made byMcLeillL and Jernigan

1351

is based on somepre-definedperformancecriteri a.

(35)

\7 2.5.2 Speckl e- sp ecifi cEdgeDet e ctionTech niq u es Mostedgedetectorsareofthegradient type i.e. the detectorsarcbased onIhe differencebetweenpixelvalues.This type ofedgedetector whenappliedtospl'('kll·d image s yieldsverypoo r resultsbecausespeckleis mult iplica tive in nature.ItJ,'gra ll.-!

theimage insuchaway thatthe-SNR becomes verylow andtheimageloses ihline details. Henc e,itisverydifficulttodete ctedg esinspeckledimages aud,1\11111.<\

impossib leusing the usualedge detector s. Since it is obvio ustha t spl'Cklcdepends onthesignal,speckleis more promi nentinhigherintens ity homogeneousllrt', tsthan in darker areas. Thus a ratiobetweenpixel valuesshould work bette rthanthd r difference. Whendealing with noisy images,it is bette rtotaketheratio oftheaverage pixelvaluesin two adj acentneighborhoods , oppositetothepixelof interest.,\ratlo magnitu de imageis thus formedand thresholding finally providestheedge map . This isthebasicidea behind thesimple RatioofAverage (ROA)edgedetector!!J]which is particularlyusefulfor detectingimagescorruptedwith a pointwis emult.iplicative noise. However.the probleminvolvedwit h speckle isevenmore difficulthenl\Jsc speckle isnot only multiplicative,but also has an inter- pixel spatialcorrclll.tilJlItil some extent.An efficientedgedete ctorshould beba.sed on the speckle model.

Verylittl e workhas been reported so farin thisarea ,although it is necessary topay attention to the problem of edge detectio nonspeckleimagery. Asimple method named the Coefficientof Variation(COV) hasbeenproposed[60]!lased011

the ConstantFalseAlarmRate (CFAR)conceptwhichusesacoefficient orvariation which can p..ovidc the edgestrength measures [60]. Frostetal. {17Jproposedall edgedetectingtechnique forSAR images.Themethodapplies the maximumLike- Iihood Ratio(LR)asthe measure oredge strength.Themaximum likelihoodratio iscom puted based on a SAR image model. Bovlk's Ratio of Averages (ROA) [9] is

(36)

IS anotherapproachwhich att em pt sto solve th is problem. He suggestedacombina tion ofthe ROA and the GaussianSmoothedLaplacian(CSL) methods. Accordi ng to Bovik, the ROA edgedetectorisquiteefficientonspeckle-degrad ed imagesbut hasa drawba ck of generating verythick edges .Onthe otherhand,a generaledge detec tor such as theGSLgives fine edgesbutalsogivesriseto many false edges whichisnotat 0\11 desirab le.Acombination ofthesetwo (a logicalAND operationontheresulting ima ges obtainedfrom the outpu tof these two detectors)givesa much betterresult than eitherof theindividualedgedetectors.However,itisworth-mentionin gthat theROA edgedete ctor is optimalifapoint wise multipli cativemodel having either nega tive exponentialor Gaussian firstorderstatis tics, isconsidered . Mor e recent work whichis simply anextensionoCthe ROAdetectorisproposed byTouzietat.

[60J, using the ConstantFalseAlarmRate(CFAR)concep t. Thismethodismodel based and designed particu larly{or SAR images.Ithasbeen shown that theperfer, ma nce oftheROA detec tor depend sonthe sizeoCtheneighborhood, the number oC independent looks,andthe ratio of meanpowers.This detect or usessomestat ist ical estim ates 10 calculateedgestrengt h. The effectofedge orientationand neighbor- hoodsizes arc alsoimportantconsider atio ns and have been takeninto account inthis particularedge detector.

2.6 Concluding remarks

This chapterattemptsto providean overallpictureof the previous workon areas relatedtothe problemofinteres t.Sincethere arc atleast four major areas involved, it is impossible toprovidemore than a briefbackground for each of thesetopics.

Butit isIclt thata moreelaboratedescriptionof the filters ofinterest(especially

(37)

19

theQVF ).the specklemodel. speckle-specificcd&cdetectors and filterc\'1l1nal ioll criteria -whicharc all direcrly relatedto them&inobjective orthis thesis-sholllJ be provided which aredealt within the nextchapter.

(38)

Chapter 3 Approach to the solution

3.1 I n troducti on

The problemdealtwithin thisthesis originates with coherentenergyimagingsystems whichprod uce imagescorruptedwit hillspecial type ofgranularnoisecalled'speckle'.

SAR ieone ofthema ny examplesofthe sourceof spec kle.Itis veryimportantto keep in mind thatthese images aremeantfor humanor computer int erpreter for interpretation..ndana lysisby extractingusefulinformation. To servethis purpose, itisnecessaryto obt ainreasonablygood qua lityimages .However,frequently, good qualit yimages arc not obtaine d directly{rom thepractical imagingsystemsandhence the question ofimage restora tione.lsce.

Thesolut ionoCthe problemrequiressome knowledge of mathematic al modelling of image degrad ation and of lincarandnonlinear filteringtechniques . It isalso important topreservethe finedetails of an imagewhile smoothingthe noisewhich is a difficult criterionto meet.Edge detectio nonspeckled imagesis essentialin many applications but it seemsfromthepreviousattemptsthat it isvery difficult to obtain a good edge map fromspeck ledimages. One ofth.'major goalsofth isthesis is to investiga tethe

20

(39)

:!l

effect of filteringon the speckledimagesin terms of reten tio nof fine de t ails. ,\Icw pap ers thathave been collect edin thearea ofedgedet ection onspeck ledilllagl'swill be discussedbrieflylaterinth is ch ap te r.However , firstit iswo rth whil e lookingiUI,) somewellknownlinear and nonlinear filteri ngtechn iques that mighthel pin ut'siguillg an edgeprese rvingspecklesmoothi ngfilter.

3.2 Speckle model

A modelbased filt erforsupp ressinganytype ofnoise isalways abetterchoicethana filterdevelop ed in anad hocmanner.So itisveryimp ortantto havea rea listic speckle modelin orderto designfilte rs forreducin g speckle.ItiJwellest ablish edtha tsp eckll' noiseint ens ity is propor tio naltothe unde rlyingimageintens ity, givinga 5iguitltu NoiseRat io(S~R)in anobserve dspeckledimage, equalsto oneforfu lly dcveluped sp eck le.This implies that a pointwisemulti plicati vemodelwou ldbe ableto l!l'Srri!le speckle quit e well.Alargeamount of researchrelated tothis area is fOllllJwhich assumesarnuitip ticat ive! model of speckle[16,23,301.Filtersdevelope dinitially rur mul tip lica tive noise modd have bee nusedinspe ckle redu ctionapplicatj ontoo.Thlls themodelcan berep re sented in(i,j)sp a tialcoordin at e systemas,

y(i,j)

=

x(i,j)'n(i,i) ^(:1.1)

whe rexis theorigina limage,yis the recorded imageandnis the random noiseprol:ess hav inga Ray le igh distrib ut ion fun cti on with mean one andst a ndard de via tiont~l1lal to[(4/.)

- IJ"'1[19J.

However , ithasbeenpointedoutby Tur111.at{62l t hat amultip licativ enoise I For simplicit y,rromnowon"multiplicative" willimply~poin t wiM mu [ti plicalivc~Unletl8other- wisestated.

(40)

22 modelj5not always a good choice.The main disadvantageof thismodel is thatitdoes nottakeinto accountthe correlationof specklewhichis animport ant consideration insome cases.The correlationresults mainly fromthe PointSpread Funct ion (PSF ) oftheimaging system. A benetmodelcan berealized bytaking thepointspread Iuucricninto account.Thus,the mode!of Equation3.1as shown in Fig. 3.1canbe wrtucn as:

y(i,j )

=

[.rU,j) n(i .j )].h(i ,j) where. denot es"convolutio n".

x(;,i ) 0\

r::-:::l

^{y(i, j)}

-~

n(i,j)

I

Figure 3.1: Speckle model

(3.2)

Themodeldescribed byEquation3.2isused by Frostetai.[161with theassump- tion thatthe PSF for SARsystems is animpulse.Morerecent work donebyHudson andJernigan[19Jassumesa SAR PSF whichhasacircularlysymmetric Gaussian shape.Hudsonand Jernigan used a one-lookSAR image model throughouttheir work which is not a generalizedmodel,due tothe commoncasetha tan image obtained from SAR is the average of multi pleindependentlooks.Insuchcases,the mode!of Equat ion3.2remainsthesame butwith a slight changein theProbabilityDensity Function(pdf)ofn(:r,g) .The noisepdfnow takesax-squareddistributionwithN degreesofIrccdcm.Lee (251showed how themeanandvarianceof thenoiseprocess variedfor one-look and mult i-leek SARimagemode ls.Thevariati ondue to intensity and amplit ude im ages isalso add ressedbyLee.

(41)

r

3.3 Image restoration filt ers

Most image restoration filters deal with additive white Gaussian noise (AWGNl.Bilt there arecertalncases where the noise is not additive, rather muhlplicatlvc,1I11I/o[

convolutional or some other type andmay not be Gaussian and/orwhite.Thctll- ters for AWGN are based on the well-establishedlinearsystemstheory.Thl'St, tilll'rs assume the stationarity of hoth image and noise which may also not bei\good<IS- sumption. Forexample,itis very likelythat images containsome highfr('(lucne}'rum- ponents - the edges for example. Therefore, althoughlinearfiltcusuch as III\\"P,(SS filters smooth out noise,they giverise to blurred images due to attenuationIIfhigh- frequency edge conten•.Nonlineartilteringappearstobe an alternativewhencdgt!

preservationbecomes important.Also,forsign al depende ntnoise, such;L~ ~llccklt'.

nonlinearfilteringmaybe unavoidable.At tem ptsto deal with the signaldependency of noise indesigning filterssuffer from various limit at ions . One of the majorlilll;l;~' tiona is thatitis often verydifficultto design afilte r whichis optimalwill!respe ctto a statistical parametersuchas MSE or SNR, because cstimerlon or suchparumeturs might requirethesolutionof complexmathematical equationsor maynol beexplic- itlyfound.Anotherapproachthatcouldbe adoptedin designingfiltersis lo lbign locall y adap tivefilters.Alt hough a relativelynew technique,there have beeuIllMIY applications of this technique in digitalimage restorationwhichhavebeen folHll] tu bequ it e successful. The mainidea behind this approachis that a finite cxlenl filter operateson a finite supportof a corrupted image withitsfilter coefficientsdepending on localstat ist ics of this support.The image is assumed to be 5tatiQIlaryuver the finite supportused for filt eri ng at any time.The resultsaresignificantly improved overwould have been ob tainedbyglobally processing theimage with one fixedfiller.

(42)

k(i,j )is the gain factordefined between 0 and1 asgiven below;

k(i , j)==

Q(i~j~'~)Ull

where

Q(i ,j)

=

E[(y(i,j)-f(i,j))lj_Ull

and1111isan estima teofthe additivenoisevarian ceandE[.Idenotes the operator givingtheexpectedvalueorthemea.nvalue.

3.3.2 Nonline arFilterin g

Although linearfiltersare simple and easy to designandimplement ,theyalso~lI l fc r from some limita tionsasmentioned earlier. Fornoisewhich isnotadditive,il.nonlinear filter may he the onlysolution.AlargenumberofncnllnearfiltershavobL'C1l

developed so far andresearch isstill goingoninthis area.Thissectiondescribes some oCthe commonlyused nonlinear filters which can begroupedintothe twocategorics,

1.Generaland 2.Model based

•Pointwisemultiplicat ive

•SpeckleReducing

a) Med ia nFilter:Thisisone ofthe mostclassicaland commonlyusedgeneral- purpose nonlinearfilters.Thisalgorithmwas firstproposed by Tukey(43\and used extensivelyin timeseries analysis. Later on,itfound application in digitalimage processing. Theprincipleof operationof this filteris verysimple. The filteris

(43)

Although,theimprovement achieved isat the priceof complexityand timerequ ire- mentindesign andimplementation,in most cases,itisw.orthwhiletodesign an adaptivefilter.Thefollowingsection willfocus on some ofthe commonlyused linear andnonlinear filters.As men tionedbefore,sincetheadapt ive quadraticVolterrafilter is ofmaininteresthere and has bothlinearandnonlinear par ts, itseemsimportant tounderstandhowthese twoparts functionin:o;rr;oothingnoiseand preservingedge structure. For this reason, alinear filteris alsoincludedin the following study.Itis intended lo make a comparat ivestudyof the performance of severalnonlinearadap- tlvoflhcrsincluding thequa dr atic Volterrafilter inedge-preserving noisesmoothing of speckledimages. lIenee,itseemsworthwhileto brieflydescribesome of the exist- ingnonlinearmetho ds which willbe usedlater . Thegeneraltheory andpropertiesof the quadra ticVolterr afilter will also bediscussed.

3.3 .1 Linear Filte rs

Linear filteringhasbeen usedextensivelyfor removing Gaussianaddit ivenoise from images. The popularityof linear filteringstemsfrom thefact that itis verysimple instruct ureandhence easy to designandimplement . Thefilter developed byLee [271 foraddit iveGaussiannoiseof zero mean isprobablyagoodexampleofadaptive linearfilters.Itsinput/ output relatio nshipcan be represent ed by

xli,;)~x(i,j)+k(i,;Hy(i,;1-x(i,;I] (3,3)

wherei(i.j) is the estimatedpixel intensityat(i,j), y(i,j) is noise corru pted image pixel, basedonadditive noise modely(i,j)=x(i,j)

+

n(i,j ),andn(i,j) isthe noise component .t(i,j) is themean of the un-corrup ted image,given byx(i,j)=y(i,j),

(44)

26

composedof a slidingwindow encompassing an odd numberofpixels.The pixel of interca tis replaced by the median of the pixels within the window. Although simple, tile filte rhasbeen prove n to be a good image enhancer speciallyinthecase ofin-pulse orshotnoise.

b} HomomorphicFilter : Its technique could be usedquiteeffectively for image restorationwhentheimage is subjectto multiplicativeinterferenceordeg radation.

/I"logarithm ic"opera ti onperformed on thenoisy imageconverts multi plicatively combinedsignalsto a sumof signals to which linearfilteringtechniquecan be applied beforea linal"exponential"operation givesbackthe estimate of the originalimage (11·Tilealgorithmis best illustrated in Fig.3.2. Atthispoint, it isworth ment'oning thalit is quitedifficultto designthe linear filter inthe middle ofthe cascade because once the"logarithmic"operation is performed,the noise statisticsare changedand

~impli fyingassumptionscouldlead to a. poor overallestimate.

,=x~

.J:":I __ r.:-l

~

Figure 3.2:Homomorphi c filtering

c)Lee'sMult iplicativeFilter : Thisis a verywellknown adaptivenonlinear filter proposedbyLee(271 whichisbased on amultiplica tive noise model. As wit h Lee'sadditive filter,themean andvarianceof theoriginal image can be estimated fromlocalmean andvariance.The multi plicativemodeluses the followingrelation,

y{i,jl

=

x(i,j).n(i,}) (3.6)

(45)

1;

where y,z and ndenote thencisyimage,original image andnoiseprec esses . n"'I>I.'C·

tively. The est imated image ilcalculatedas:

i(i ,j)

=

f(i, j )+k(i ,j )[y(i ,j)-z(i ,i )]

wherethe gain factork(i ,i)canbe obt ainedas

inwhichiiand17"arethe meanand st andard deviationof the noiseproces s,rc- spect ively. Also sincethesignal and noise ate Itatisti callyindependent,Ji(i.j )

=

f(i,j )·ii(i,i>,with theassumptionofn(i , il

=

J,which leads1.0 f(i,i )

=

y(i,i)

Q(i, j)=a

2

q,: ~~~j)

^_^t'(i,j)

(3.9)

I'I.IU)

d) Two-pointTaylorFilter: This algorilhm [38] is developedrOtcstilllalillll image3 from their noisyobservationthat arecorrupted by multip licativenoise.h ishued on the calculationof localstatisticsestimates, Taylorseriesapproxillldiu ll!l tooptimal formulas and normaliza tionof databylocalmultiplicative samplemeans.

The degradationmodel canbeexpressedas,

y(i, n=z(i,j)'n(i,j)

wherexis the originalimage,yis the noisy image,and nisthe noise process.

Apixel estimatei(i , j) isdefinedwithtwo pointestimatorsall,

(1.11)

(3.\ ')

(46)

28

wherez(k, l)is a neighbouring pixel ofy(i, j ),anda

=

a(i,j)andb=b(i ,j )are two estimatingpower coefficients.The errorfunctionto be minimizedthus takesthe followingform,

J(a,b)=

EI (z- ,' ''11

(3.13)

Thetermy. z·canbe expanded usingTaylor series withthebivariateexpansion of

!I~Z6

=

I+alny + blnz+ a blnyln z+..·,'v'alny,bln.o: (3.14)

Asimple versionof thetwo point estimatoronlylakesthe first threeterms ofthe seriesi.e.y<1z~=1

+

aIn11

+

binz,Minimizingthe abovefunction withrespect to aandbi.e setting~

=

0 and_~

=

0thefollowing systemof equat ionsis obtained. The truncat ion oftileseriesensuresthat the systemofequation s is linear.

aE(102y)

+

bE(lnylnz)=E(zlny) -E(loy ) a8(loy 1n: )

+

6E(102_z)

=

E(xlnz)-£(10:)

which when solved'yields,

E(lo':)[E(%loy)- E(lo1/)]- E(lo y ln z )(E (.2:/ n z - EOnzl]

a

=

E(ln'y)(ln:l:)_E((lny lnz)]1 b_E(ln'y)jE(:dn z)- E(lnz)J- E(lnylnz)[E(:dny-E(lny)]

- E(ln' y)(ln'z)- E((lnylnz)]'

(3.15)

(3.16)

Speckle ReducingFilters:There is a largenumberofnonlinea r filters(16, 19, 23.2·1,281whicharc developedparticularlyforreducing speckle. Anymodel-based fillershouldbe derivedfrom an accurate model ofthe degradedimage.Meetofthe specklesupp ression metho dsusing nonlinearfilters assumespeckletobeapointwise mult iplicat ive processwhichin some casesdoes notboldtrue.Ithasbeen quitewell

'Forcomplet~d~tail.,see(38J.

(47)

establishedthat speckle in an imageis sparlallycorrelated.However,a fewofthe multiplicative filtersas described inprevioussect ions havebeen usedandfoundtu be effective.Thefollowing sectionsprovidean overviewonthree lUtt'rsbasctlon speckledimagemodel.

e)Sigm afilter: Lee[281sl1~C'>.edthisriltueforsmoothingspeckleillSAR imageswho observed thatSARsp...cklesuppression techniquesfallintomajortwo categories.First.SAR imagesmay be improvedby averaging severalframesfrom lion-overlappingspectra. But when such meansarenot available,speckle Hlllouthillg could be done after theimageh. sbeen formedby methodsbased on the statisticsof image andnoise.Lee'sSigmafilterisbased onthe sigma probabilityof theGaussinn dist ribution. Unlikeusuallinearfiltering, in thiscase.the pixelto be "roresse,l isreplaced byrhe averageintensityof onlythosepixelswhich hevcallillwnsily within twonoisestandarddeviat ions fromtheintensityofthe centre pixel.Thus,the estimatedpixelintensityx(i, j)cal.beexpressed

1 ~+i ...+;

'U,j)~

No

.'f",.~.^!(k,/)'^{y(k, l)} ^{('. 17)}

where11isthe noisedegradedimage,N.is the totalnumberornon-serepointsillthe followingsummation . ando(k,l)isdefinedas:

{

, II(1-2.(i,j))'y(k ,l )<y(i .j)<(I+2.li,j)) · ylk,l ) .

6(k,I)_ (.!.I8 )

o

otherwise

C)hostfHt er:Althoughtheybeganwith a ccnvolutlcnel-rnultiplicetlvemu<JcI or specklenoise,Frostet al.[16]concluded thatthe SAR transfer Iunction canbe assumedas const antoversomefinite bandwidthand hence the SAlt impulseresponse canbe assumedasan impulse.Iff(i,j)denotestheimpulseresponse of the filler,

(48)

30

the sig nalestimateis:

x(i,j )=y(i,j). f(i,j) (3.19)

Hased on the simp lifiedmodel ofspeckle (Equation 3.1),Frostetat. proposed a tocally adaptive speckle suppressing filter.The basicidea behindthis techniqueis similarto any otherfiltering techniquebasedonamultiplicativenoise model. The followingequationrepresentstheimpulseresponse of thefilter:

f(i, j)

=

K·aJ(i,j)'cxp(-Q(i,j)~) ^(3.20)

whereKisaconstantchosensuchthatthefilterprcvideazero flat field respon se [i.c.all filler coefficientsmustsum to zero).The parameterOJisthe decay const ant whichdependsonthelocal mean andvarianceandis given bythe followingequation,

(3.21)

whereListhe number oflooksused to formtheimage,(7=and17..arerespectivel y the localstandar d deviation of the image andthe noise, and e is thelocal meanor the image.It isquite evidentfrom equations (3.:.!U) and(3.~1 )thaLthe filterweights decrease exponent iallywithdistancefrom the centreandthatthe relativeweight is controlledby thevariance to meansquared rat io. In uniform region s(smaller variance]and brighter region(great ermean),thespeckle is moreprominentand hence the weights aremoreevenly spaced thanthey areindar ker andnon-uniform regions.

g) Kuan filter:Kuanetai. [231proposedamultiplicat ivefilter whichcanhe used for suppressingspeckle,ifit is considered a'l multiplicativenoise.Hudson and Jernigan[19Jconsideredthie modelas oneof thebest in terms of textureclassification

(49)

31

forsp eckledimages. The algorithn: is similar10Lee's multiplicativefilter 1191.The recorded signaly(i , j).5consideredas•

y(i,j)=z(i,j)+u(i,j) (:1.22)

where the noise process u(i,j) dependsal lthe signal.r(i ,j )bythefollowing equation, .(i,j)= ,(i,jlln(i,) ,-11

Assum inglocal.;tationarity of the image,the result ingestimate 01 theimageis,

i(i, j)

=

ili,j)

+

k[y(;" -, ,jll wit h

3.3,3 QuadraticVolterraFilter(QVF)

(UI )

(3.25)

Arela tivelynewtechnique fornonlinearfiltc, ing isbasedon tru ncatedVolt erraecrlcs.

Volt erra seriesisinherently.onlinear innature which has sti mulated res earche rs to applythem fornonlinear filte ringpurposes. To avoid thecom plexity ofusing higher order terms,theseriescan betruncate datsecond orde r, lead ing totheQuadrati c Volt err a Filter(QVF).Ramp ouiet al.[411-[50]madean extens ivestudyoftIlis filter and provedit s robustnessin restoring imagescorru ptedbymultiplicati venoise all discusse d in deta ilsin Secti on2.4.Based on Ramponi'swork[SOl,aquadratic Volterr a filte r ispropo sedin present workto resvrespeckled images.Beforeapp rc achlng the design strategy for a specklespecificfilte r based onthe quad rat icVolte rraserk'll, it see ms necessary tohavean idea about thegeneraltheory andprop ert iesofthe quadr a ticVol te rraseries aswill be discuss ed next.

(50)

32 3.3.3.1Th eoryandproperty

The I/Orelationshipof a quadraticVolterra$Criescan bedescribed by the[a llowin!

cxprcuion:

(3.26)

where 11 andiarcrespectively theinputand outputor the system,(n I18,)is the pixel ofinterestand hoisthe constanttermoftheVolterrafilter.

Thelinea roperatorIi.and quadrat icoperatorIi,are givenby, h-l [y(nl,n 2)]

= :L

hl(i1> i,)y(nl -illn,-it)

(',,;,E5)

h·,[y(n"

",)J = :L E

h,(it.i,. i..i,)y(n l-ill "'-i,).

(i••i. £s) (j , ,n E 5)

whereS isthe finite suppo rt

or

the filter. Forexample,tbefinitesupport ofsize N )( Nin20space is defined u,

;,~-(N- 11I2,j,$(N- 11I2,

;,~-(N -11I2, j,$(N- 1)/2, forNtak ento be odd.

Equa tio n(3.26)canhe describedmoreconvenientlyin matrix3 form: (3.28)

(3.29) whereIIIandY\arcNxNmat rices,Y,andH,areN'xN'mat rices;tr{.}indicat es thetrace ofamatrixand

{.rr

^denotes^matrixtransposition.The element(il.i~)of 3To avoid ne!:&1iveindu inc inIhema tric",HI& Hr.allindiasareoffaetby( ¥). (Nodd).

(51)

33 the input matrixYIcorrespondstothe inputsample(lit-ittn, - i, ). The matrix Y, illobtained(rom Y1using the followingfor mula,

('1.:10) where0 denotes the'Kronecker'product .ThematrixY2 conta insthe products of all possiblepairsofelements in}).

Asseenin the previousanalysis,thenumberof linear and nonlinear cocfllcicuts fora.finitesupportof Nx Narc N' and N4respectively.Thus,even for N :: 3, thenumberofcoefficients for thequadratic part ofthe filter is 81,alargenumtrcr.

Fortu nately,somevalid conditionscanbe exploited withoutlosing thegCIl~'ral i lyor thefilter,so thatthe numberof independentcoefficientscanbereducedtoalarge exte nt under thefollowingconditions:

i)uniformityin grey levelwhich impliesforahomogeneous region,the linearCI) -

efficientaand nonlinearcoefficientsshouldsumupto one and zero respectivelyi~

expressedmathematicallyby :

L:

h1(i1> i2)=0

;"i, ES

L: L:

h,(i" i" i l,i2)=0

;J,i,ESj.ohES

ii)symmetrycondition whichis obviousfromthefollowingequations:

(:1.:11)

(:1.32) which reduces the number ofcoefficients by half;and

iii)isotropy conditionwhichmeans that the orientationof thefiltermaskmus t not make any differencetotheoutput;thisparticul arcond ition givesala rgereduction inthe nc-nber of coefficients.

(52)

34

Exploitingtheseconditio nsfor N ""3the totalnumberof independentlinearand nonlinear termsdropto 6 and II,respectivel y.

hshouldbe clearthatthe quadrati cVolterrafilterhasmainly twoparts- alinear partand aquadrat icpartandthefilt er responseisthe combination ofthese twoparts exce pt for a possibleconstant offset.The linearpart 's role is smoo t hingthe noise, wherees,the quadraticpartcompensatesfor thedamage cause d by the low-p assfilter on fine detailsofthe image.Thisseems to give an improvedperformanceprovided thatthe Iilters.bothlinea rand quadraticcanbe welldesigned.However, thisis not an easy task.Thefollowing analysisis donebase d onRamponi'swork[50lwhich arra nges the quadratic coefficients into three majorclasses,referred to asType0, Type I andType2 coefficients.This conc eptwill be used laterin designin gthe qua.draticpartof thefilter usedin this resea rch.

Thequadratic co efficients canbe arranged in a3²x 3²lexicographi cally ordered matrixH2 • Thus,H2canbecon sideredasamatrix whichis composedof3x 3clements,witheach element a su b-mat rixofH2denoted byH2[i,ij ,wher ei,i representsthe row and colu mn of matrixH2 .Each sub-matrixH2[i ,iJof matrix 112is in factamatrix composedof 3x3 elementsdenotedbyH2[i ,i](k,I),where k,Ireferstothe row and columnofthe sub-matrix1l2(i,i ]. Thus,the element of If'l' Jl2[i,j ](k,/)acts on the productof th e imagepixelsy(i -

¥ ,j - ¥ )

and y(k-¥,l-l!f) .

A pcesib leset>.of the 11 independe ntcoefficients are:

A

=

{H,/I, I)II,1),11,[1.1](1,2 ),H,II,1](1,3), 1I,[I,I ](2,2) ,I/,[I,I](2,3),H, [I,1)(3,3 ), 11,[1, 211',2),1/,[1,2)(2,I),H,[1,2)(2,2),

(53)

H,[I,21(3,2),H,[2,2)(2 , 2)) (3.33)

Depending onth erelati ve positionofthe pairofpixelson whichthe coefficients act,these can be dividedinto 6SfOUPSand theindepe nd entcoefficientsthenIur thcr reduced to 6.The set ofallquadraticcoefficientsAca nnow be definedas,

where

4H,[I,I)(I,1)+4H,[1,2)(I , 2)

+

H,(~ 21(2,2) 4H,(I,11(1,2)

+

4H,[I,2)(2,2) 2H,\I ,l)(l,3)

+

H,11 ,2)(3,2) 2H,[I,1)(2,2)

+

2H,(I,2)(2,1) D~ 2H,[I,II(2,3)

H,[I,II(3 ,3)

(3.3-1)

(3.35)

Torenderthe subsequent.designsimpler, theresponse,areagain classifiedintothfl.'e majorcatego riesbasedon thedistancemeasuredusing8connectivity u ametric between the coupleof pixelson ....hich the correspondi ng quadraticcoefficientact,. Thedistancebetweentwopixeb(i,i)and(ktl)denotedby dist« i,j),(k,t))isdefined

tli.d {(i, i), (k,/)) ::::maz(. b.t(i-.I:),4b,,(j-I)) (3.36) And beeee thethreety pca are,

Type0: Ao

=

{a} havingdilJ!((i,j ),(k,I))::::0 Type1: AI::::{P, t} bavingdi"I((i,i).(.I:,I))==I Type2,A,={I, e,"1 h,,;ngdi,t«i,j ), (' ,I))

=

2 (3 .3 7)

(54)

36

3.4 Speckle-specific edge detection techniques

AsmentionedinSection 2.5,edgedetectorsaremainlyofthe Stament typei.e.the dct.«tonarebasedonthedifferencebetween pixelvaluell.Tbiltypeofedge det ector , whenapplied tospeckledima~.yieldsverypoorresults becauseof multiplicative- convolutional nature ofsp eckle noise.Veryfew speckle-sp ecificedgedetect ion meth - 00' havebeendevelopedasdiscussed in deta il inSection2.5.2. Brief analytical descripticna of someof thesemethod sareinclude d here.

3.4.1 Coefficientof Va ri at ion

Itiswell known thAtforradarimagery,the coefficient ofvariation(olp=~IE( x)) isconstanti.e.independ ent of mean power overahomogeneous area {50,611.The Probabilityof FalseAlarm(P/a)illdefinedas the proba.b ilitythata.pixel ofa.be- rnogeneous area. is detected uanedge pixel (61). UPI ai.depende nton the mean powerasiti.intheta&eofspeckled image,iti.bettertouseanedgedetectorbued onanoperator whichisindependentofthe meanpowerandsuchanoperator hasa.

propertyof Const ant Fa.lse AlarmRate(CFRA).An edgedetectorof CFARcould bedevelopedusingthecoefficienta/IJasan edgestrensthmeasure.In practice,an estimate ofcoefficient of va.riation[i.e,thestandarddeviat iontomeanratiodenoted by u/ fJ)iscomp uted over awindo wofNneighboringpixels and thecentre pixelof thewindowisrepleeed bytbisvalue.a and}Jarecomputedas,

(3.38)

and

(3.39)

(55)

37

Inpractice since the distributionof11/Jlis around

If.

the thresholdis setto a value T

= .;tii +

c,wherefis verysmallnumberandListhe number ofindependen t looks.Sothelar gerulp.is,the greateris the probability thatthe pixel of interesti!l apartof anedge.The pixel isassignedto an edgeifthe edgestrength~?:T, 3.4.2 Frost'CFAR edgedetector

Frostet al.[17]proposeda.nedge detectingtechn iquefor SARimages.The method applies maxim umLikelihood Ratio(LR)as the measureofedge strength.Maxi·

mumlikelihood ratio is computedbased on a SARimagemodeland works quite satisfactorily.Th e likelihoodratiois givenby,

A(YIY:lY3"'YN)= N •

;Nnf-.l iJoL expt.-~; IPo) .

^(3.'l0)

[li=l(fJi exp(-y;/Pl)+{1i cXP(- y;/ P1)) whereY.theith pixelvalue and

Po=M,/L, (3.41)

withalocal mean ofMilusing allpixels Y,Y2' • YN,andLindependentnumberof looks.

DI = Po- RJ p,~Po+ RI andRJisdefined as,

(3.12)