IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL.. NO., 1 Nonparam(4)

Abstract — We propose a nonparametric statistical snake technique that is based on the minimization of the stochastic complexity (minimum description length principle). The probability distributions of the gray levels in the different regions of the image

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0.012FREQUENCY

IEEETRANSACTIONSONIMAGEPROCESSING,VOL.??.NO.??,????

HISTOGRAM (256 BINS)STEP FUNCTION (q=20)

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BHATTACHARYYA DISTANCE

Fig.3.ANMPasafunctionoftheBhattacharyyadistanceBforsegmentationresultsobtainedwiththelevelsetsnakeandthetwostrategiesontheimageFig.2apertubedwithGaussian,GammaandPoissonnoises.EachANMPhavebeenestimatedon20noiserealizations.

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thatthesimpleproposedapproachesforestimatingthegraylevelprobabilitydistributionandfortheoptimizationstrategyprovidegoodresultsandleadtoasimpleandfastsegmentationtechnique.

A.In uenceoftheoptimizationstrategy

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(d)FINAL CONVERGENCE(e)

Fig.2.Illustrationofthe3-stagestrategy.(a)Syntheticimagewithoutnoise(115×83pixels).(b)NoisyversionquantizedwithQ=256levelsandwithinitialcontour.Onecanseethehistogramsandthestepfunctionsofparameterq=20associatedtotheobjectandthebackgroundoftheimage(b).(c)Initialconvergenceperformedconsideringstepfunctionsofparameterq=20.Onecanseethehistogramsassociatedtotheimage(c)andthestepfunctionswithnumberofstepsestimated(q=4)aftertheinitialconvergence.(d)Finalconvergenceperformedfromthecontourobtainedattheendoftheinitialconvergenceandwiththe

estimatedGLPDs.(e)Finalcontourobtainedconsideringq=Q=256.

Inordertocomparethetwooptimizationstrategiesin-troducedinII-E,theaveragenumberofmisclassi edpix-els(ANMP)aftersegmentationiscomputed.ThesyntheticimagesconsideredarenoisyversionsoftheimageFig.2aperturbedwithGaussian,GammaandPoissonnoisesfordifferentvaluesofthecontrastbetweenthetworegionsintheimage.ThiscontrastcanbemeasuredwiththeKullback-Leiblerdivergence[23]ortheBhattacharyya[23]distancebetweenthedistributionsofthebackgroundandtargetgraylevels.However,ithasbeenshown[36]that,forsmalltargets,theBhattacharyyadistanceisabettermeasureofcontrast.Inparticular,differentnoisycon gurationswiththesamevalueofBleadtosimilarvaluesoftheANMPfordifferenttypesofnoisesforpolygonalsnakes[36]andforlevelsetsnakeswithparametricpdfmodels[16].Inthecontinuouscase,theBhattacharyyadistancebetweenpdfPtandPbreads

B= logPt(z)Pb(z).

TheNMPafterasegmentationisdeterminedfromthe nalcontourbycountingthenumberofpixelsthatbelongtothetruetargetshapebutlieoutsidethecontourΓ,andthosethatbelongtothetruebackgroundbutlieinsidethecontourΓ.Inthefollowing,thevaluesoftheNMPwillbenormalizedbythenumberofpixelsinthetrueshapeofthetarget.

Forthelevelsetsnakeimplementation,onecanseeinFig.3thatthetwooptimizationstrategiesleadtoequivalentsegmen-tationresults.Thisresulthasbeencon rmedwithdifferentexperimentsandwithpolygonalsnakeimplementations.The3-stagestrategyismuchfaster(afewtenssecondsinsteadofmanyminutes).Forexample,theimageFig.2bhasbeensegmentedin12minuteswiththefull-iterativestrategyandonlyin41secondswiththe3-stagestrategy.So,thisfasterstrategywillonlybeconsideredinthefollowing.B.Evaluationofthecontourregularity

Weshowinthissubsectionthattheminimizationofthestochasticcomplexityleadstocontourwiththeappropriate

functionswithq=20suchthateachstepisofequallength.Then,noconvergenceofthecontourisperformedbetweenfusionofstepsoftheprobabilitydistribution.However,whentheajandqhavebeenestimated,a nalconvergenceofthecontourisperformedfromthecontourobtainedbeforethestepfusions.

III.EXPERIMENTALRESULTS

Thissectionprovidesanevaluationofourmethod.Syntheticimagesare rstconsideredsincetheyallowonetogetaprecisedeterminationofthenumberofmisclassi edpixels(NMP).Realimagesarealsoconsideredtoshedlightintheperformanceinapracticalcase.Theseresultsdemonstrate

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