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

4

0.012FREQUENCY

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

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

OBJECT

0.004

ANMP(%)

50

100

150

200

250

BINS’ INDEXSTEP FUNCTION (q=20)BACKGROUND

0.008

(a) 0 0

BHATTACHARYYA DISTANCE

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

250

50 100 150 200

BINS’ INDEX

(b)NOISY IMAGE WITH INITIAL CONTOUR

ENCY

0.016 0.012 0.008 0.004

0 0

50

HISTOGRAM (256 BINS)STEP FUNCTION (q=4)

OBJECT

thatthesimpleproposedapproachesforestimatingthegraylevelprobabilitydistributionandfortheoptimizationstrategyprovidegoodresultsandleadtoasimpleandfastsegmentationtechnique.

A.In uenceoftheoptimizationstrategy

100 150 200BINS’ INDEX

HISTOGRAM (256 BINS) 250

0.008

FREQUENCBACKGROUND

(c)INITIAL CONVERGENCE

0.006 0.004 0.002

0 0

50

100 150 200BINS’ INDEX

250

(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