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

IEEETRANSACTIONSONIMAGEPROCESSING,VOL.??.NO.??,????1

Nonparametricstatisticalsnakebasedonthe

MinimumStochasticComplexity

PascalMARTIN,PhilippeREFREGIER,Fr´ed´ericGALLANDandFr´ed´ericGUERAULT

EDICS:SEGM,NOIS

niqueAbstractplexitythat—isWebasedproposeontheanonparametricminimizationofstatisticalthestochasticsnaketech-com-distributions(minimumdescriptionlengthprinciple).Theprobability

imagearearedescribedofthegraywithlevelsinthedifferentregionsoftheaestimated.Thesegmentationstepfunctionsisthusobtainedwithparametersbyminimizingthatthecriteriontypesuser.ef ciencyofimagesWethatillustratedoesnotincludeanyparametertobetunedbywithleveltherobustnesssetandofthistechniqueonvariousparametricofstatisticalthisapproachtechniques.

isalsopolygonalanalyzedcontourincomparisonmodels.withTheset,Indexsnakes,Termsminimum—Imagedescriptionsegmentation,lengthstochasticprinciple.

complexity,levelA

I.INTRODUCTION

Nimportantgoalofcomputationalvisionandimageobjectsprocessingfromvariousistotypesautomaticallyofimages.recoverOverthetheyears,shapemanyofapproacheshavebeendevelopedtoreachthisgoal.Inthispaper,contourswe(snakes).

focusonthesegmentationofobjectsusingactiveafunctionThe rstinsnakesorder[1]tomoveweredriventhemtowardsbythedesiredminimizationfeatures,ofusuallyedges.Theseapproachesareedgebasedinthesensethatarewelltheinformationadaptedtoausedcertainisclassstrictlyofproblems,alongtheboundary.buttheycanTheyfailinthepresenceofstrongnoisealthoughseveralimprovementsandlimitationsreformulations[2][3](andhavereferencesbeenproposedtherein).toAnotherovercomestrategytheseconsistsinconsideringnotonlytheedges,butalsotheinnerand[6],[7],theouter[8].

regionsde nedbytheactivecontour[4],[5],toInminimizetheregion-basedacriterionapproaches,thatisthethesumcontouroftwoistermsdeformed[9],[10],[11],[12]:theexternalenergy,thatisbasedonthegraylevelsenergy,ofthattheallowsimageandoneontoaregularizestatisticalthemodel,contour.andtheIthasinternalbeenshownleadstothatasatisfyingtheminimizationtradeoffofbetweenthestochasticthesetwocomplexityenergies[13]forvarioustypesofcontourmodels(spline[14],polygonal[15],levelpropertiesset[16]).intheThecontextresultingofstatisticalsnakesestimationpresentcleartheoryoptimalifthe

processingPh.R´efr´ed’Ing´group,gier FresnelandFr´eInstituted´ericGallandUMRCNRSTICarewiththe6133,PhysicsEcoleandG´eImageMarseilleenieursdeMarseille,DomaineuniversitairedeStJ´en´eralisteeric.galland@fresnel.fr.Cedex20,France.r ome,13397Sacoman,F.Gu´eE-mail:philippe.refregier@fresnel.fr,fred-Martiniswith13016theMarseilleraultiswithSimagD´eveloppement,2all´eeboth.E-mail:France.pascal.martin@fresnel.fr.

E-mail:frederic.guerault@simag.fr.P.apriorigraylevelprobabilitydistribution(GLPD)modeliswelladaptedtothedata.

TheGLPDmodelsthatbelongtotheexponentialfam-ily[10]allowonetodealwithmanyapplications(radarimages,modelsmaylowphotonfailtoprovide ux,...).aNevertheless,fairdescriptionsuchofparametrictheGLPDinsomepracticalcasesanddifferentapproacheswerede-velopedproposedtotoovercomeestimatethesetheGLPDlimitations.ontheInwhole[17],imagetheauthorswithacorrespondsGaussianmixturetoaregion.suchAlthoughthateachthiselementapproachofistheinterestingmixtureandprovidesgoodresultsondifferenttypesofimages,wewillregion.seeInthat[18],itisasupervisedpreferabletomethodestimateisproposedtheGLPDforintextureeachsegmentationtasks.Thisapproachrequirestrainingwhichisanpaper.importantIn[19],difference[20],thewithauthorsthetechniqueproposedproposedanonparametricinthisstatisticalwithParzenapproachwindowsbased[21].onAthelevelestimationsetimplementationoftheGLPDinwhichthevarianceσPoftheGaussiankernelisautomaticallyestimatedapproacheshas[19],also[20],been[22]developedthecriterion[22].toHowever,optimizeincontainstheseatuningparameterinordertobalancethecontributionoftheinternalandoftheexternalenergy.

isWebasedproposeontheinminimizationthispaperaofsegmentationacriterionwithouttechniquetuningthatparameterandthatisnotdedicatedtoaparticularprobabilitydistributionandofthebackgroundfamily.ForarethatdescribedpurposethewithGLPDstepfunctionsoftheobjectwithparametershand.Thisisandannumberimportantofdifferencestepsestimatedtothepreviousfromthecitedimagenon-inparametricstatisticalsnaketechniquesandtoourknowledge,thisacriterionisthe rstwithoutdemonstrationtuningparameterofsnakeandsegmentationthatisnotdedicatedbasedontoaparticularGLPD.Itwillbestudiedwhentheresultsareequivalentmodeladaptedtothetoonesthe uctuationsobtainedwhenpresentaparametricintheimagestatisticalisused.Furthemore,ofthetechniqueweshallproposedalsodemonstrateinthispaper.

thestrongerrobustnessinThesectiongeneralII.ExperimentalmodelofthestochasticresultsarecomplexityprovidedinissectionpresentedIIIonsyntheticandrealimages.

II.MINIMUMSTOCHASTICCOMPLEXITYAPPROACHInthissection,thestochasticcomplexitythatcorrespondstoimagethecriterionwithsnakethatmodelswillbeisminimizedde ned.

inordertosegmentthe