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