IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL.. NO., 1 Nonparam(3)

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

0}[25],[26],[27],[28],[29],[30],[12],[31].Thiscontourmodelallowstosegmenttheimageintworegions(i.e.R=2)notthenecessarilynumberofsimplynatsrequiredconnected.tocodeForsuchthecontourcontourmodels,canbeapproximatedby[16]

LSC=log(8)|Γ|,

(7)

where|Γ|isthelengthinpixelunitsofthecontour.

Foranuniqueandsimplyconnectedobjecttosegmentintheimage,itcanbeadvantageoustoconsiderpolygonalcontourmodelsofthestochastic[10],[15].Itcomplexityhasbeenshowncanlead[15]tothatef cienttheminimizationtechniquewithoutgrayleveltuning uctuationsparameterfollowintheaparametricoptimizedprobabilitycriterionwhendensitythefunction(pdf)thatbelongstotheexponentialfamilyandthatisadaptedhasbeentogeneralizedthe uctuationstomultiregionpresentinthesnakeimage.inThis[32]approachandthenumberofnatsneededtocodesuchamultiregionpolygonalcontourcanbeapproximatedby

PC=nlogN+(n+1)logp+p[2log(2e)+log(m

xwherem x(respectivelym y)isthemeanvalueofhorizontalm y)](8)

(resp.vertical)distancesbetweenadjacentnodes,nisthenumberandpitsofnumberEulerianofgraphssegments.

ofthemultiregionpolygonalsnakecontourOfcoursemodelsthissuchapproachassplinecoulddescriptorsbegeneralizedforexampletoother[14]orsakemultiregionofsimplicity,level,setthistechniquespaperfocuses[33],[34].onlevelHowever,setandforpolygonalsnakesforthesegmentationintworegions.Themoreknowngeneralbutarbitrarycaseofnumbermultiregionofregionspolygonalwillbealsosnakesconsideredwithaasanillustration.E.Optimizationstrategy

theThestochasticsegmentationcomplexityofthe image.ThisiscriterionobtaineddependsbyminimizingonthecontourΓ(i.e.,theparameterofinterest)andontheqparametersdistributionajthatareintroducedforthedescriptionoftheandqoftheprobabilitiesstepfunction)Pu.canThesebeparametersobtainedby(Γminimizing,a1,...,aq .Forthatpurpose,Γisestimatedbyminimizing with xedaj.Then,theparametersajandqaredeterminedbyminimizingiteratedifnecessary.

forthegivenvalueofΓ,andtheprocessis1)Levelsetcontourestimation:Inthissubsection,theimplementationoftheminimizationalongΓisdescribed.Thistechniquerefertopublishedisstandardworksinlevelforfurthersetimplementation,details[26],[31],thus[16].weTheequation φ(x,yevolution)

isgiven[26]bythepartialdifferentialequation3

itis

thesumof3termsF Γ.S(s(x,y))=

AccordingtoEq.1,S

Γ

andFCLS(s(x,y))=

LSC

N

insteadof(q 1)log

√

| φ|

.Usingtheresults

in[16],onecanshowFS(s(x,y))=

thattheexpressionforFS(s(x,y))is

qj=1{H(n1(j)) H(nH2(j))

+ (n1(j))(10) n2(j)[Rj(s(x,y)) n2(j)]

,whereH(z)= zlog(z)andnu(j)=

Nu(j)