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

2A.Generalmodel

Lets={s(x,y)|(x,y)∈[1,Nx]×[1,NimagetosegmentwithNwithy]}denotethex×Nypixelsandgraylevelsquantizedexample,Qon=Q256levels.).OneOneassumesthushasthats∈the[1,Qimage]withisQcomposed∈N(forofRregions uwithu=1,...,R(notnecessarilysimplyconnected).ThegraylevelsTheofnumber ofpixelsof uwillbenotedNu.uareassumedtobespatiallyuncorrelatedandindependentlystatisticalregion-baseddistributedwithsnakes,GLPDthePuWithcriterion.

thathastoobtainedbeoptimizedbydetermininginorderthetostochastic ndthe nalcomplexitycontourofΓthecanim-beage[14],[15],[16].Withthisapproach,onehastodeterminethethedatasumandofthefornumberthedescriptionofbitsneededofthemodelfortheofdescriptionthedata[13].ofSincetheGLPDthemodelPu,theofstochasticthedataincludescomplexitythecancontourbewrittenmodelasandthesumof3terms

= S+ P+ C

(1)

wherelevels1of SRtheisimagethenumberwithbothofbitsthecontourneededΓtoandcodethetheGLPDsgrayP,...,P xed, PisthenumberofbitsneededtocodethecontourGLPDsΓ.Inandthe following,Cisthenumberthesequantitiesofbitsneededwillbetomeasuredcodetheinnats(i.e.naturallogarithmwillbeconsidered).Wedetailinterms.

thefollowingtheparticularexpressionofthesedifferentB.Graylevelcoding

Thenumberofnats SneededtodescribethegraylevelsofswithgivenGLPDsP1,...,PRissimplyequalto

S=

R)])

(2)

u=1(x,y

log(Pu[s(x,y)∈ u

sincein uthenumberofnatsneededtocodethevalue

s(x,y)withanentropiccode[23]is log(Pu[s(x,y)]).ThechoiceoftheGLPDestimationtechniquewehavedonesimilarissegmentationbasedontwoperformancesconstraints.toThethe rstonesoneobtainedistowithgettechniquesbasedonparametricstatisticalmodelsadaptedtotheatechniquegraylevelwhich uctuations.canleadThetolowsecondcomputationalconstrainttime.istodevelopForthatpurpose,weproposetoestimatetheGLPDineachregion withanirregularstepfunctionPqu

u

(s)withqsteps(Fig.1)

Pqu

(s)

=

qpu(j)Rj(s),(3)

j=1

whereRj(s)=1ifs∈[aj,aj+1[andRj(s)=0otherwise,

withajandwhere∈[1the,Qa],ja1=1,aq+1=Q+1andaj>aiifj>areidenticalforeachdistributionPu

i

thushasPqu

(s)=p.Thisisaq.One

u(j)ifs∈[aj,aj+1[generalmodelthatcandescribeanydistributionofrandomvariablequantizedonQlevels.Inparticular,weshallshowonrealimagessegmentationthatthisforapproachwhichtheallows uctuationsonetoperformare

multiplicative.

SARimageIEEETRANSACTIONSONIMAGEPROCESSING,VOL.??.NO.??,????

Y

CNEUQERFjj+1BINS’INDEX

Fig.(GLPD).1.Illustrationsolidline:ofhistogram,theestimationdottedofthelinegray:steplevelsfunction.

probabilitydistributionIndeed,oncethenoiseispresentineachregionoftheimage,thewaytherandomgraylevelshavebeengeneratedisnomoreimportanttheIndifferentandeachregionregionsonlythedifferencebetweenthehistogramsof isrelevant.

uandforgivenvaluesoftheaj,theMaximumLikelihoodestimationp u(j)ofthepu(j)is

p u(j)=

1

b,

(4)

jNu

wherebj=aj+1 usuchas ajandwhereNu(j)isthenumberofpixelsins∈[aj,aj+1[.Onethusgets

S=

Ru=1 qN(j)logj=1

Nu(j)

uNu)nats.Furthermore,onealsoneedstothatcodecodingq 1thevaluesvaluebbj(since q

j=1bj=Q).Weconsiderjrequiresapproximatelylog(1+bj)natssincebj=1needstobecodedwithatleastonebit.Sincethevaluesbjareequalineachregion,onegets

P=

R(q 1)log

u=1