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

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%( PM 5.5NA 4.5 3.5 0.5

1 1.5 2 2.5 3 3.5REGULARIZATION COEFFICIENT

Fig.implemented4.ANMPasafunctionofλobtainedwiththelevelsetsnakePoissonwiththe3-stagestrategyontheimageFig.2aperturbedwithrealizations.noiseasGaussianSimilar(B=orGamma.

results0.55).areEachobtainedANMPwithhasotherbeennoiseestimateddistributionson20noisesuch(a)

(b)

(c)

HISTOGRAM (256 BINS) 0.01STEP FUNCTION (q=5)

STEP FUNCTION (q=5)

0.008

OBJECT

Y

0.008Y

CCNN 0.006

E 0.006EUUQ 0.004QE 0.004ERRFF 0.002

0.002

0 0

50

0 0

50

100 150 200 250

(d)

BINS’ INDEX

100 150 200 250

BINS’ INDEX

(e)

Fig.(b)5.(a)Image(128×128pixels)withoutnoiseandwithinitialcontour.stageNoisyversionofimage(a).(c)Polygonalcontourestimatedwiththe3line)ofstrategy.theobject(d),and(e)Histogramsbackground.

(solidline)andestimatedGLPDs(dotted-regularity.setandofForthethatpolygonalpurpose,snakesasuccessiveadaptedanalyzistoimagesofwiththeleveltworegionsAccordingisperformed.

tosectionII-D,thestochasticcomplexityforlevelsetsnakesleadsto LSC=λ|Γ|withλ=log(8).Fig.4establishesthisvalueλopt=log(8) 2isindeedoptimalifdifferentsegmentationsareperformedwithdifferentvaluesofλ.

Asegmentationresultobtainedwithapolygonalcontourmodelandthe3-stagestrategyonanimagequantizedwithQ=256levelsisshowninFig.5.ThenoisyimageinFig.anobject5bwasandgeneratedabackgroundwithgrayapolygonleveldistributionswith16nodesgeneratedandwithstepfunctionswithq=5steps.ThehistogramsandtheestimatedThesegmentationdistributionsresultareisshownshownininFig.Fig.5c5dandandcorrespondsinFig.5e.toanestimatedpolygonalsnakewith16nodes(i.e.equaltothetruevalue).

C.In uenceoftheGLPDsmodelization

InordertoanalyzetherelevanceofestimatingtheGLPDswithstepfunctionswhoseparametersqandajareestimatedbyobtainedminimizingwiththethelevelstochasticsetsnakecomplexity,onanoisysegmentationimagequantized

results5

HISTOGRAM (256 BINS) 0.006STEP FUNCTION (q=1)

Y

CN 0.004EUQER 0.002

FIMAGE

0 0

50

100 150 200 250

BINS’INDEX

Fig.whole6.imageSolidofline:Fig.histogram,2.

dottedline:estimatedGLPDobtainedonthewithQ=256levelsareshowninFig.2.Thenoisyimagehasbeengeneratedwithanobjectandabackgroundgrayleveldistributionsthatcorrespondtostepfunctionswithq=4.pdfFromeitherFig.with2candq=in20Fig.(initial2e,itconvergenceisclearthatofestimatingthe3-stagethestrategy)oftheestimatedorwithitscontours.histograms(i.e.leadswhentoqsigni cantis xedto uctuations256andisnotestimated).Whenthe3-stagestrategyisimplementedtheestimatedvalue)andvaluethecorrespondingofqisequaltosegmentation4(i.e.isequalresulttoisthegreatlytrueimproved,WenowseeproposeFig.2d.

toshowthatestimatingtheparametersajandmentingqofthethe3-stageGLPDapproachonthewholedevelopedimageaboveinsteadmayofnotimple-allowoneresulttoillustratesgetsatisfactorytheimprovementsegmentationofresults.theproposedInparticular,approachThisincomparisontotheonedevelopedin[17],thatconsistsinobjectperformingandbackgroundtheestimationregionsofonthethewholegraylevelimagepdfbeforeofthethesegmentation.Forthatpurpose,theparametersajandqoftheGLPDfollowingarestochasticestimatedcomplexityonthewhole[24]

imagebyminimizingthe I

(s)= q

j=1N(j)log N(j)

N +

q 1

j=1log(1+bj),

(12)

whereN(j)isthenumberofpixelsintheimagesuchthats∈[aj,aj+1[.ThisapproachisanalogoustotheonedevelopedinsectionTheGLPDII-CbutofwiththewholeauniqueimageregionshownforintheFig.GLPD2bisestimation.presentedininFig.dotted6aline.anditsTheestimationminimizationwithaofstepEq.function12leadistorepresentedq=1anddoesnotallowonetoseparatetheobjectandthebackgroundwhereasstrategy.

goodresultsareobtained(Fig.2d)parisonwithparametricstatisticalapproach

Whenthegraylevelsofthedifferentregionsoftheimageareef cientdistributedsnakewithbasedpdftechniquesthatbelongthattorelietheonexponentialtheminimizationfamily,ofthestochasticcomplexity[15],[16]canbedeveloped.Ouraimobtainedinthiswithsubsectiontheseparametricistocomparestatisticaltheapproachessegmentation[15],results[16]totheonesobtainedwiththeproposednonparametricstatis-ticalisused.

approachofthispaperwhenalevelsetimplementation