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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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)
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OBJECT
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0.008Y
CCNN 0.006
E 0.006EUUQ 0.004QE 0.004ERRFF 0.002
0.002
0 0
50
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100 150 200 250
(d)
BINS’ INDEX
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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
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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