IEEE TRANSACTIONS ON IMAGE PROCESSING, 2-SEGM 1 RO

IEEETRANSACTIONSONIMAGEPROCESSING,2-SEGM1ROBUSTREAL-TIMESEGMENTATIONOFIMAGESANDVIDEOSUSINGASMOOTHING-SPLINESNAKE-BASEDALGORITHMF.Precioso,M.BarlaudT.Blu,M.UnserLaboratoireI3S-UPRES-A6070CNRSBiomedicalImagingGroup,STI/IOAUniversit´edeNice-SophiaAntipolisSwissFederalInstituteofTechnology,Lausanne2000routedesLucioles-F-06903Sophia-AntipolisFRANCECH-1015LausanneEPFL,SWITZERLANDPhone:[phone],Fax:[phone]Phone:[phone],Fax:[phone]{Frederic.Precioso,Michel.Barlaud}@i3s.unice.fr{Thierry.Blu,Michael.Unser}@epfl.chAbstract—Thispaperdealswithfastimageandvideoseg-mentationusingactivecontours.Regionbasedactivecontoursusinglevel-setsarepowerfultechniquesforvideosegmentationbuttheysufferfromlargecomputationalcost.AparametricactivecontourmethodbasedonB-Splineinterpolationhasbeenproposedin[26]tohighlyreducethecomputationalcostbutthismethodissensitivetonoise.Here,wechoosetore-laxtherigidinterpolationconstraintinordertorobustifyourmethodinthepresenceofnoise:byusingsmoothingsplines,wetradeatunableamountofinterpolationerrorforasmoothersplinecurve.Weshowbyexperimentsonnaturalsequencesthatthisnewflexibilityyieldssegmentationresultsofhigherqualityatnoadditionalcomputationalcost.Hencerealtimeprocessingformovingobjectssegmentationispreserved.I.INTRODUCTIONWeaddresstheproblemofimageandvideosegmentationusingregion-basedactivecontours.Thegoalistoextractimageregionscorrespondingtosemanticobjects.ImageandVideosegmentationcanbecastinaminimizationframeworkbychoosingacriterionwhichincludesregionandboundaryfunctionals.BoundaryfunctionalswerefirstproposedbyKassetal.[21]andgeodesicactivecontoursbyCasellesetal.[3],[4]foractivecontoursegmentation.Region-basedactivecon-tourswerefirstintroducedbyRonfardetal.[30]andCohenetal.[10].Chakrabortyetal.[5]combinedbothboundaryandregioninformationformedicalimagessegmentation.ThenChesnaudetal.[9],Chanetal.[6],Zhuetal.[35],Paragiosetal.[24]andDebreuveetal.[12]introduceregion-basedstatisticdescriptorsforimageorvideosegmentation.Jehan-Bessonetal.[17],[20]addressthesegmentationproblemwherefeaturesoftheregiontobesegmentedareembeddedinregionfunctionals.Inthisframework,Gastaudetal.[13]proposeanewapproachintroducingshapeprior.Thismethodusesavariationalapproachasopposedtopreviousworkonshapeprior,basedonprobabilisticmethods[11].Theshapepriorallowsfreeformdeformation[13]andisnotrestrictedtoaparametricdeformationasin[8].Allthesecontourorregion-basedmethodsusealevel-setapproachwhichisaccuratebuttimeconsuming.Inthispaper,weproposeaparametricactivecontourevolutionbasedonacubicsplinecontour[2].InSection2,wepresentasurveyoftheregion-basedcriterion,thederivationofthecriterionandcomputationofthevelocityvector.InSection3,weproposeacubicB-splineimplementation.CubicB-splinespreserveC2regularityandhaveexcellentapproximationproperties[32]whichmeansthat,foragivenaccuracy,fewersamplesareneededthanwithotherparametricmethods;moreover,fastalgorithmsareavailableforB-splines,whichgreatlyreducesthecomputationcost.Unfortunately,interpolationmethodsarenotrobusttonoise.Thisiswhyweproposetousesmoothingsplines[33]intheB-splineinterpolationapproachof[27].ThesecurvespreservetheimplementationadvantagesastheB-splineswhilesofteningtheinterpolationconstraint.Therelaxationoftheinterpolationconditionistradedforanoptimalincreaseofthesmoothnessofthesplinesnake.Asmoothnessparametercontrolstheamountofrelaxation.InSection4,wecomparetheinfluenceofthesmoothingsplineparameterwiththecurve-lengthregularizationcoef-ficient.Finally,weshowsomeexperimentsonrealvideosequences.II.REGION-BASEDACTIVECONTOURSA.CriterionandvelocityLetusdefineageneralsegmentationcriterion.Foreachframeofthesequence,wesearchabackgroundregionΩout,andobjectregionsΩinwithacommonboundaryΓ(Fig.1).Thusthecriterionincludesbothregionandboundaryfunc-tionals:J(Ωout,Ωin,Γ)=
Ωoutkout(Ωout)dσ+
Ωinkin(Ωin)dσRegionterms+
ΓβdsBoundaryterm(1)IEEETRANSACTIONSONIMAGEPROCESSING,2-SEGM2Fig.1.DomaindefinitionInthiscriterion,koutisthedescriptoroftheunknownbackgrounddomainΩout,kinisthedescriptoroftheunknownobjectdomainΩinandβistheweightoftheregularization.Sinceweusestatisticaldescriptors(forkinandkout)suchasmean,varianceorregionhistogram,theentropydescriptorsaregloballyattachedtotheregionΩ[9].InthevariationalEulerianmethodproposedbyJehan-Bessonetal.[20]foraregion-basedactivecontoursegmentation,theauthorsin-troduceadynamicalscheme(shapegradientmethod)inthecriterion.Henceregionsbecomecontinuouslydependentonanevolutionparameterτ.ThecriterionJ(Ωout(τ),Ωin(τ),Γ(τ))isdenotedbyJ(τ).Thusthecomputationofthederivativeprovides:J
(τ)=
Ωout(τ)∂kout∂τdσ+
Ωin(τ)∂kin∂τdσ(a)+
Γ(τ)(kout−kin)(v.N)ds(b)+
Γ(τ)(−β.κ+∇β.N)(v.N)ds(c)(2)whereκisthecurvatureofthecontour,visthevelocityofΓ(τ)andNistheunitinwardnormaltoΓ(τ).Theterms(a)arededucedfromthevariationofthedescriptorswiththeregion.Theterm(b)isdeducedfromthevariationoftheregion.Andtheclassicalterm(c)comesfromthederivationoftheBoundarytermin(1)[4].Completeproofsareavailablein[17],[19],[20].TheactivecontourΓ(τ)evolvesfromaninitialpositionΓ(0)towardsth