Stochastic differential equations and geometric fl

1StochasticDifferentialEquationsandGeometricFlowsGozdeUnal,HamidKrim,andAnthonyYezzixElectricalandComputerEngineeringDept,NorthCarolinaStateUniversity,Raleigh,NC27695.E-mail:fgbozkur,[e-mail]OSRgrantF49620-98-1-0190andbyONR-MURIgrantJHU-72298-S2andbyNCSUSchoolofEngineering.xSchoolofElectricalEngineering,GeorgiaInstituteofTechnology,Atlanta,GA30332.E-mail:[e-mail]curveevolution,appliedtoasinglecontourortothelevelsetsofanimageviapartialdif-ferentialequations,hasemergedasanimportanttoolinimageprocessingandcomputervision.Curveevolutiontechniqueshavebeenutilizedinproblemssuchasimagesmoothing,segmentation,andshapeanalysis.Wegivealocalstochasticinterpretationofthebasiccurvesmoothingequation,thesocalledgeometricheatequation,andshowthatthisevolutionamountstoatangentialdiffusionmovementoftheparticlesalongthecontour.Moreover,assumingthataprioriinformationabouttheshapesofobjectsinanimageisknown,wepresentmodificationsofthegeometricheatequationdesignedtopreservecertainfeaturesintheseshapeswhileremovingnoise.Wealsoshowhowthesenewflowsmaybeappliedtosmoothnoisycurveswithoutdestroyingtheirlargerscalefeatures,incontrasttotheoriginalgeometricheatflowwhichtendstocircularizeanyclosedcurve.KeywordsGeometricimageandshapeflows,stochasticdifferentialequations,nonlinearfiltering,shapeanalysis2I.INTRODUCTIONInrecentyearscurveevolutionhasemergedasanimportantapplicationofpartialdifferentialequations(PDE’s)inimageprocessing,computervision,andcomputergraphics.Curveevolutiontechniqueshavebeenappliednotonlytoindividualcurves,forapplicationssuchasedge-detection,skeletonization,andshapeanalysis,buthavealsobeenconsideredfortheirsimultaneousactiononthelevelsetsofanimageinanumberofgeometricallybasedanisotropicsmoothingalgorithms.OsherandSethian[1,2]extendedthislatterperspectivetothetreatmentofindividualcurvesthroughasetofalgorithms,knownaslevelsetmethods,whichenabletheimplementationofcurveandsurfaceevolutiononafixedgrid.Thesetechniqueshaveaidedanumberofresearchersinpushingtheapplicationofcurveevolutiontonewlimitsbyprovidingasimpleframeworkfortreatingcertaintypesofsingularitiessuchasshocksandtopologicaltransitions[1,3].Muchoftheresearchincurveevolutiontheoryhascenteredaroundthesocalledgeometricheatequation[4]inwhichacurveisevolvedalongthenormaldirectioninproportiontoitssignedcurvature.Thisflowiswellknownforitssmoothingproperties[5–7]andthefactthatitcorrespondstothegradientevolutionforarclength(therebyearningthenamecurveshorteningflow).Becausecurvatureisapurelygeometricquantity(invarianttorotationandtranslation),curvature-basedmotiongivesrisetoaEuclideaninvariantscalespace[8–10],allowingonetotracefeaturesinacurvefromfinertocoarserscalesastheevolutionproceeds.Anaffineinvariantscalespacecanbeobtainedfromarelatedcurvatureflowwhichdependsuponthecuberootofthecurvature(see[8,11,12]).Whenappliedtothelevelsetsofanimage,theseflowshaveapowerfuldenoisingeffectwhenrunforashortamountoftime.Ifrunfortoolong,however,evenlargescalefeatureswillbedestroyed.Thereasonstemsfromthefactthatasthegeometricheatflowshrinksanyclosedcurve,thecurvebecomesmoreandmorecircular(ellipticalinthecaseoftheaffineflow)andwilleventuallycollapseintoasinglepoint[4].Itisthereforenotalwayspossibletopreservedesiredfeaturesintheshapesofobjects(cornersforexample)iftoomuchevolutionisrequiredtoremoveasignificantlevelofnoise.Furthermore,itisnotwellunderstoodhowthesecurvature-basedfiltersareaffectedbydifferentnoisedistributionsandwhenthissortofproblemmayoccur.Tothebestofourknowledge,andasidefrom[13,14],nonlineardiffusioninthepreviousliteraturewasdis-cussedfromapurelydeterministicperspective.Inthispaperweprovideastochasticformulationofthegeometricheatequationandusetheresultinginsightstodevelopanewclassofcurvature-basedflowsandanisotropicdiffu-sionfilterswhichpreservedesiredfeaturesintheshapeofanobject.Underthesenewflows,evolvingcurvestake3thelimitingformofapolygon(see[15]forevolutionsofpolygonsrelatedtothegeometricandaffinegeometricheatflows,and[16]forevolutionsofpolygonsgloballythroughanelectricfieldconcept).Theresultingdiffusionmodelsmaythereforebeappliedformuchlongerperiodsoftimewithoutdistortingtheshapesofpolygonalobjectsintheimage,therebymitigatingthetradeoffbetweennoiseremovalandshapedistortion.Polygonalstructuresareubiquitousinimagesofman-madeobjects(buildings,roads,vehicles,etc...),whichcontainmanystraightlines,oftenorientedinparticulardirections(e.g.horizontalandvertical),thatcometogethertoformsharpcorners.Theabilitytopreservesuchdistinctivefeaturesisnotonlydesirablewhenfilteringanimagewhichcontainsthesetypesofshapes,butisalsoimportantwhenapplyinglowlevelsmoothingtoanextractedshapesincesuchfeaturesconstituteimportantandpowerfulcuesforrecognizingobjectsinhigherlevelvisionalgorithms.Wewillpresentbothapplicationsinthispaper.Fromadualperspectivetoourcontour-basedapproachtoshaperepresentation,skeletonizationapproachesmayalsoallowshapeanalysiswithoutdisplacementofcorners[7,17–21].Inthispaper,wedevelopanewclassofcurveevolutions,whichareobtainedbyamodificationo