Level

LevelsetmethodsforwatershedimagesegmentationErlendHodneland1,Xue-ChengTai2∗,JoachimWeickert3,shtliev1,ArvidLundervold1,andHans-HermannGerdes11DepartmentofBiomedicine,UniversityofBergen,JonasLiesvei91,5009Bergen,Norway2DepartmentofMathematics,UniversityofBergen,JohannesBrunsgate12,5007,Bergen,Norway3FacultyofMathematicsandComputerScience,Bldg.E11,SaarlandUniversity,66041Saarbr¨ucken,Germany*Correspondingauthor:tai@paperewpreviousstudiesaddressthetaskofregularizingtheobtainedwatershresentformulation,theto-pographicaldistancefunctionisappliedinalevelsetformulationtoper-formthesegmentation,andtheregulnthewell-knownFour-Colortheorem,ismodel,itispossibletosegmentany2Dimageorithmhasbeentestedonreal2Dfluorescencemicroscopyimagesdisplayingratcancercells,andthealgorithmhasalsobeencofixedsetofmarkersandafixedsetofchallengingimages,thecomparisonofthesetwomethodsshowsthatthepresentlevelsetformulationperformsbetterthanastandardwatershedsegmentation.1IntroductionSegmentationisamajorchallengeinimageanalysis,refelapproacheshavebeenproposed,andtheyaremainlydividedintotwocategories:energy-drivensegmentation[1–7]andwatershed-based[8–11].Energy-drivensegmentationnor-mallyconsistsoftwoparts,atermassuresasolutionwhichissufficientlyclosetothedesiredhedsegmentation[8–11]isaregionally,thewatershedtechniqueshavebeenconductedwithoutasmooth-ingterm,butrecentprogresshasresultedinenergy-basedwatershedsegmen-

ollrgy-drivensegmentationmethodsaremainlydividedintotwoclasses,contour-based(snakes)tourbasedmethodsrelyonstrongedgesorridgesasastoppingterminackeapproachhasbeenstudiedin[1,7].Cremers[6]includedstatisticalshapeknowledgetotheMumford-ShahfunctionalandXu[12]introducedthegradientvectorflow(GVF)incorporatingaglobalandexhemostwell-knownregion-basedmethodistheMumfordandShahmodel[13].InChan-Vese[5,14],theOsher-Sethianlevelsetidea[15]wascomly,somevariantsoftheOsher-SethianlevelsetideawasproposedbyTaietal.[4,16].Inthiswork,ershedsegmentationhasproventobeapowerfulandcipal,water-shedsegmentationdependsonridgestoperformapropersegmentation,aprop-ertywhichisoftenfulfilledinconion-basedsegmentationitispossibletoconverhedisnormallyimplementedbyregiongrowingbasedonasetofmarkerstoavoidsevereover-segmentation[10,11,17].Differentwatershedmeth-odsuseslightlydifferentdistancemeasures,buttheyallsharethepropertythatthewaters[9]usethetopographicaldistancefunctionforsegmentingimagesusingwatershedsegmentation,whileNajmanandSchmitt[8]presentthewater-sheddiffetal.[17]usetheshortestpathcostbetweentwonodeswhichisdefinedasthesmallestlexiographiccostofallpathsbetweentwopoints,whichreflectstheflcessofawatershedseunately,thestandardwatershedframeworkhasaverylimitedflample,r,recentprogressallowsaregularizationofthewatershedlines[18]withanenergy-basedwatershedal-gorithm(watersnakes).Incontrasttothestandardwatershedandthewater-snakes,ourworkisbasedonpartialdiffer,themethodisflample,itcouldallowrmore,itwouldbedesireabletodevelopmethodswhichcouldoptimizethenumberofopertyisimportantsincecreatingmarkersautomaticallyoftenresultsinse-

vereover-segmentationduetosuperflhodwouldpermitanoptimizationonthenumberofmarkers.2Marker-controlledwatershedsegmentationbylevelset2.1CreatingmarkersThemarker-controlledwatershedsegmentationhasbeenshowntobearobustandflexiblemethodforsegmentationofkerimageusedforwatershedseg-mentationisabinaryimageconsistingofeithersinglemarkerpointsorlargermarkeritialmarkerhasaone-to-onerelationshiptoaspecificwatershedregion,thusthenumberofmarkerswillequalthefiegmentation,theboundariesofthewatershedregionsarearrangedonthedesiredridges,kerscanbemanuallyorautomaticallyselected,buthighthroughputexperimentsofteresentwork,anadaptivethresholding[19]andfiarcombinationoftheseoperatorswasusedin[20].First,anadaptivethresholdingrasttoglobalthresholding,anadaptivethresholdingyimagefbwasthusconstructed,,allsmallobjectsinfbwereremovedsincetheywereconsideredtobeinsignifiletocloseminorgapsinthebinarystructuresoutliningtheapproximateboundaries,hiterativeclosingstep,alargerstructuralelyaftereachclosingstepamor-phologicalfillingwasperformedtofifilledregionsthathadnointersectionwithearlierfisingwasperformediterativelyinordertoobtainmarkersthatwereascloseaspossibletothedesiredboundaries,anditwasrepeatedapredefi1demonstratestheprocessofcreatingmarkersfromadaptivethresholdingandfilling,wheretheimagein(a)wasusedtocreateabinaryimage(b)llestobjectswereremovedanditerativeclosingandfillingwereappliedto(b)toobtainthefinalbinarymarkerimage(c)wherethemarkerregionsarelabeledwhite(value1)andthebackgroundisblack(value0).

(a)(b)(c)Fig.1:vethresholdingwasappliedtotheimagein(a)toconstructabinaryimage(b).Thereafter,morphologicalclosingandfillingwasappliedtoachievethefinalmarkerimage(c)whichwasusedinthewatershedsegmentation.2.2TopographicaldistancefunctionFollowing[9],weapplythetopographicaldistancefunctiontoobtaiographicaldistancefunctionbetweentwopointsxandyisaccordingto[18]definedas:Defioothfunctionf(x):Rn→Rthetopographicaldistancebetweentwopointsxandyisdefinedasthegeodesicdistanceweightedbythegradient|∇f|,i.e.