Topological correction of subcortical segmentation

Abstract. We propose a method for automatically correcting the spherical topology of any segmentation under any digital connectivity. A multiple region growing process, concurrently acting on the foreground and the background, pides the segmentation into

TopologicalCorrectionofSubcortical

Segmentation

FlorentS´egonne1,2,EricGrimson1,andBruceFischl1,2

2MITAILabAthinoulaA.MartinosCenter-MGH/NMRCenter1

Abstract.Weproposeamethodforautomaticallycorrectingthespher-icaltopologyofanysegmentationunderanydigitalconnectivity.Amul-tipleregiongrowingprocess,concurrentlyactingontheforegroundandthebackground,pidesthesegmentationintoconnectedcomponentsandsuccessiveminimumcostdecisionsguaranteeconvergencetocor-rectsphericaltopology.Incontrasttoexistingproceduresthatsupposespeci cinitialsegmentation(fullconnectivity,nocavities...)andarede-signedforaparticulartask(corticalrepresentation),noassumptionismadeontheinitialimage.Ourmethodappliedtosubcorticalsegmenta-tionsallowsustocorrectthetopologyoffourteennon-corticalstructuresinlessthanaminute.

1Introduction

Excludingpathologicalcases,mostmacroscopicbrainstructuresarefullycon-nectedanddonotpossessanytopologicalartifactsuchashandlesorcavities:theyhavethesimpletopologyofasphere.Manyrecentsegmentationalgorithmsareabletoidentifyandpreciselylocatethesestructures,althoughwithoutcon-strainingthetopology.Beingabletoachieveaccurateandtopologicallycorrectrepresentationsofdi erentbrainstructuresiscertainlyanimportantgoalinmedicalimaging(shapeanalysis,visualization...)

Onlyafewautomatictechniqueshavebeenproposedtoproducetopologi-callycorrectsegmentations.Severalapproacheshavetriedtodirectlyincorporatetopologicalconstraintsintothesegmentationprocess[1,2,3,4,5].Aninitialre-gion,carryingthecorrecttopology,

isusuallydeformedbyaddition/deletionofpointsby

minimizingaglobalenergyfunctionwhilepreservingthecorrecttopol-ogy.Theproblemwiththesemethodsisthatlocaltopologicalconstraintscanleadtostronggeometricalerrors,andthatthe nalsegmentationcanstronglydependontheorderinwhichpointsareadded.

Recently,newapproacheshavebeendevelopedtoretrospectivelycorrectthetopologyofasegmentedimage.Thesemethodscanbepidedintwomainclasses:volume-basedmethodsthatworkdirectlyonthevolumelatticeandcor-rectthetopologybyaddition/deletionofvoxels,andsurface-basedmethodsthataimatmodifyingthetessellationbylocatingandcuttinghandles/ llingholes.Mostvolume-basedapproacheshavebeenspeci callydesignedtocorrectthetopologyofthecorticalsurface.ShattuckandLeahyexaminetheconnectivityR.E.EllisandT.M.Peters(Eds.):MICCAI2003,LNCS2879,pp.695–702,2003.cSpringer-VerlagBerlinHeidelberg2003

Abstract. We propose a method for automatically correcting the spherical topology of any segmentation under any digital connectivity. A multiple region growing process, concurrently acting on the foreground and the background, pides the segmentation into

696F.S´egonne,E.Grimson,andB.Fischlofthesegmentationtodetecttopologicaldefectsandminimallycorrectthem

[6].Inspiredbytheirwork,Hanetal.developedanalgorithmtoremoveallhandlesfromabinaryobjectunderanydigitalconnectivity[7].Successivemor-phologicalopeningscorrectthesegmentationatthesmallestscales.KriegeskorteandGoebleusearegiongrowingmethodprioritizedbythedistance-to-surfaceofthevoxelsinordertoforcethecutstobelocatedatthethinnestpartofeachtopologicaldefect[8].Thesameprocessisappliedtotheinverseobject,o eringanalternativesolutiontoeachcut.Anempiricalcostisthenassignedtoeachsolutionandthe naldecisionistheoneminimizingtheglobalcostfunction.Othertypesofapproachesoperatedirectlyonthetriangulatedsurfacemesh.Fischletal.proposedanautomaticproceduretolocateandcorrecttopologicaldefectsbyhomeomorphicallymappingtheinitialtriangulationontoasphere[9].Topologicaldefectsarelocatedasnon-homeomorphicregions,constitutedofoverlappingtriangles.Agreedyalgorithmisthenusedtoretessellateincorrectpatches,constrainingthetopologyonthespherewhilepreservinggeometricalaccuracybymaximumlikelihoodoptimization.Anotherapproachisproposedin[10].Handlesinthetessellationarelocalizedbysimulatingwavefrontpropaga-tiononthetessellation:theyaredetectedwherewavefrontsmeet.Onepotentialdrawbackofthismethodisthatitdependsonthevertexusedtoidentifythedefect.

Whilethesemethodscanbee ective,theycannotbeusedtocorrectthetopol-ogyofarbitrarysegmentations,astheymakeassumptionsaboutthetopologyoftheinitialbinaryimage.Mostfrequently,fully-connectedvolumesareassumedandcavitiesaresupposedtoberemovedasapreprocessingstep.Inthesub-corticalsegmentationtopologyproblem,themodi cationofasmallnumberofvoxelsperstructureisusuallysu cienttocorrectthem.However,duetothepresenceofimagingartifact,anatomicalvariability,varyingcontrastpropertiesandpoorregistration,noassumptionscanbemadeabouttheinitialtopologyofthesegmentation.

Inthispaper,wedevelopacompletelyautomatedvolume-basedmethodtocorrectthetopologyofanysegmentationunderanydigitalconnectivity.Thenoveltyofourapproachcomesfromthefactthatanyinitialsegmentation(dis-connectedregions,handles,cavities,holes...)willstillbecorrected.Ateachstepofouriterativetopologicalcorrection,minimumcostdecisionsaretakenandconvergenceisguaranteed.

2BackgroundandDe nitions

Inthissection,somebasicnotionsofdigitaltopologyarepresented.WerefertotheworkofG.Bertrandformoredetails[11].TheinitialsegmentationisabinarydigitalimageI Z3composedofaforegroundobjectFandaninversebackgroundobjectB=.Followingtheconventionalde nitionofadjacency,3typesofconnectivitymightbeconsidered:6-,18-and26-connectivity.Forin-stance,twovoxelsare6-adjacentiftheyshareaface,18-adjacentiftheyshareatleastanedgeand26-adjacentiftheyshareatleastacorner.Inordertoavoidtopologicalparadoxes,di erentconnectivities,nand,mustbeusedforFand

B.Thisleavesuswithfourpairsofcompatibleconnectivities:(6,26),(6,18),

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