Deformable Fourier models for surface finding in 3

DeformableFouriermodelsforsurfacefindingin3DimagesLawrenceH.StaibtandJamesS.Duncant,*DepartmentsoftDiagnosticRadiologyand*ElectricalEngineeringYaleUniversity,NewHaven,CT06510,U.S.A.ABSTRACTThispaperdescribesanewglobalshapeparametrizationforsmoothlydeformablethree-dimensionalobjects,suchasthosefoundinbiomedicalimages,whosediversityandirregularitymakethemdifficulttorepresentintermsoffixedfeaturesorparts.Thisrepresentationisusedforgeometricsurfacematchingtothree-dimensionalimagedata.Theparametrizationdecomposesthesurfaceintosinusoidalbasisfunctions.Fourtypesofsurfacesaremodeled:tori,opensurfaces,closedsurfacesandtubes.Thisparametrizationallowsawidevarietyofsmoothsurfacestobedescribedwithasmallnumberofparameters.Surfacefindingisformulatedasanoptimizationproblem.Resultsofthemethodappliedtosyntheticandmedicalthree-dimensionalimagesarepresented.1.INTRODUCTIONThisworkisaimedatextractingthesurfacesofstructuresfoundinthree-dimensionalimages.Three-dimensionalimagesareavailablefromthemedicalimagingmodalities:magneticresonanceimaging(MRI),computedtomography(CT),singlephotonemissioncomputedtomography(SPECT)andpositronemissiontomography(PET).Confocalmicroscopyisalsoagrowingsourceofthree-dimensionalimages.Segmentationinthree-dimensionalimagesisacrucialstepforvisualizationeithertostripawayobscuringstructuresortorenderthesurfaceofthestructuredelineated.Theproblemsofmanualsegmentationintwo-dimensionalimagesareexacerbatedinthree-dimensionalimageswhenitisdoneslicebyslice.Inaddition,surfacesformedslicebysliceeithermanuallyorautomaticallyarelikelytocontaininconsistenciesthatwilltendtocorrupttherendering.Smoothlydeformableobjectsdonotnecessarilyhaveanobviousdecompositionthatcanbeexploited.Auniformshaperepresentationthatdescribestheentireshapeisthereforeneededanditshoulddescribearelativelybroadclassofshapes.Representationsforobjectsareneededinordertocharacterizeandunderstandshape.Agreatdealofresearchhasbeendoneintheareaofshaperepresentation.Forarepresentationtobeusefulformodelingitshouldbeconcise.Methodsbasedonexplicitlylistingpointsorpatchesonthesurfaceareverbosebecauseoftheimplicitredundancy.Parametricrepresentationscapturetheoverallshapeinasmallnumberofparameters.Thismeansthattheoptimizationofamatchmeasurebetweendataandamodelcanoccurinalowerdimensionalspace.Boundaryfindingusingonlylocalinformationhasoftenbeenfrustratedbypoor-contrastboundaryre-gionsduetooccludingandoccludedobjects,adverseviewingconditionsandnoise.Amodel-freeinterpretationisdoomedbytheunderconstrainednatureoftheproblem.Imperfectimagedatacanbeaugmentedwiththeextrinsicinformationthatageometricshapemodelprovides.Inordertoexploitmodel-basedinformationtothefullestextent,itshouldbeincorporatedexplicitly,specifically,andearlyintheanalysis.Inaddition,theboundarycanbeprofitablyconsideredasawholebecauseittendstoresultinamoreconsistentsolutionoverall.90ISPIEVol.1808VisualizationinBiomedicalComputing19920-8194-1008-X/92/$4.00Apowerfulpropertyfordistinguishinganobjectfromitssurroundingsinanimageisoverallorglobalshape.Inordertotakefulladvantageofshape,theproblemofobjectidentificationisapproachedasaprocessofboundaryfindingordelineationusingaboundarymeasureandincorporatingglobalshapeinformation.2.RELATEDWORKINBOUNDARYFINDINGLocaledgedetectorsappliedtorealimagesproducespuriousedgesandgaps.Theseproblemscanonlybeovercomebytheincorporationofinformationfromhigherscaleorganizationoftheimageandmodelsoftheobjectssought.Contextualinformationhasbeenusedforboundarydeterminationviagrouping[1],relaxationlabeling[2]andscale-spacemethods[3].Thesemethods,bythemselves,willnotnecessarilyfindcompleteboundaries.Pixelsearchmethodsassociateedgeelementsbyfindinganoptimalpaththroughatwo-dimensionalimage,basedoncriteriadesignedtofindboundaries.Thetypicalobjectivefunctioncombinesboundarystrengthandlowoverallcurvature[4].Pixelsearchdoesnotgeneralizeobviouslytothreedimensionsbecausethereisnonaturalorderingofvoxelsinasurface.Otherinvestigatorshaveconsideredwhole-surfacemethodsthatadjustatentativesurfacemeshinordertomatchtotheimage.Byconsideringthesurfaéeasawhole,astructureisimposedontheproblemthatbridgesgapsandresultsinoverallconsistency.Terzopoulosetal.[5]usedenergy-minimizingmeshesthatareattractedtoimagefeaturessuchaslinesandedgeswhileinternalsplineforcesimposeasmoothnessconstraint.Thegoalwastofindsurfacesimpliedbysilhouettesintwo-dimensionalimages.Thisideahasalsobeenusedforfindingsymmetrysurfacesfromscalespacestacksoftwo-dimensionalimages[6],surfacesinrangeimages[7,8]andsurfacesinthree-dimensionalimages[9,10].Otherwhole-boundarymethodsoptimizeinaparameterspace.Parametricrepresentationsareusefulformodelingbecausetheycapturetheoverallshapeconcisely.Thismeansthattheoptimizationofamatchmeasurebetweendataandamodelcanoccurinalowerdimensionalspace.Widrow[11]usedparametrizedtemplatescalledrubbermaskstomodelobjects.Theparametersaresizesandrelationshipsbetweensubparts.Yuilleetal.[12]usedasimilarmethodforfindingfeaturesinimagesoffaces.Bothofthesemethodsdescribetheoverallshapeofthestructureusingveryfewparameters.How