Global Segmentation and Curvature Analysis of Volu

1GlobalSegmentationandCurvatureAnalysisofVolumetricDataSetsUsingTrivariateB-splineFunctionsOctavianSoldea,GershonElber,andEhudRivlinAbstractThispaperpresentsamethodtogloballysegmentvolumetricimagesintoregionsthatcontainconvexorconcave(elliptic)iso-surfaces,planarorcylindrical(parabolic)iso-surfaces,andvolumetricregionswithsaddle-like(hyperbolic)iso-surfaces,regardlessofthevalueoftheiso-surfacelevel.Theproposedschemereliesonanovelapproachtogloballycompute,bound,andanalyzetheGaussianandmeancurvaturesofanentirevolumetricdataset,usingatrivariateB-splinevolumetricrepresentation.Thisschemederivesanewdifferentialscalarfieldforagivenvolumetricscalarfield,whichcouldeasilybeadaptedtootherdifferentialproperties.Moreover,thisschemecansetthebasisformorepreciseandaccuratesegmentationofdatasetstargetingtheidentificationofprimitiveparts.Sincetheproposedschemeemployspiecewisecontinuousfunctions,itispreciseandinsensitivetoaliasing.IndexTermsGaussianandmeancurvature,symboliccomputation,globalanalysis,segmentation.I.INTRODUCTIONTheavailabilityinrecentyearsofabroadvarietyofrangeandvolumetricimageshaspresentednewproblemsandchallengesforthescientificcommunity.Inthiscontext,segmentationisstillamajorconundrumandcentralresearchtopic[8],[9],[32],[30],[46].Forexample,in[8],theauthorsemployedsegmentationprocessesinordertoreconstructplanarsurfacesoccludedbyobjectsinrangeimages.In[46],surfacesegmentationisemployedintasksofidentificationandsearchingofobjectsinadatabaseofthree-dimensionalobjects.In[32],rangeimagesegmentationisusedintasksofclassificationofobjects,aprocessthatisalsoknownasgenericrecognition.In[9],theroleofsegmentationforreconstructionofvolumetricpartsfromrangedataisdiscussed.In[30],afiniteelementsegmentationtechniqueisemployedinthevisualizationandanalysisofaheartcapturedinasequenceofvolumetricimages.OctavianSoldeaiswiththeTechnion,IsraelInstituteofTechnologyGershonElberiswiththeTechnion,IsraelInstituteofTechnologyEhudRivliniswiththeTechnion,IsraelInstituteofTechnology2In[29],theauthorspresentedasegmentationprocessthatdetectsprimitivessuchasplanes,spheres,cylinders,cones,andtorifromrangedataimages.Inthiscontext,volumetricsegmentationcanbeemployedaswell.Apotentialapplicationofvolumetricsegmentationforuseinidentifyingcylindersisillustratedinthiswork.Mostofthecurvaturecomputationliteraturedealswith3Dmeshes,whicharetypicallyobtainedfrom3Dscanners.Incontrast,volumetricimagesarecommonlyobtainedfromdevicessuchasCTs,MRIs,andSEMs.Segmentationofvolumetricimagesrepresentsanewresearchfieldwithimmenseandunexploredpotentialforcomputervisionapplications.Ourschemerepresentsanovelapproachtovolumetricsegmentation,employingaglobalanalysisoftheinputdata,andcanbeemployedinclassicalsegmentationof3Dimagesaswell.Onewayofhandlingthesegmentationproblemisbycomputingorestimatingdifferentialgeometricalpropertiesoftheanalyzedobjects;see,forexample,[3],[9],[18],[39],[43].Amongthedifferentialpropertiesthatarewidelyused,Gaussianandmeancurvaturestakecenterstage,beingfundamentalprescriptionsofanintrinsicsurfaceproperty[26].Inthispaper,wepresentatechniquetogloballyandsimultaneouslycomputetheGaussianandmeancurvaturesfor(alliso-surfacesof)volumetricimages.Givenavolumetricdatasetf(u;v;w);weareabletocomputeK(u;v;w)andH(u;v;w);thescalarfieldsthatrepresenttheGaussianandmeancurvaturesoftheiso-surfaceat(u;v;w);aswellasotherdifferentialcharacteristics.Theintroducedcomputationalcapabilityopensthewayforamorepreciseandrobustglobalcurvature-basedsegmentationofvolumetricdatasets.Tothebestofourknowledge,allcontemporaryalgorithmscomputethecurvaturepropertiesindiscretelocationsonly.Inthepresentedapproach,weareabletoglobally,continuously,andsimultaneouslycomputecurvaturepropertyfunctionsovertheentirevolume.Thepresentedschemeoffersadditionaladvantagesinitsimprovedaccuracyindetectingboundariesofcurvature-basedsegmentedregions.Anotherofitsbenefitsisitsabilitytoperformglobalcurvatureanalysisthatisinsensitivetoaliasingaswellasignorantofaspecificiso-level.Thisisincontrasttoother3methodsthatusediscreteapproaches.Thispaperisorganizedasfollows.InSectionII,wedescribethecurrentstate-of-the-artinsegmentationandcurvaturecomputationwork.InSectionIII,weprovidesomenecessarymathematicalbackgroundandinSectionIV,wedescribethemechanismforevaluatingtheGaussianandmeancurvaturesusedinourpresentedapproach.InSectionV,segmentationofthevolumeaswellascurvature-basediso-surfaceextractionsareconsideredusingtheintroducedscheme.Afewadvantagesovertraditionalvoxel-basedschemesarealsopresented.InSectionVI,severalexamplesofouralgorithmappliedtovolumetricand3Dscannedimagesareportrayedandfinally,inSectionVII,weconclude.II.RELATEDWORKBelowweprovideanoverviewofthecurrentstate-of-the-artincurvatureestimationovertwo-manifolds.Theinterestofthescientificcommunityinsegmenting3Dimageshasresultedinaplethoraoftechniquesforsurfacesegmentation.WedescribeseveralsurfacerelatedworksinSectionII-B.SeveralvolumetricrelatedstudiesthatanalyzedvolumetricdatasetsaredescribedinSectionII-C.A.EvaluatingCurvatureBeingorientationindependentofsurfacecharac