Affine shape analysis and image analysis 1 Summary

Affineshapeanalysisandimageanalysis1SummaryAf?neshapeanalysisandimageanalysisVicPatrangenaru,KantiV.MardiaDepartmentofMathematicsandStatistics,GeorgiaStateUniversity,Atlanta,USADepartmentofStatistics,UniversityofLeeds1SummaryAstudyofaf?neshapeisneededincertainproblemsthatariseinbioinformaticsandpatternrecognition.Inparticularaf?neshapeanalysisisusefulintheanalysisof2Delectrophoresisimagesandinthereconstructionofalargerareafromanumberaerialimages.TwoaerialimagestakenfromdifferentdistancesofagroundscenearegiveninFigure1.;theproblemaskstoreconstructalargercontiguousimagethatcontainstheinformationinboththeseimages.Figure1.Aerialphotographsoftwosectionsofagroundscene.Anotherclassicalexampleconsistsinmatchingtwolabelledelectrophoresisgels(seeFigure2,basedondatafrom).(1)DEFINITION2.1.a.Theaf?neshapeinstatisticalshapeanalysisofistheorbitofviatheaction(1).Anotherde?nitionofshapeisgivenbyA.Heyden(1995)andG.Sparr(1996):DEFINITION2.1.b.Theaf?neshapeincomputervisionofisthelinearsubspaceofgivenby(2)Wecanprovethefollowing:PROPOSITION2.1.Thereisanaturalonetoonecorrespondencebetweenaf?neshapesinstatisticalshapeanalysisandaf?neshapesincomputervision.DEFINITION2.2.Theaf?neshapespace,orspaceofaf?ne-adsinisthequotient.TheGrassmannmanifoldof-dimensionalvectorsubspacesofwillbedenotedbyNamely,rankwhereisamatrix.AsaconsequenceofProposition2.1onemayshowthatTHEOREM2.1.Theaf?neshapespacehasastrati?cation,wherethe-stratumisdiffeomorphictotheGrassmannmanifoldof-dimensionalvectorsubspacesofInparticular,thestratumofaf?neshapesofadsingeneralpositionisdiffeomorphictotheGrassmannmanifold3Extrinsicmeansofaf?neshapesandreconstructionoflargerplanarscenesAf?neshapedistributionshavebeenconsideredbyGoodallandMardia(1993),Leung,BurlandPerona(1998),BerthilssonandHeyden(1999)etal.InviewofTheorem2.1,extrinsicmeansofdistributionsofaf?neshapescanbedeterminedusingthegeneralapproachinBhattacharyaandPatrangenaru(2021),forcertainconvenientequivariantembeddingsofinanEuclideanspace.Suchanembeddingcanbede?nedasfollows:letbethesetofsymmetricmatricesendowedwiththecanonicalEuclideansquarenormanaturalembeddingofintoisobtainedbyidentifyingeach-dimensionalvectorsubspacewiththematrixoforthogonalprojectioninto.Dimitric(1996)provedthatthisembeddingisequivariant,hasparallelsecondfundamentalformandembedstheGrassmannianminimallyintoahypersphere.ThisisanextensionoftheVeronese-Whitneyembeddingofprojectivespaces(seeBhattacharyaandPatrangenaru(2021)),thatiscommonlyusedforaxialdata(seeMardiaandJupp,1999),orformultivariateaxialdata(MardiaandPatrangenaru,2021).Assumeaprobabilitydistributionofaf?neshapesofconFigurationsingeneralpositionisnonfocalw.r.t.thisembeddingInthiscase,themeanofthecorrespondingdistributionofsymmetricmatricesofrankhastheeigenvaluessuchthatTheextrinsicmeanofisthevectorsubspacespannedbyuniteigenvectorscorrespondingtothe?rsteigenvaluesofAssumeisasampleofsizeof-vectorsubspacesinandthesubspaceisspannedbytheorthonormalunitvectorsandsetTheextrinsicsamplemeanisofthissample,whenitexists,isthe-vectorsubspaceception.Theextrinsicsamplemeanisusefulinaveragingimagesofremoteplanarscenes,byadaptingthestandardmethodofimageaveragingofDrydenandMardia(1998)asshowninMardiaetal.(2021).ThismethodcanbeusedinreconstructionoflargerplanarscenesasinFaugerasandLuong(2021),asshowninFigure3.Figure3.ReconstructionofalargerviewofthesceneinimagesinFigure1,basedonanextrinsicmeanaf?neshape.4DiscussionInsummary,inaf?neshapeanalysis,therealGrasmannmanifoldsplaythekeyr?oleinthesamewayasthatofthecomplexprojectivespacesinsimilarityshapeanalysis(Kendall,1984).Whilethe?rstlargesampleresultsonGrassmannmanifoldsareaboutdistributionswithoutanextrinsicmean,datadrivenanalysisisneededforconcentrateddistributionsonsuchmanifolds.ThereforelargesampleandnonparametricbootstrapmethodsshouldbedevelopedforextrinsicsamplemeansonGrassmannmanifoldsandappliedinpractice.Themethodofimagewarpingwasus