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1、ApproximateThinPlateSplineMappingsGianlucaDonato1andSergeBelongie21DigitalPersona,Inc.,RedwoodCity,CA94063gianlucad@digitalpersona.com2U.C.SanDiego,LaJolla,CA92093-0114sjb@cs.ucsd.eduAbstract.Thethinplatespline(TPS)isaneffectivetoolformodelingcoordinatetransformationsthathasbeenappliedsuccessfu
2、llyinseveralcomputervisionapplications.Unfortunatelythesolutionrequiresthein-versionofap×pmatrix,wherepisthenumberofpointsinthedataset,thusmakingitimpracticalforlargescaleapplications.Asitturnsout,asurprisinglygoodapproximatesolutionisoftenpossibleusingonlyasmallsubsetofcorrespondingpoints.Web
3、eginbydiscussingtheobviousapproachofusingthesubsampledsettoestimateatransformationthatisthenappliedtoallthepoints,andweshowthedrawbacksofthismethod.Wethenproceedtoborrowatechniquefromthemachinelearningcom-munityforfunctionapproximationusingradialbasisfunctions(RBFs)andadaptittothetaskathand.Us
4、ingthismethod,wedemonstrateasignificantimprovementoverthenaivemethod.Onedrawbackofthismethod,however,isthatisdoesnotallowforprincipalwarpanalysis,atechniqueforstudyingshapedeformationsintroducedbyBooksteinbasedontheeigenvectorsofthep×pbendingenergymatrix.Toad-dressthis,wedescribeathirdapproxima
5、tionmethodbasedonaclassicmatrixcompletiontechniquethatallowsforprincipalwarpanalysisasaby-product.Bymeansofexperimentsonrealandsyntheticdata,wedemonstratetheprosandconsofthesedifferentapproximationssoastoallowthereadertomakeaninformeddecisionsuitedtohisorherapplication.1IntroductionThethinplate
6、spline(TPS)isacommonlyusedbasisfunctionforrepresentingcoordinatemappingsfromR2toR2.Bookstein[3]andDavisetal.[5],forexam-ple,havestudieditsapplicationtotheproblemofmodelingchangesinbiologicalforms.Thethinplatesplineisthe2Dgeneralizationofthecubicspline.InitsregularizedformtheTPSmodelincludesthe
7、affinemodelasaspecialcase.OnedrawbackoftheTPSmodelisthatitssolutionrequirestheinversionofalarge,densematrixofsizep×p,wherepisthenumberofpointsinthedataset.Ourgoalinthispaperistopresentandcomparethreeapproximationmethodsthataddressthiscomp
