Givenasetofsignals,aclassicalconstructionofanoptimaltruncatablebasisforoptimallyrepresentingthesignals,istheprincipalcomponentanalysis(PCAforshort)approach.Whentheinformationaboutthesignalsonewouldliketorepresentisamoregeneralproperty,likesmoothness,adifferentbasisshouldbeconsidered.OneexampleistheFourierbasiswhichisoptimalforrepresentationsmoothfunctionssampledonregulargrid.ItisderivedastheeigenfunctionsofthecirculantLaplacianoperator.Inthispaper,basedontheoptimalityoftheeigenfunctionsoftheLaplace-Beltramioperator(LBOforshort),theconstructionofPCAforgeometricstructuresisregularized.Byassumingsmoothnessofagivendata,onecouldexploittheintrinsicgeometricstructuretoregularizetheconstructionofabasisbywhichtheobserveddataisrepresented.TheLBOcanbedecomposedtoprovidearepresentationspaceoptimizedforbothinternalstructureandexternalobservations.Theproposedmodeltakesthebestfromboththeintrinsicandtheextrinsicstructuresofthedataandprovidesanoptimalsmoothrepresentationofshapesandforms.
