Nonparametric Density Estimation on Homogeneous Spaces in High Level Image Analysis.

Author

Jeff Lee
Robert Paige L., Missouri University of Science and TechnologyFollow
Vic Patrangenaru
Frits H. Ruymgaart

Abstract

Thelandmarkdatareductionapproachinhighlevelimageanalysishasledtosignificantprogresstoscenerecognitionviastatisticalshapeanalysis(DrydenandMardia,1998).Whileanumberoffamiliesofsimilarityshapedensitieshaveprovenusefulindataanalysis,onlyafewpara-metricmodelshavebeenconsideredonlyrecentlyinthecontextofprojectiveshape(MardiaandPatrangenaru,2004),oraffineshape.Shapespacesofinteresthavethegeometricstructureofsymmetricspaces:planarsimilarityshapespacesarecomplexprojectivespaces(Kendall,1984),affineshapespacesarerealGrassmannmanifolds(Sparr,1992),andspacesofplanarprojectiveshapesofconfigurationsofpointsingeneralpositionareproductsofrealprojectivespaces(MardiaandPatrangenaru,2004).Therefore,datadrivendensityestimationofshapes,regardedaspointsonsymmetricspacesandarisingfromdigitizinglandmarksinimages,isnecessary.Recently,Pelletier(2004)consideredkerneldensityestimationon"general”Rie-mannianmanifolds;hisresultshoweverholdonlyinhomogeneousspaces.Thisissufficientforimageanalysis,sinceanysymmetricspaceishomogeneous.PelletierestimatorsgeneralizethedensityestimatorsoncertainhomogeneousspacesintroducedbyRuymgaart(1989),byH.Hendriks,J.H.M.JanssenandRuymgaart(1993),andbyLeeandRuymgaart(1998).Inthispaper,weproposeaclassofadjustedPelletierdensityestimators,onhomogeneousspaces,thatconvergeuniformlyandalmostsurelyatthesamerateasnaivekerneldensityestimatorsonEuclideanspaces

Recommended Citation

J. Lee et al., "Nonparametric Density Estimation on Homogeneous Spaces in High Level Image Analysis.," Bioinformatics, Images, and Wavelets, Texas Tech University, Jan 2004.

Department(s)

Mathematics and Statistics

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2004 Texas Tech University, All rights reserved.

Publication Date

01 Jan 2004