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

Jeff Lee; Robert Paige; Vic Patrangenaru; Frits H. Ruymgaart

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

Jeff Lee
, Robert Paige
, Vic Patrangenaru
, Frits H. Ruymgaart

Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

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

Original language	American English
Journal	Bioinformatics, Images, and Wavelets
State	Published - Jan 1 2004

Disciplines

Mathematics
Statistics and Probability

Cite this

@article{0242d423af16421f93528eb02cc4e21d,

title = "Nonparametric Density Estimation on Homogeneous Spaces in High Level Image Analysis.",

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{\textquotedblright}Rie-mannianmanifolds;hisresultshoweverholdonlyinhomogeneousspaces.Thisissufficientforimageanalysis,sinceanysymmetricspaceishomogeneous.PelletierestimatorsgeneralizethedensityestimatorsoncertainhomogeneousspacesintroducedbyRuymgaart(1989),byH.Hendriks,J.H.M.JanssenandRuymgaart(1993),andbyLeeandRuymgaart(1998).Inthispaper,weproposeaclassofadjustedPelletierdensityestimators,onhomogeneousspaces,thatconvergeuniformlyandalmostsurelyatthesamerateasnaivekerneldensityestimatorsonEuclideanspaces",

author = "Jeff Lee and Robert Paige and Vic Patrangenaru and Ruymgaart, \{Frits H.\}",

year = "2004",

month = jan,

day = "1",

language = "American English",

journal = "Bioinformatics, Images, and Wavelets",

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TY - JOUR

T1 - Nonparametric Density Estimation on Homogeneous Spaces in High Level Image Analysis.

AU - Lee, Jeff

AU - Paige, Robert

AU - Patrangenaru, Vic

AU - Ruymgaart, Frits H.

PY - 2004/1/1

Y1 - 2004/1/1

N2 - 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

AB - 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

UR - https://scholarsmine.mst.edu/math_stat_facwork/78

M3 - Article

JO - Bioinformatics, Images, and Wavelets

JF - Bioinformatics, Images, and Wavelets

ER -

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

Abstract

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