A Study of the Relationship Between Spectrum and Geometry Through Fourier Frames and Laplacian Eigenmaps

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dc.contributor.advisorBenedetto, John J.en_US
dc.contributor.authorDuke, Kevin W.en_US
dc.contributor.departmentMathematicsen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.date.accessioned2012-10-11T06:17:32Z
dc.date.available2012-10-11T06:17:32Z
dc.date.issued2012en_US
dc.description.abstractThis thesis has two parts. The first part is a study of Fourier frames. We follow the development of the theory, beginning with its classical roots in non-uniform sampling in Paley-Wiener Spaces, to its current state, the study of the spectral properties of finite measures on locally compact abelian groups. The aim of our study is to understand the relationship between the geometry of the supporting set of a measure and the spectral properties it exhibits. In the second part, we study extensions of the Laplacian Eigenmaps algorithm and their uses in hyperspectral image analysis. In particular, we show that there is a natural way of including spatial information in the analysis that improves classification results. We also provide evidence supporting the use of Schrödinger Eigenmaps as a semisupervised tool for feature extraction. Finally, we show that Schrödinger Eigenmaps provides a platform for fusing Laplacian Eigenmaps with other clustering techniques, such as kmeans clustering.en_US
dc.identifier.urihttp://hdl.handle.net/1903/13260
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledCantor measuresen_US
dc.subject.pquncontrolledFourier framesen_US
dc.subject.pquncontrolledhyperspectral imageryen_US
dc.subject.pquncontrolledLaplacian Eigenmapsen_US
dc.subject.pquncontrolledsampling theoryen_US
dc.titleA Study of the Relationship Between Spectrum and Geometry Through Fourier Frames and Laplacian Eigenmapsen_US
dc.typeDissertationen_US

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