UMD Theses and Dissertations

Permanent URI for this collectionhttp://hdl.handle.net/1903/3

New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a given thesis/dissertation in DRUM.

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    Balayage of Fourier Transforms and the Theory of Frames
    (2011) Au-Yeung, Enrico; Benedetto, John; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    Every separable Hilbert space has an orthogonal basis. This allows every element in the Hilbert space to be expressed as an infinite linear combination of the basis elements. The structure of a basis can be too rigid in some situations. Frames gives us greater flexibility than bases. A frame in Hilbert space is a spanning set with the reconstruction property. A frame must satisfy both an upper frame bound and a lower frame bound. The requirement of an upper bound is rather modest. Most of the mathematical difficulty lies in showing the lower bound exists. We examine the theory of Beurling on Balayage of Fourier transforms and the role of spectral synthesis in this theory. Beurling showed that if the condition of Balayage holds, then the lower frame bound for a Fourier frame exists under suitable hypothesis. We extend this theory to obtain lower bound inequalities for other types of frames. We prove that lower bounds exist for generalized Fourier frames and two types of semi-discrete Gabor frames.
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    Enumeration of Harmonic Frames and Frame Based Dimension Reduction
    (2009) Hirn, Matthew John; Benedetto, John J; Okoudjou, Kasso A; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    We investigate two aspects of frame theory, one of a theoretical nature, the other very much on the applied side. In the former, we enumerate all harmonic frames of prime order, and develop partial proofs concerning the structure of the symmetry group for this subset of frames. In the latter, we develop frame theory in the context of kernel eigenmap methods, merging the two theories in a practical manner and applying new algorithms to hyperspectral imagery data for the purposes of material classification. These two problems, while seemingly separate, are united by frame theory and serve to illustrate both the beautiful theoretical nature of frames as well as their practicality in dealing with real world problems.
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    Dimensionality reduction for hyperspectral data
    (2008-05-09) Widemann, David P; Benedetto, John J; Czaja, Wojciech; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    This thesis is about dimensionality reduction for hyperspectral data. Special emphasis is given to dimensionality reduction techniques known as kernel eigenmap methods and manifold learning algorithms. Kernel eigenmap methods require a nearest neighbor or a radius parameter be set. A new algorithm that does not require these neighborhood parameters is given. Most kernel eigenmap methods use the eigenvectors of the kernel as coordinates for the data. An algorithm that uses the frame potential along with subspace frames to create nonorthogonal coordinates is given. The algorithms are demonstrated on hyperspectral data. The last two chapters include analysis of representation systems for LIDAR data and motion blur estimation, respectively.