Theses and Dissertations from UMD

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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 give thesis/dissertation in DRUM

More information is available at Theses and Dissertations at University of Maryland Libraries.

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    Constant Amplitude Zero Autocorrelation Sequences and Single Pixel Camera Imaging
    (2018) Magsino, Mark; Benedetto, John J; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    The primary focus is on Constant Amplitude Zero Autocorrelation sequences, or CAZAC sequences for short. We provide both an exposition and some new results relating to CAZAC sequences. We then apply them to Gabor frames and prove a condition for checking when Gabor systems generated by a CAZAC sequence are tight frames. We also construct multiple examples using this condition. Finally, we explore natural generalizations of CAZAC sequences to the torus and the real line and provide examples highlighting the differences between the generalizations. In the last section, we highlight some work done in image processing. In particular, we present results in single-pixel camera imaging and the demosaicing problem in image processing.
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    Nonlinear Analysis of Phase Retrieval and Deep Learning
    (2017) Zou, Dongmian; Balan, Radu V; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    Nonlinearity causes information loss. The phase retrieval problem, or the phaseless reconstruction problem, seeks to reconstruct a signal from the magnitudes of linear measurements. With a more complicated design, convolutional neural networks use nonlinearity to extract useful features. We can model both problems in a frame-theoretic setting. With the existence of a noise, it is important to study the stability of the phaseless reconstruction and the feature extraction part of the convolutional neural networks. We prove the Lipschitz properties in both cases. In the phaseless reconstruction problem, we show that phase retrievability implies a bi-Lipschitz reconstruction map, which can be extended to the Euclidean space to accommodate noises while remaining to be stable. In the deep learning problem, we set up a general framework for the convolutional neural networks and provide an approach for computing the Lipschitz constants.
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    Frame Multiplication Theory for Vector-valued Harmonic Analysis
    (2014) Andrews, Travis David; Benedetto, John J; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    A tight frame is a sequence in a separable Hilbert space satisfying the frame inequality with equal upper and lower bounds and possessing a simple reconstruction formula. We define and study the theory of frame multiplication in finite dimensions. A frame multiplication for a frame is a binary operation on the frame elements that extends to a bilinear vector product on the entire Hilbert space. This is made possible, in part, by the reconstruction property of frames. The motivation for this work is the desire to define meaningful vector-valued versions of the discrete Fourier transform and the discrete ambiguity function. We make these definitions and prove several familiar harmonic analysis results in this context. These definitions beget the questions we answer through developing frame multiplication theory. For certain binary operations, those with the Latin square property, we give a characterization of the frames, in terms of their Gramians, that have these frame multiplications. Combining finite dimensional representation theory and Naimark's theorem, we show frames possessing a group frame multiplication are the projections of an orthonormal basis onto the isotypic components of the regular representations. In particular, for a finite group G, we prove there are only finitely many inequivalent frames possessing the group operation of G as a frame multiplication, and we give an explicit formula for the dimensions in which these frames exist. Finally, we connect our theory to a recently studied class of frames; we prove that frames possessing a group frame multiplication are the central G-frames, a class of frames generated by groups of operators on a Hilbert space.
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    The Multiplicative Zak Transform, Dimension Reduction, and Wavelet Analysis of LIDAR Data
    (2010) Flake, Justin Christopher; Benedetto, John J; Czaja, Wojciech; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    This thesis broadly introduces several techniques within the context of timescale analysis. The representation, compression and reconstruction of DEM and LIDAR data types is studied with directional wavelet methods and the wedgelet decomposition. The optimality of the contourlet transform, and then the wedgelet transform is evaluated with a valuable new structural similarity index. Dimension reduction for material classification is conducted with a frame-based kernel pipeline and a spectral-spatial method using wavelet packets. It is shown that these techniques can improve on baseline material classification methods while significantly reducing the amount of data. Finally, the multiplicative Zak transform is modified to allow the study and partial characterization of wavelet frames.
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    Sigma Delta Modulation and Correlation Criteria for the Construction of Finite Frames Arising in Communication Theory
    (2004-04-29) Kolesar, Joseph Dennis; Benedetto, John J; Mathematics
    In this dissertation we first consider a problem in analog to digital (A/D) conversion. We compute the power spectra of the error arising from an A/D conversion. We then design various higher dimensional analogs of A/D schemes, and compare these schemes to a standard error diffusion scheme in digital halftoning. Secondly, we study finite frames. We classify certain finite frames that are constructed as orbits of a group. These frames are seen to have subtle symmetry properties. We also study Grassmannian frames which are frames with minimal correlation. Grassmannian frames have an important intersection with spherical codes, erasure channel models, and communication theory. This is the main part of the dissertation, and we introduce new theory and algorithms to decrease the maximum frame correlation and hence construct specific examples of Grassmannian frames. A connection has been drawn between the two parts of this thesis, namely A/D conversion and finite frames. In particular, finite frames are used to expand vectors in $\RR^d$, and then different quantization schemes are applied to the coefficients of these expansions. The advantage is that all possible outcomes of quantization can be considered because of the finite dimensionality.