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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Subject: search.filters.subject.Superresolution

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    Uniqueness for continuous superresolution by means of Choquet theory and geometric measure theory
    (2021) Cinoman, Ryan M; Benedetto, John J; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    The problem of superresolution is to recover an element of a vector space from data much smaller than the dimension of the space, using a prior assumption of sparsity. A famous example is compressive sensing, where the elements are images with a large finite resolution. On the other hand, we focus on a continuous form of superresolution. Given a measure $\mu$ on a continuous domain such as the two dimensional torus, can we recover $\mu$ from knowledge of only a finite number of its Fourier coefficients using a total variation minimization method? We will see that the answer depends on certain properties of $\mu$. Namely, a necessary condition is that $\mu$ be discrete.We use methods from geometric analysis to investigate the continuous superresolution problem. Tools from measure theory relate properties of the support of a measure, such as Hausdorff dimension, to properties of its Fourier transform. We also use measure theory to investigate the possibility of alternatives to total variation that may allow us to recover surface measures defined on space curves. There is a theorem of Choquet concerning representations of points in convex sets as sums of their extreme points. As it turns out, we can apply this to the superresolution problem because the extreme points of the set of measures with total variation $1$ are precisely the set of delta measures. We consider superresolution as a convex optimization problem, where the goal is to find representations of the initial data as sums of delta measures. Choquet theory provides tools to investigate the previously unresolved problem of uniqueness. We use this to give a novel sufficient condition for a measure to be uniquely superresolved, given data on a known finite set of frequencies.
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    Multiscale and Directional Representations of High-Dimensional Information Content in Remotely Sensed Data
    (2015) Weinberg, Daniel Eric; Czaja, Wojciech; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    This thesis explores the theory and applications of directional representations in the field of anisotropic harmonic analysis. Although wavelets are optimal for decomposing functions in one dimension, they are unable to achieve the same success in two or more dimensions due to the presence of curves and surfaces of discontinuity. In order to optimally capture the behavior of a function at high-dimensional discontinuities, we must be able to incorporate directional information into our analyzing functions, in addition to location and scale. Examples of such representations are contourlets, curvelets, ridgelets, bandelets, wedgelets, and shearlets. Using directional representations, in particular shearlets, we tackle several challenging problems in the processing of remotely sensed data. First, we detect roads and ditches in LIDAR data of rural scenes. Second, we develop an algorithm for superresolution of optical and hyperspectral data. We conclude by presenting a stochastic particle model in which the probability of movement in a particular direction is neighbor-weighted.
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    Anisotropic Harmonic Analysis and Integration of Remotely Sensed Data
    (2015) Murphy, James Michael; Czaja, Wojciech; Benedetto, John J.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    This thesis develops the theory of discrete directional Gabor frames and several algorithms for the analysis of remotely sensed image data, based on constructions of harmonic analysis. The problems of image registration, image superresolution, and image fusion are separate but interconnected; a general approach using transform methods is the focus of this thesis. The methods of geometric multiresolution analysis are explored, particularly those related to the shearlet transform. Using shearlets, a novel method of image registration is developed that aligns images based on their shearlet features. Additionally, the anisotropic nature of the shearlet transform is deployed to smoothly superrsolve remotely-sensed image with edge features. Wavelet packets, a generalization of wavelets, are utilized for a flexible image fusion algorithm. The interplay between theoretical guarantees for these mathematical constructions, and their effectiveness for image processing is explored throughout.