ASU Electronic Theses and Dissertations
This collection includes most of the ASU Theses and Dissertations from 2011 to present. ASU Theses and Dissertations are available in downloadable PDF format; however, a small percentage of items are under embargo. Information about the dissertations/theses includes degree information, committee members, an abstract, supporting data or media.
In addition to the electronic theses found in the ASU Digital Repository, ASU Theses and Dissertations can be found in the ASU Library Catalog.
Dissertations and Theses granted by Arizona State University are archived and made available through a joint effort of the ASU Graduate College and the ASU Libraries. For more information or questions about this collection contact or visit the Digital Repository ETD Library Guide or contact the ASU Graduate College at gradformat@asu.edu.
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- All Subjects: Mathematics
- Creators: Espanol, Malena
Description
This thesis addresses the problem of approximating analytic functions over general and compact multidimensional domains. Although the methods we explore can be used in complex domains, most of the tests are performed on the interval $[-1,1]$ and the square $[-1,1]\times[-1,1]$. Using Fourier and polynomial frame approximations on an extended domain, well-conditioned methods can be formulated. In particular, these methods provide exponential decay of the error down to a finite but user-controlled tolerance $\epsilon>0$. Additionally, this thesis explores two implementations of the frame approximation: a singular value decomposition (SVD)-regularized least-squares fit as described by Adcock and Shadrin in 2022, and a column and row selection method that leverages QR factorizations to reduce the data needed in the approximation. Moreover, strategies to reduce the complexity of the approximation problem by exploiting randomized linear algebra in low-rank algorithms are also explored, including the AZ algorithm described by Coppe and Huybrechs in 2020.
ContributorsGuo, Maosheng (Author) / Platte, Rodrigo (Thesis advisor) / Espanol, Malena (Committee member) / Renaut, Rosemary (Committee member) / Arizona State University (Publisher)
Created2023
Description
During the inversion of discrete linear systems, noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion. This is a process called regularization. The influence of the provided prior information is controlled by the introduction of non-negative regularization parameter(s). Many methods are available for both the selection of appropriate regularization parame- ters and the inversion of the discrete linear system. Generally, for a single problem there is just one regularization parameter. Here, a learning approach is considered to identify a single regularization parameter based on the use of multiple data sets de- scribed by a linear system with a common model matrix. The situation with multiple regularization parameters that weight different spectral components of the solution is considered as well. To obtain these multiple parameters, standard methods are modified for identifying the optimal regularization parameters. Modifications of the unbiased predictive risk estimation, generalized cross validation, and the discrepancy principle are derived for finding spectral windowing regularization parameters. These estimators are extended for finding the regularization parameters when multiple data sets with common system matrices are available. Statistical analysis of these estima- tors is conducted for real and complex transformations of data. It is demonstrated that spectral windowing regularization parameters can be learned from these new esti- mators applied for multiple data and with multiple windows. Numerical experiments evaluating these new methods demonstrate that these modified methods, which do not require the use of true data for learning regularization parameters, are effective and efficient, and perform comparably to a supervised learning method based on es- timating the parameters using true data. The theoretical developments are validated for one and two dimensional image deblurring. It is verified that the obtained estimates of spectral windowing regularization parameters can be used effectively on validation data sets that are separate from the training data, and do not require known data.
ContributorsByrne, Michael John (Author) / Renaut, Rosemary (Thesis advisor) / Cochran, Douglas (Committee member) / Espanol, Malena (Committee member) / Jackiewicz, Zdzislaw (Committee member) / Platte, Rodrigo (Committee member) / Arizona State University (Publisher)
Created2023
Description
Atmospheric turbulence distorts the path of light passing through the air. When capturing images at long range, the effects of this turbulence can cause substantial geometric distortion and blur in images and videos, degrading image quality. These become more pronounced with greater turbulence, scaling with the refractive index structure constant, Cn2. Removing effects of atmospheric turbulence in images has a range of applications from astronomical imaging to surveillance. Thus, there is great utility in transforming a turbulent image into a ``clean image" undegraded by turbulence. However, as the turbulence is space- and time-variant and statistically random, no closed-form solution exists for a function that performs this transformation. Prior attempts to approximate the solution include spatio-temporal models and lucky frames models, which require many images to provide a good approximation, and supervised neural networks, which rely on large amounts of simulated or difficult-to-acquire real training data and can struggle to generalize. The first contribution in this thesis is an unsupervised neural-network-based model to perform image restoration for atmospheric turbulence with state-of-the-art performance. The model consists of a grid deformer, which produces an estimated distortion field, and an image generator, which estimates the distortion-free image. This model is transferable across different datasets; its efficacy is demonstrated across multiple datasets and on both air and water turbulence. The second contribution is a supervised neural network to predict Cn2 directly from the warp field. This network was trained on a wide range of Cn2 values and estimates Cn2 with relatively good accuracy. When used on the warp field produced by the unsupervised model, this allows for a Cn2 estimate requiring only a few images without any prior knowledge of ground truth or information about the turbulence.
ContributorsWhyte, Cameron (Author) / Jayasuriya, Suren (Thesis advisor) / Espanol, Malena (Thesis advisor) / Speyer, Gil (Committee member) / Arizona State University (Publisher)
Created2021