Doctor of Philosophy, The Ohio State University, 2022, Electrical and Computer Engineering

This dissertation addresses two separate engineering challenges: image-inverse problems and novelty detection. First, we address image-inverse problems. We review Plug-and-Play (PnP) algorithms, where a proximal operator is replaced by a call of an arbitrary denoising algorithm. We apply PnP algorithms to compressive Magnetic Resonance Imaging (MRI). MRI is a non-invasive diagnostic tool that provides excellent soft-tissue contrast without the use of ionizing radiation. However, when compared to other clinical imaging modalities (e.g., CT or ultrasound), the data acquisition process for MRI is inherently slow, which motivates undersampling and thus drives the need for ac- curate, efficient reconstruction methods from undersampled datasets. We apply the PnP-ADMM algorithm to cardiac MRI and knee MRI data. For these algorithms, we developed learned denoisers that can process complex-valued MRI images. Our algorithms achieve state-of-the-art performance on both the cardiac and knee datasets. Regularization by Denoising (RED), as proposed by Romano, Elad, and Milanfar, is a powerful image-recovery framework that aims to minimize an explicit regular- ization objective constructed from a plug-in image-denoising function. Experimental evidence suggests that RED algorithms are state-of-the-art. We claim, however, that explicit regularization does not explain the RED algorithms. In particular, we show that many of the expressions in the paper by Romano et al. hold only when the denoiser has a symmetric Jacobian, and we demonstrate that such symmetry does not occur with practical denoisers such as non-local means, BM3D, TNRD, and DnCNN. To explain the RED algorithms, we propose a new framework called Score-Matching by Denoising (SMD), which aims to match a “score” (i.e., the gradient of a log-prior). Novelty detection is the ability for a machine learning system to detect signals that are significantly different from samples seen during training. Detecting novelties is (open full item for complete abstract)

Committee: Philip Schniter (Advisor); Rizwan Ahmad (Committee Member); Lee Potter (Committee Member) Subjects: Electrical Engineering

PHD, Kent State University, 2022, College of Arts and Sciences / Department of Mathematical Sciences

This work is concerned with the development of accurate and efficient iterative methods for the solution of linear discrete ill-posed problems when the matrix involved is nonsymmetric. These problems often arise in science and engineering through the discretization of Fredholm integral equations of the first kind. The matrices that define these problems are typically severely ill-conditioned and may be rank-deficient. Because of this, the solution of linear discrete ill-posed problems may not exist or are very sensitive to perturbations caused by errors in the available data. These difficulties can be reduced, for example, by applying iterative regularization techniques. Krylov subspace projection strategies have been used in tandem with iterative methods to form efficient and accurate solution methods. Specifically, the Arnoldi iteration is a well known iterative process that constructs an orthonormal basis of a Krylov subspace. The opening focus is on the development of a novel "approximate Tikhonov regularization" method based on constructing a low-rank approximation to the matrix in the linear discrete ill-posed problem by carrying out a few steps of the Arnoldi process. The subsequent chapter focuses on the description of three iterative methods that modify the generalized minimum residual (GMRES), block GMRES, and global GMRES methods for the solution of appropriate linear systems of equations. The primary contribution to this field of this work is through the introduction of two block variants for when there are multiple right-hand sides in the linear system. In the final chapter the limitations of applying block GMRES methods to linear discrete ill-posed problems are discussed. While block algorithms can be executed efficiently on many computers, the work herein shows that available block algorithms may yield computed approximate solutions of unnecessarily poor quality. A termed "local'' block GMRES method that can overcome the problems associated with b (open full item for complete abstract)

Committee: Alessandro Buccini (Advisor); Lothar Reichel (Advisor) Subjects: Applied Mathematics

PHD, Kent State University, 2021, College of Arts and Sciences / Department of Mathematical Sciences

This work is concerned with structure preserving and other techniques for the solution of linear discrete ill-posed problems with transform-based tensor-tensor products, e.g., the t-product and the invertible linear transform product. Specifically, we focus on two categories of solution methods, those that involve flattening, i.e., reduce the tensor equation to an equivalent equation involving a matrix and a vector, and those that preserve the tensor structure by avoiding flattening. Various techniques based on Krylov subspace-type methods for solving third order tensor ill-posed problems are discussed. The data is a laterally oriented matrix or a general third order tensor. Regularization of tensor ill-posed problem by Tikhonov's approach and truncated iterations are considered. Golub-Kahan bidiagonalization-type, Arnoldi-type, and Lanczos-type processes are applied to reduce large-scale Tikhonov minimization problems to small-sized problems. A few steps of the t-product bidiagonalization process can be employed to inexpensively compute approximations of the singular tubes of the largest Frobenius norm and the associated left and right singular matrices. A less prohibitive computation of approximations of eigentubes of the largest Frobenius norm and the corresponding eigenmatrix by a few steps of the t-product Lanczos process is considered. The interlacing of the Frobenius norm of the singular tubes is shown and applied. The discrepancy principle is used to determine the regularization parameter and the number of iterations by a chosen method. Several truncated iteration techniques, e.g., SVD-like, and those based on the above processes are considered. Solution methods for the weighted tensor Tikhonov minimization problem with weighted global and non-global bidiagonalization processes are discussed. The weights on the fidelity and regularization parts of this problem are suitably defined symmetric positive definite (SPD) tensors. The computation of the inverse of (open full item for complete abstract)

Committee: Lothar (Advisor) Reichel (Committee Chair); Xiaoyu Zheng (Committee Member); Barry Dunietz (Committee Member); Sergij Shyanovskii (Committee Member); Jing Li (Committee Member) Subjects: Applied Mathematics

Master of Sciences, Case Western Reserve University, 2015, Applied Mathematics

This paper presents a fast algorithm for total generalized variation (TGV) based image reconstruction of magnetic resonance images collected by a technique known as partial parallel imaging (PPI). TGV is a generalization of the commonly employed total variation (TV) regularizer. TV reconstructs piecewise constant images and is known to produce oil-painting artifacts, while TGV reconstructs images with piecewise polynomial intensities and largely avoids this issue. The proposed algorithm combines the Bregman Operator Splitting with Variable Stepsize (BOSVS) approach derived by Chen, Hager, et al. [8] with the closed-form expressions for the TGV subproblem that arises in the alternating directional method of multipliers, derived by Guo, Qin and Yin [13]. The ill-conditioned inversion matrix that comes from PPI is approximated according to a stepsize rule similar to that in BOSVS. The stepsize rule starts with a Barzilai-Borwein step, then uses a line search to ensure convergence and eciency. The proposed regularizer is shown to achieve better results than TV, especially for reconstructing smooth details, in sampling conditions as low as 7.87%.

Committee: Weihong Guo PhD (Advisor); Steven Izen PhD (Committee Member); Julia Dobrosotskaya PhD (Committee Member) Subjects: Applied Mathematics