The rise in the number of wireless applications in the past few decades has led to the need for developing radar and wireless communications systems that coexist in the same frequency band. This thesis explores two aspects of such shared spectrum signal processing. The first part of the thesis focuses on the orthogonal time frequency space (OTFS) modulation scheme and its effectiveness in providing a dual-function radar-communications (DFRC) waveform that can achieve both radar and communications functionalities. A simple "signals and systems" view of OTFS modulation based on the Zak transform is presented. Dual use of the OTFS signal processing chain as a pulse-Doppler radar system is discussed. A 2D root-MUSIC algorithm is presented for delay-Doppler parameter estimation using an OTFS frame containing multiple pilot symbols. With the help of numerical results, it is shown that multiple pilot symbols allow for transmission at a lower peak transmitted power as compared to the single pilot case while maintaining accurate delay-Doppler estimation performance. The second part of the thesis is focused on predicting the asymptotic detection performance of a radar that operates in dynamic environments such as a shared spectrum environment. Specifically, a framework based on Wilks' theorem is proposed to predict the asymptotic detection performance of a generalized likelihood ratio test (GLRT) radar detector operating in the presence of multiple cooperative communications users in the same frequency band. The framework is developed for two scenarios that differ in the amount of side information available to the radar regarding the communications symbols. The derived GLRTs demonstrate very good agreement in performance with the performance predicted by Wilks' theorem, and it is shown that utilizing additional knowledge about the communications symbols provides significant improvement in detection performance. Lastly, the asymptotic distribution of a general GLRT statistic is derived for a scenario where the assumed data distribution is different from the true distribution. Such a result can be utilized for radar performance prediction in general model mismatch scenarios including dynamic shared spectrum environments where accurate model information is not available.

In conventional radar signal processing, a structured model for the target response is used, while clutter and interference are characterized by the covariance matrix of the data distribution. In contrast, the channel matrix-based model represents both target and clutter returns as responses to corresponding channels, resulting in a more versatile model that can incorporate various scenarios. Optimal receive architectures for target detection within a channel matrix-based model are explored using likelihood ratio tests (LRT) and average LRT (ALRT) tests. Generalized likelihood ratio test (GLRT) statistics are derived for the channel matrix-based MIMO radar data model under the assumption of complex multivariate elliptically symmetric (CMES) data distribution, considering both known and unknown covariance matrices of the waveform-independent colored noise (WICN). For the known covariance case, the GLRT statistic follows a chi-square distribution, while for the unknown covariance case, it aligns with Wilks' lambda distribution. The GLRT statistic for the known WICN covariance case, when the maximum likelihood estimate of the covariance matrix replaces the true covariance matrix, matches the Bartlett-Nanda-Pillai trace statistic under the null hypothesis and follows a non-central Lawley-Hotelling $T_0^2$ distribution under the alternative hypothesis. Asymptotically, all derived statistics converge to the known covariance case. Monte Carlo simulations and the saddle point approximation method are employed to generate receiver operating characteristic (ROC) curves for a simple numerical example, supplemented by experimental results and high-fidelity simulations. The potential of deep learning techniques for radar target detection is investigated, with a proposed deep neural network (DNN) architecture benefiting from both model-based and data-driven approaches. The asymptotic distribution of the GLRT statistic for adaptive target detection is non-central chi-squared with a non-centrality parameter that depends on the waveform information. This provides a basis for the design of optimal waveforms fortarget detection. The waveform optimization problem is formulated as a semidefinite programming instance, and an algorithm is proposed to maximize the non-centrality parameter, thereby enhancing the probability of target detection. This algorithm also incorporates power and peak-to-average power ratio (PAPR) constraints, essential for ensuring practical and efficient radar operation.

The need to recover a signal from incomplete or corrupted measurements is a central challenge in signal processing. A particular problem of this type is recovery of a signal after its Fourier magnitude or its Fourier phase is lost. This problem has a rich history that originated in the field of x-ray crystallography and continues to be of substantial interest in molecular imaging and numerous other applications. It has been observed that Fourier phase is typically more important in representing recognizable features of one-dimensional signals (e.g., audio waveforms) and two-dimensional signals, such as images. Classical experiments illustrating this observation are reproduced in this thesis, and practical iterative algorithms for recovering a signal from either its phase or magnitude are demonstrated. Unsurprisingly, it is typically more difficult to compensate for the loss of phase information, and recovery of a signal from its Fourier magnitude is seen to be less effective than recovery from its Fourier phase. A partitioning method is introduced to improve image recovery from magnitude information, and the phase unwrapping problem for one-dimensional signals is discussed briefly.

Little is known about the state of Arctic sea ice at any given instance in time. The harshness of the Arctic naturally limits the amount of in situ data that can be collected, resulting in gathered data being limited in both location and time. Remote sensing modalities such as satellite Synthetic Aperture Radar (SAR) imaging and laser altimetry help compensate for the lack of data, but suffer from uncertainty because of the inherent indirectness. Furthermore, precise remote sensing modalities tend to be severely limited in spatial and temporal availability, while broad methods are more accessible at the expense of precision. This thesis focuses on the intersection of these two problems and explores the possibility of corroborating remote sensing methods to create a precise, accessible source of data that can be used to examine sea ice at local scale.

Deforestation in the Amazon rainforest has the potential to have devastating effects on ecosystems on both a local and global scale, making it one of the most environmentally threatening phenomena occurring today. In order to minimize deforestation in the Ama- zon and its consequences, it is helpful to analyze its occurrence using machine learning architectures such as the U-Net. The U-Net is a type of Fully Convolutional Network that has shown significant capability in performing semantic segmentation. It is built upon a symmetric series of downsampling and upsampling layers that propagate feature infor- mation into higher spatial resolutions, allowing for the precise identification of features on the pixel scale. Such an architecture is well-suited for identifying features in satellite imagery. In this thesis, we construct and train a U-Net to identify deforested areas in satellite imagery of the Amazon through semantic segmentation.

Recent satellite and remote sensing innovations have led to an eruption in the amount and variety of geospatial ice data available to the public, permitting in-depth study of high-definition ice imagery and digital elevation models (DEMs) for the goal of safe maritime navigation and climate monitoring. Few researchers have investigated texture in optical imagery as a predictive measure of Arctic sea ice thickness due to its cloud pollution, uniformity, and lack of distinct features that make it incompatible with standard feature descriptors. Thus, this paper implements three suitable ice texture metrics on 1640 Arctic sea ice image patches, namely (1) variance pooling, (2) gray-level co-occurrence matrices (GLCMs), and (3) textons, to assess the feasibly of a texture-based ice thickness regression model. Results indicate that of all texture metrics studied, only one GLCM statistic, namely homogeneity, bore any correlation (0.15) to ice freeboard.

Deforestation in the Amazon rainforest has the potential to have devastating effects on ecosystems on both a local and global scale, making it one of the most environmentally threatening phenomena occurring today. In order to minimize deforestation in the Amazon and its consequences, it is helpful to analyze its occurrence using machine learning architectures such as the U-Net. The U-Net is a type of Fully Convolutional Network that has shown significant capability in performing semantic segmentation. It is built upon a symmetric series of downsampling and upsampling layers that propagate feature information into higher spatial resolutions, allowing for the precise identification of features on the pixel scale. Such an architecture is well-suited for identifying features in satellite imagery. In this thesis, we construct and train a U-Net to identify deforested areas in satellite imagery of the Amazon through semantic segmentation.

In this work, the author analyzes quantitative and structural aspects of Bayesian inference using Markov kernels, Wasserstein metrics, and Kantorovich monads. In particular, the author shows the following main results: first, that Markov kernels can be viewed as Borel measurable maps with values in a Wasserstein space; second, that the Disintegration Theorem can be interpreted as a literal equality of integrals using an original theory of integration for Markov kernels; third, that the Kantorovich monad can be defined for Wasserstein metrics of any order; and finally, that, under certain assumptions, a generalized Bayes’s Law for Markov kernels provably leads to convergence of the expected posterior distribution in the Wasserstein metric. These contributions provide a basis for studying further convergence, approximation, and stability properties of Bayesian inverse maps and inference processes using a unified theoretical framework that bridges between statistical inference, machine learning, and probabilistic programming semantics.

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.

Solving partial differential equations on surfaces has many applications including modeling chemical diffusion, pattern formation, geophysics and texture mapping. This dissertation presents two techniques for solving time dependent partial differential equations on various surfaces using the partition of unity method. A novel spectral cubed sphere method that utilizes the windowed Fourier technique is presented and used for both approximating functions on spherical domains and solving partial differential equations. The spectral cubed sphere method is applied to solve the transport equation as well as the diffusion equation on the unit sphere. The second approach is a partition of unity method with local radial basis function approximations. This technique is also used to explore the effect of the node distribution as it is well known that node choice plays an important role in the accuracy and stability of an approximation. A greedy algorithm is implemented to generate good interpolation nodes using the column pivoting QR factorization. The partition of unity radial basis function method is applied to solve the diffusion equation on the sphere as well as a system of reaction-diffusion equations on multiple surfaces including the surface of a red blood cell, a torus, and the Stanford bunny. Accuracy and stability of both methods are investigated.