Many problems in digital image processing are problems in digital image restoration. Our goal in this technical report is to examine what benefits are incurred when we impose an intensity constraint in the context of total variation regularization, which is a common image restoration tool that works be encouraging smoothness of the restored image--Abstract.

Mild Cognitive Impairment (MCI) is the stage between the declining of normal brain function and the more serious decline of dementia. Alzheimer’s disease (AD) is one of the leading forms of dementia. Despite the fact that MCI does not always lead to AD, an early diagnosis of MCI may be helpful in finding those with early signs of AD. The Alzheimer’s Disease Neuroimaging Initiative (ADNI) has utilized magnetic resonance imaging (MRI) for the diagnosis of MCI and AD. MCI can be separated into two types: Early MCI (EMCI) and Late MCI (LMCI). Furthermore, MRI results can be separated into three views of axial, coronal and sagittal planes. In this work, we perform binary classifications between healthy people and the two types of MCI based on limited MRI images using a deep learning approach. Furthermore, we implement and com- pare two various convolutional neural network (CNN) architectures. The MRIs of 516 patients were used in this study: 172 control normal (CN), 172 EMCI patients and 172 LMCI patients. For this data set, 50% of the images were used for training, 20% for validation, and the remaining 30% for testing. The results showed that the best classification for one model was between CN and LMCI for the coronal view with an accuracy of 79.67%. In addition, we achieved 67.85% accuracy for the second proposed model for the same classification group.

Critical to accurate reconstruction of sparse signals from low-dimensional observations is the solution of nonlinear optimization problems that promote sparse solutions. Sparse signal recovery is a common problem of many different applications ranging from photography to tomography and from radiology to biology. Within the compressive imaging community, minimizing the $\ell_1$-norm or the total variation (TV) seminorm penalized least-squares problem is the most conventional approach for sparse signal recovery. The least-squares data-fidelity term assumes a Gaussian noise model. However, there are many real-world applications that do not follow Gaussian noise statistics. For an instance, when the number of observed photon counts is relatively low at the camera detector, they are corrupted by Poisson noise. This phenomenon can be seen in a variety of different applications including astronomy, night vision, and medical imaging. Therefore, the contribution of the dissertation to sparse signal recovery is two-fold. First, we propose several nonconvex algorithms operate on Poisson statistics to promote sparsity. Second, we will present methods based on trust-region and alternating minimization techniques for sparse signal recovery under Gaussian statistics.

While convex optimization for low-light imaging has received some attention by the imaging community, non-convex optimization techniques for photon-limited imaging are still in their nascent stages. Theoretically, the non-convex $\ell_p$-norm regularization $(0 \leq p < 1)$ would lead to more accurate reconstruction than the convex $\ell_1$-norm relaxation. In this dissertation, we explore sparse Poisson intensity reconstruction methods using gradient based optimization approach with the nonconvex regularization techniques: $\ell_p$-norm, TV$_p$-seminorm, and the \textit{generalized} Shannon entropy. The proposed methods lead to more accurate and high strength reconstructions in medical imaging and computational genomics. In particular, we developed a stage-based nonconvex approach to solve time dependent bioluminescence and fluorescence lifetime imaging problems in the Poisson noise context.

In Gaussian noise context, we solve the $\ell_2$-$\ell_1$ and $\ell_2$-$\ell_p$ sparse recovery problems by transforming the objective function into an unconstrained differentiable function and apply a limited-memory trust-region method. Unlike gradient projection-type methods, which uses only the current gradient, our approach uses gradients from previous iterations to obtain a more accurate Hessian approximation. Numerical experiments with simulated compressive sensing 1D and 2D data are provided to illustrate that our proposed approach eliminates spurious solutions more effectively while improving the computational time to converge in comparison to standard approaches. Moreover, we employ nonconvex $\ell_p$-norm regularization for better recovery and demixing of sparse signals arise in image inpainting and source separation applications.

The collection of data has become an integral part of our everyday lives. The algorithms necessary to process this information become paramount to our ability to interpret this resource. This type of data is typically recorded in a variety of signals including images, sounds, time series, and bio-informatics. In this work, we develop a number of algorithms to recover these types of signals in a variety of modalities. This work is mainly presented in two parts.

Initially, we apply and develop large-scale optimization techniques used for signal processing. This includes the use of quasi-Newton methods to approximate second derivative information in a trust-region setting to solve regularized sparse signal recovery problems. We also formulate the compact representation of a large family of quasi-Newton methods known as the Broyden class. This extension of the classic quasi-Newton compact representation allows different updates to be used at every iteration. We also develop the algorithm to perform efficient solves with these representations. Within the realm of sparse signal recovery, but particular to photon-limited imaging applications, we also propose three novel algorithms for signal recovery in a low-light regime. First, we recover the support and lifetime decay of a flourophore from time dependent measurements. This type of modality is useful in identifying different types of molecular structures in tissue samples. The second algorithm identifies and implements the Shannon entropy function as a regularization technique for the promotion of sparsity in reconstructed signals from noisy downsampled observations. Finally, we also present an algorithm which addresses the difficulty of choosing the optimal parameters when solving the sparse signal recovery problem. There are two parameters which effect the quality of the reconstruction, the norm being used, as well as the intensity of the penalization imposed by that norm. We present an algorithm which uses a parallel asynchronous search along with a metric in order to find the optimal pair.

The second portion of the dissertation draws on our experience with large-scale optimization and looks towards deep learning as an alternative to solving signal recovery problems. We first look to improve the standard gradient based techniques used during the training of these deep neural networks by presenting two novel optimization algorithms for deep learning. The first algorithm takes advantage of the limited memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm in a trust-region setting in order to address the large scale minimization problem associated with deep learning. The second algorithm uses second derivative information in a trust region setting where the Hessian is not explicitly stored. We then use a conjugate based method in order to solve the trust-region subproblem.

Finally, we apply deep learning techniques to a variety of applications in signal recovery. These applications include revisiting the photon-limited regime where we recover signals from noisy downsampled observations, image disambiguation which involves the recovery of two signals which have been superimposed, deep learning for synthetic aperture radar (SAR) where we recover information describing the imaging system as well as evaluate the impact of reconstruction on the ability to perform target detection, and signal variation detection in the human genome where we leverage the relationships between subjects to provide better detection.

In today’s complex networked world, graph-structured data has emerged as a fundamental element across a multitude of fields, including social networks, biological systems, and communication infrastructures. The ability of graphs to represent complex relationships and interactions makes them invaluable for understanding and leveraging data in these domains. As artificial intelligence continues to advance, deep learning has become a pivotal technology, driving breakthroughs across various applications. In particular, deep learning techniques have revolutionized data analysis by automating feature extraction and enhancing predictive accuracy. Among these techniques, Graph Neural Networks (GNNs) have gained prominence for their ability to effectively handle graph-structured data. GNNs extend the power of deep learning to graph-based tasks such as node classification, link prediction, and graph generation, by leveraging the intrinsic relationships between nodes and edges. These models, through their message-passing mechanisms, enable sophisticated learning from the rich structural information present in graphs. Despite their success, GNNs face significant challenges, notably their vulnerability to adversarial attacks. These attacks, which involve deliberate modifications to graph structures—such as adding, removing, or rewiring edges—can severely compromise model performance and security, particularly in sensitive applications like financial systems and network security. Addressing these vulnerabilities, this dissertation proposes innovative methodologies to enhance the robustness of GNNs against adversarial threats. It proposes novel attack models that leverage meta-learning and convex relaxation techniques to generate and evaluate adversarial perturbations effectively. These models utilize advanced optimization strategies, including Focal Loss Projected Momentum and Projected Metattack, to address the vulnerabilities of GNNs. In parallel, a defense mechanism based on Non-Negative Matrix Factorization is developed to purify graph structures. This approach employs matrix decomposition techniques to separate the true graph structure from adversarial noise, demonstrating significant improvements in resilience against poisoning attacks. Additionally, the dissertation explores the interpretability of learning methods through the development of graph kernels based on Optimal Transport (OT) theory. This method enhances node embeddings and predictive modeling by offering a clearer understanding of graph data, particularly in biological applications. The OT-based kernels provide a transparent and efficient alternative to traditional deep learning methods, facilitating better insights into the decision-making processes of graph-based systems. Collectively, these contributions advance the state-of-the-art in robust graph learning by addressing critical issues related to model security, interpretability, and optimization, thereby enhancing the reliability and applicability of graph-based systems across various domains. In addition to the core contributions on robust and interpretable graph learning, the dissertation explores advancements in other related fields. This includes applying deep unrolling techniques to improve high dynamic range (HDR) imaging from low dynamic range (LDR) images, developing a novel algorithm for ghost-free HDR synthesis. It also introduces a new self-supervised learning method that combines contrastive learning with optimal transport for more efficient and scalable clustering. Moreover, it addresses challenges in genomics by presenting an optimization framework for detecting structural variants, enhancing the accuracy of detection in low-coverage sequencing scenarios. Collectively, these contributions extend the dissertation's impact beyond graph-based learning, advancing both theoretical and practical aspects across various domains. The developed methods were rigorously evaluated on real-world graph and imaging datasets, demonstrating their effectiveness and robustness. Comparative studies with state-of-the-art approaches reveal that our methods not only meet but often exceed the performance of existing solutions. These evaluations highlight the enhanced capability of our techniques in addressing complex challenges in robust and interpretable learning.

In this work we develop a framework to detect structural variants (SVs) in the genomes of related individuals. In particular, we consider low-coverage regimes that are more inexpensive than in high-coverage settings but are more susceptible to sequencing errors. To improve our ability to accurately predict SVs, we incorporate statistical models with familial relationship constraints and sparsity promoting penalties. We use simulated data to run experiments. Previous detection methods have used Poisson statistical models. The main contribution of this thesis is the use of the more general negative binomial distribution model in one-parent/one-child and two-parent/one-child frameworks. We extend the existing SPIRAL algorithm, which uses a Poisson log-likelihood objective function, and implement a negative binomial log-likelihood objective function. The genomes tested are haploid, meaning there is only one copy of each chromosome.

Neural networks generally require large amounts of data to adequately model thedomain space. In situations where the data are limited, the predictions from these models, which are typically obtained from stochastic gradient descent (SGD) minimization algorithms, can be poor. In addition, the data is commonly corrupted due to poor imaging appatus. In these cases, the use of more sophisticated optimization approaches and model architectures becomes crucial to increase the impact of each training iteration. Second-order methods can capture curvature information, providing a more informed guess on the direction and step length. However, they require vast amounts of storage and can be computationally time demanding. To address the computational issue, we propose an optimization algorithm that uses second-derivative information, exploiting curvature information for avoiding saddle points. We utilize a Hessian-free approach where we do not explicitly store the second-derivative matrix, by applying a conjugate gradient method. The algorithm uses a trust-region method, which does not require the Hessian to be positive definite. We present numerical experiments which demonstrate the improvement in classification accuracy using our proposed approach over a standard SGD approach. We propose using a limited-memory symmetric rank-one quasi-Newton approach which further addresses the time and space computational complexity. The approach allows for indefinite Hessian approximations, enabling directions of negative curvature to be exploited. Furthermore, we use a modified adaptive regularized using cubics approach, which generates a sequence of cubic subproblems that have closed-form solutions with suitable regularization choices and investigate the performance of our proposed and compare our approach to state-of-the-art first-order and other quasi-Newton methods. To incorporate the benefits of an exponential moving average algorithm to a quasi-Newton approach, we propose a quasi-Adam approach. Judicious choices of quasi-Newton matrices can lead to guaranteed descent in the objective function and improved convergence. In this work, we integrate search directions obtained from using these quasi-Newton Hessian approximations with the Adam optimization algorithm. We provide convergence guarantees and demonstrate improved performance through an extensive experimentation on a variety of applications. Finally, to mitigate the issue of data corruption, we propose a variety of architectures for various applications in image processing. We propose a blind source signal separator, which involves separating image signals which have been superimposed by a common observing apparatus. We propose novel deep learning architectures for low photon count image denoising, which contains Gaussian noise in a low-photon count setting. Then we propose a novel architecture for lowphoton count and downsampled imaging, where the signal is interfered with some Gaussian noise, Poisson noise and then downsampled. Finally, we propose a novel adversarial detection method for white-box attacks using Radial basis function and Discrete Cosine Transforms.

This dissertation describes optimization methods and matrix factorizations for large-scale quasi-Newton trust-region methods. The proposed methods are applicable to convex and non-convex optimization problems, and are described in algorithms for software implementations.