Constrained imaging: denoising and sparse sampling

Magnetic resonance imaging (MRI) is a powerful tool for studying the anatomy, physiology, and metabolism of biological systems. Despite the fact that MRI was introduced decades ago and has already revolutionized medical imaging, current applications are still far from utilizing the full potential of the MR signal. Traditional MRI data acquisition and image reconstruction methods are based on simple Fourier inversion, leading to undesirable trade-offs between image resolution, signal-to-noise ratio (SNR), and data acquisition time. Classical approaches to addressing these trade-offs have relied on improved imaging hardware and more efficient pulse sequences. In contrast, our work addresses the limitations of MR using relatively less-explored signal processing approaches, which have recently become practical because of increasing computational capabilities. This dissertation concerns the use of constrained imaging models to guide the design of both data acquisition and image reconstruction, leading to improved imaging performance in the context of both noise-limited and resolution-limited scenarios. To address noise limitations for high-resolution imaging, we introduce a quasi-Bayesian edge-preserving smoothness prior for modeling correlated image sequences. The prior models the correlated edge structures that are observed in the image sequence, and is used within a penalized maximum likelihood framework to reduce image noise while preserving high-resolution anatomical structure. In contrast to many constrained imaging methods, we demonstrate that the proposed method is relatively simple to analyze and is robust to model inaccuracy when reconstruction parameters are chosen appropriately. Resolution and SNR analysis shows that the proposed formulations lead to substantial improvements in SNR with only a moderate decrease in spatial resolution. An examination of resolution and SNR trade-offs is presented, which serves as a guide for the optimal design of data acquisition and image reconstruction procedures in this context. To address limited spatial resolution in high-SNR scenarios, we design specialized data acquisition and image reconstruction procedures to enable image reconstruction from sparsely-sampled data. Specifically, we leverage prior information that the image has sparse or low-rank structure to significantly reduce sampling requirements in two different contexts. In the first context, we assume that the image is sparse in a known transform domain, and develop a novel non-Fourier data acquisition scheme to enable high-quality reconstruction from undersampled data. The second context is specific to spatiotemporal imaging, and it is assumed that the temporal evolution of the spatiotemporal image is highly correlated at different spatial positions. This correlation leads to the formulation of a novel low-rank matrix recovery problem, which we demonstrate can be solved efficiently and effectively using special algorithms. Applications of the proposed techniques are illustrated with simulated and experimental data from a variety of different MR imaging scenarios