Constrained Magnetic Resonance and Computed Tomographic Imaging: Models and Applications

Tomographic imaging systems like CT and MRI allow the visualization of the interior of objects and are invaluable in diverse applications such as medical imaging and security scanning. These systems usually consist of two separate sub-systems: data acquisition and image reconstruction. Traditional imaging usually involves sampling at the Nyquist rate, and places the overwhelming burden of image formation on the data acquisition side. The reconstruction module simply acts as a mapping from the measurement space to image space. Acquiring data at the full sampling rate can be prohibitively expensive in several applications. There is tremendous interest in forming images from subsampled data to improve scanning throughput and reduce scan times. In diagnostic imaging, there are additional motivating factors to lower scan times: reduced patient discomfort, lowered motion sensitivity, and enabling novel imaging applications. Constrained imaging (CI) seeks to form images from subsampled data by enforcing suitable constraints on the images. A major difference from traditional imaging is that CI shifts a significant part of the image formation burden from the data acquisition system to the reconstruction system. MRI is a valuable tool for diagnostic applications but is popularly used as a qualitative imaging modality. However, MRI can also support the quantification of tissue specific parameters making it valuable for tissue characterization and pathological assessment. The mapping of tissue relaxation parameters like T1 and T2 is an emerging area of interest in quantitative MRI called parameter mapping. Classical parameter mapping experiments are plagued by extremely long scan times due to the need to form images with multiple contrast weightings. Using sub-sampled data for image formation is critical to allow these scans to be performed in a clinically acceptable time. This dissertation studies two different CI models for relaxation mapping. The proposed models seek to impr