Title: An information theoretic approach to compressed sensing and its utility in magnetic resonance imaging
Description: Compressed sensing (CS) is a new mathematical theory that challenges the traditional two-stage process of data acquisition followed by compression where most of the acquired transform-domain data is discarded. Instead, it proposes to acquire and compress the data simultaneously if the signal is known to exhibit transform-domain sparsity. In this thesis, we analyze the limits of CS using information theoretic tools, and devise algorithms that perform close to these limits. We also investigate the use of CS in magnetic resonance imaging (MRI). We first consider a sparse recovery problem that is analogous to CS reconstruction. Using a coding theory approach, we construct a class of measurement matrices and their polynomial-time algebraic decoding algorithms that exactly reconstruct a sparse signal from the optimal number of measurements. Then we consider a more general problem of sparsely representing a vector drawn from a uniform distribution on a hypersphere, and establish a Shannon theoretic converse rate-distortion theorem. In applications, measurement data is corrupted by additive noise. To investigate the effects of noise on CS measurements, we study the information theoretic limits on the number of measurements required to reconstruct a sparse signal from noise-corrupted compressed samples. We also construct a class of measurement matrices, referred to as low density frames, and develop reconstruction algorithms that produce accurate estimates even in the presence of additive noise.

MRI is a non-invasive and radiation-free imaging modality that is used to evaluate various diseases and guide therapies. One of the disadvantages of MRI is lengthy acquisition. Over the past decade, several methods including parallel imaging, partial acquisition, and rapid data acquisition has been proposed to reduce MRI data acquisition. We investigate the utility of CS to accelerate image acquisition in MRI, in particular in cardiac MRI (CMR). We develop and investigate CS techniques suitable for accelerating coronary artery and pulmonary vein MRA, including methods that utilize a probabilistic model for the dependencies exhibited by the wavelet coefficients of medical images, and distributed CS using multiple-coil information. Our results show the feasibility and potential of CS in CMR.