Prospective and Retrospective Retooling of Magnetic Resonance Imaging
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At its heart, signal processing can be broken down into two broad categories based on a single core principle: attaining a signal based objective (higher resolution, reduction in noise interference, signal based statistical inference, signal separation / characteristic manipulation etc.). The goal then of signal processing is to achieve this objective either once the data has been acquired (herein referred to as retrospective reconstruction) or by designing the system to achieve the desired objective (herein referred to as prospective reconstruction). In this work, we consider retrospective and prospective medical image reconstruction with special attention to magnetic resonance imaging. Convex relaxations of sparse priors have given birth to strident improvements in the way signals are recovered from under-determined systems. In the retrospective vein of image reconstruction, we seek to extend the benefits afforded by sparse regularization by invoking non-convex sparse priors for inverse problems. We develop a novel algorithmic solution, both in its design and computational efficiency, for analysis based non-convex sparse priors in tight frames. Theoretically, we show that our algorithm is guaranteed to converge to a local minimum based on the non-convex objective function being examined. Numerically, this class of non-convex regularized linear inverse problems has a range of practical applications: under-determined signal recovery, single image super-resolution and image denoising. In each of these applications, we demonstrate that our non-convex formulation can outperform both convex and non-convex state of the art counterparts. To truly achieve a desired objective, both the data acquisition methodology and reconstruction pipeline must be jointly designed. Speed of imaging is of great concern in magnetic resonance imaging (MRI). In MR systems, faster imaging translates to a range of benefits from increased temporal / spatial resolution to reduced motion artifacts. In the prospective approach, we develop a novel MR data acquisition and reconstruction framework to accelerate MR imaging beyond what is currently commercially available. This is done by leveraging phase encoding gradients during the data acquisition process thereby affording better control over the spectral distribution of the underlying encoding operator. In doing so, we recover signals at higher acceleration factors than the current state of the art method. Additionally, at state of the art achievable acceleration factors, we demonstrate that our method can recover the underlying signal with greater fidelity. We demonstrate the viability of our method through proof of concept 1D simulations and 3D phantom data acquired on a 3T human scanner. Finally, our reconstruction methodology forms a generalized framework seamless transition between 1D and 3D MR image reconstruction.