Fast TV-Regularized Large-Scale and Ill-Conditioned Linear Inversion with Application to PPI

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The research of the PIs is focused on the development of new algorithms to generate images from data acquired through the emerging Magnetic Resonance (MR) technology known as Partially Parallel Imaging (PPI). Several fast algorithms for obtaining TV (total variation) regularized images have already been developed, but for efficiency require that the underlying matrices satisfy specific properties, that do not hold for PPI acquired data. Algorithms which can be applied for a general matrix are too slow for real time practical application. The goals of the PIs' research are to both study and compare recently developed fast methods, as well as to develop novel, fast and accurate algorithms suitable for general large-scale ill-conditioned inversion problems. Image reconstruction requires the fast solution of two problems, a sparsification problem known as the basis pursuit denoising problem, and a TV problem. Efficiency for the basis pursuit denoising problem is achieved using active set techniques, while efficiency for the TV problem relies on splittings which reduce the original problem into subproblems that can be solved quickly. Convergence and statistical reliability of the algorithms will be established. The PIs' research will also provide extensions of the algorithms for the solution of related TV-based problems for obtaining more general classes of images.This research has broad impact on Partially Parallel Magnetic Resonance imaging technology. Magnetic resonance imaging is commonly used in radiology to non-invasively visualize the internal structure and function of the body. It provides better contrast between the different soft tissues than most other modalities. Due to the time needed to acquire an image, the cost of this technology can be high. Also motion effects can lead to image degradation. The algorithms to be developed by the PIs will reduce scan time, while improving the accuracy of the reconstructed images. More generally, these algorithms have the potential for impact on applications which require the solution of large, ill conditioned, nonsmooth inversion problems.