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.

Project Report

The research focused on the development of novel algorithms for solving certain classes of non-smooth convex optimization problems. A specific example of the class of problems to which the research applies is the problem of image reconstruction in parallel magnetic resonance imaging (PMRI). This medical imaging technique provides better contrast between soft tissues than most other modalities, but due to the time needed to acquire an image, the cost of this technology can be high, and motion effects during the scan can lead to image degradation. Scanner manufacturers are trying to speed up the technology by only recording part of the signal response. To recover the image, a complex mathematical inversion problem must be solved involving a large but incomplete data set. The project has developed techniques that can solve this inversion problem much faster than previous techniques, while also providing high quality images. The research demonstrated that the partial data collected by a parallel magnetic resonance scanner uniquely determines the image. The algorithms that were developed for PMRI were shown to be convergent and applicable to a broad class of problems. In addition, results on the rate of convergence were obtained. All the algorithms are publically available for both researchers and companies. An image data base was developed and posted on the following web page so that anyone can test an algorithm on a standard data set and compare to the work of others: Graduate students participated in all aspects of the research. Altogether, 8 PhD students have been involved in the research. The graduating students are continuing their research either in universities or companies.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Junping Wang
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University of Florida
United States
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