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

pi的研究重点是开发新的算法，从通过新兴的磁共振（MR）技术获得的数据中生成图像，称为部分并行成像（PPI）。已经开发了几种用于获得TV（总变差）正则化图像的快速算法，但为了效率，要求底层矩阵满足特定的属性，而这些属性不适用于PPI获取的数据。可以应用于一般矩阵的算法对于实时实际应用来说太慢了。pi的研究目标是研究和比较最近开发的快速方法，以及开发适用于一般大规模病态反演问题的新颖，快速和准确的算法。图像重建需要快速解决两个问题，一个是被称为基追求去噪的稀疏化问题，另一个是电视问题。基跟踪降噪问题的效率是利用主动集技术实现的，而电视问题的效率依赖于将原始问题分解成可以快速解决的子问题的分裂。建立算法的收敛性和统计可靠性。pi的研究还将提供算法的扩展，用于解决相关的基于电视的问题，以获得更一般的图像类别。本研究对部分平行磁共振成像技术具有广泛的影响。磁共振成像在放射学中常用来无创地观察身体的内部结构和功能。它提供了更好的对比不同的软组织比大多数其他方式。由于获取图像所需的时间，这种技术的成本可能很高。此外，运动效果可能导致图像退化。pi将开发的算法将减少扫描时间，同时提高重建图像的精度。更一般地说，这些算法对需要解决大型、病态、非光滑反演问题的应用有潜在的影响。

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.