Fast Algorithms for Tomography

层析成像，或从不同方向的线积分的集合（称为Radon变换）重建一个对象是一个研究得很好的问题。也许最重要的是，它是大多数关键诊断成像模式的基本原理，包括x射线断层扫描（CT）， PET和SPECT，某些形式的MRI，以及新兴技术，如电阻抗断层扫描（EIT）和光学断层扫描。断层扫描也是科学和工程中许多其他问题和应用的基本原理——从亚细胞结构的电子显微镜和制造业中的无损评估（NDE），到地球物理勘探和环境监测，再到合成孔径雷达（SAR）的遥感。用于解决断层问题的最广泛使用的技术被称为滤波反向投影（FBP），并且在医用x射线CT领域中无处不在。但是FBP方法是缓慢的，新技术正在以FBP方法和现有计算机技术无法处理的速度进行数据采集。这项研究涉及到解决断层成像问题的新方法，使用了FBP所需努力的很小一部分。研究者正在研究一些有前途的技术来加速FBP重建算法。他发现Radon变换可以用一种特殊的方式分解，这种新的分解导致了一个O（N^2 log N）的方法（用于从N个投影重建一个N X N的图像）来解决层析问题，而不是FBP方法，它是一个O（N^3）的方法。对于大N，通常在第三代或第四代x射线CT系统中遇到，这代表了近两个数量级的加速。这项研究还涉及使用这种分解来加速解决更复杂的层析问题，其中数据丢失或扭曲。

FAST ALGORITHMS FOR TOMOGRAPHYPI: Yoram BreslerTomography, or the reconstruction of an object from a collection of its line integrals from various directions (known as its Radon transform) is a well-studied problem. Perhaps most importantly, it is the principle underlying most of the key diagnostic imaging modalities including x-ray Tomography (CT), PET and SPECT, certain forms of MRI, and emerging techniques such as electrical impedance tomography (EIT) and optical tomography. Tomography is also the fundamental principle in numerous other problems and applications in science and engineering -- from electron microscopy of subcellular structures and nondestructive evaluation (NDE) in manufacturing, through geophysical exploration and environmental monitoring, to remote sensing by synthetic aperture radar (SAR). The most widely used technique for solving the tomographic problem is known as Filtered Backprojection (FBP), and is ubiquitous in the realm of medical x-ray CT. But the FBP method is slow, and new technologies are making data acquisition possible at rates that cannot be handled by the FBP method and existing computer technology. This research involves new ways of solving the tomography problem that use a very small fraction of the effort required by the FBP.The investigator is studying a number of promising techniques for accelerating the FBP reconstruction algorithm. He has discovered that the Radon transform can be factored in a special way, and that this new factorization leads to an O(N^2 log N) method (for reconstructing an N X N image from N projections) for solving the tomographic problems, as opposed to the FBP method, which is an O(N^3) method. For large N, as are typically encountered in third or fourth generation x-ray CT systems, this represents a speedup of nearly two orders of magnitude. This research also involves the use of this factorization to accelerate solutions to more complicated tomographic problems, in which data are missing or distorted.