A Fast Numerical Method for Imaging Small Abnormalities in Diffusion Tomography

逆问题中的一个基本挑战是在给定外部源进入该介质的辐射的边界测量值的情况下确定未知介质的组成。研究者和他的合作者利用一种新颖的方法——椭圆系统方法（ESM），从抛物线/扩散方程建模的情况下对这个问题进行了理论和数值研究。ESM涉及到一个带边界条件的耦合椭圆型偏微分方程系统的推导和求解，该系统由时间数据的标准化形式发展而来。然后根据上述归一化溶液重建介质的扩散和吸收系数，从而得到组分。该项目涉及对这种新方法在各种情况下的探索、扩展、改进和测试。它为从分散数据中重建图像的重要问题提供了一种快速、准确的新方法。在国家研究委员会最近发表的一份报告（新兴生物医学成像的数学和物理）中，呼吁开发新的有效的医学成像算法。一个例子是使用光学（近红外激光）方法对小的乳腺癌肿瘤进行早期成像。癌性肿瘤比大部分乳腺组织更能吸收光，因此人们对开发“光学”乳房x光检查的早期检测方法越来越感兴趣。困难在于，与x射线不同，光基本上是直线传播的，而光是高度散射的，这使得重建变得困难。研究人员开发了他们的新方法来解决这个问题，它使用偏微分方程的数值解的方法来实现快速和准确的重建。在其他应用中，这种方法很有可能成为x射线乳房x光检查的光学替代品，或者减少因假阳性而导致的活组织检查的数量。

Klibanov 9704923 A fundamental challenge in inverse problems is the determination of the composition of an unknown medium given the boundary measurements of radiation by an outside source into the medium. The investigator and his collaborator make a theoretical and numerical study of this problem from situations modeled by the parabolic/diffusion equation using a novel approach -- the Elliptic Systems Method (ESM). The ESM involves the derivation and solution of a system of coupled elliptic partial differential equations with boundary conditions developed from a normalized form of the temporal data. The diffusion and absorption coefficients of the medium are then reconstructed from the above normalized solution, yielding the composition. This project involves a number of explorations, extensions, improvements and testing of this new method applied to a variety of situations. It provides a fast and accurate new approach to the important problem of reconstructing images from scattered data. In a recently published report by the National Research Council (Mathematics and Physics of Emerging Biomedical Imaging), a call is made for the development of new effective medical imaging algorithms. An example is the early imaging of small cancerous breast tumors using optical (near infrared laser) methods. Cancerous tumors are more light-absorbing than the bulk of breast tissue, leading to an increased interest in developing early detection methods by "optical" mammography. The difficulty is that unlike x-rays, which travel in essentially straight lines, light is highly scattered, making reconstruction difficult. The investigators develop their new approach to this problem, which uses the methods of numerical solutions of partial differential equations to achieve rapid and accurate reconstructions. This approach, among other applications, has a very good potential to lead to an optical alternative to x-ray mammography exams or to a decrease in the number of biopsies resulting from false positives.