A Fast Numerical Method for Imaging Small Abnormalities in Diffusion Tomography

克利巴诺夫 反问题中的一个基本挑战是确定未知介质的成分，给定外部源进入介质的辐射的边界测量。 研究者和他的合作者从抛物/扩散方程建模的情况下使用一种新的方法-椭圆系统方法（ESM）对这个问题进行了理论和数值研究。 ESM涉及的推导和解决方案的耦合椭圆型偏微分方程的边界条件开发的时间数据的归一化形式的系统。 然后从上述归一化的溶液中重建介质的扩散和吸收系数，得到组成。 该项目涉及对这一适用于各种情况的新方法的一些探索、扩展、改进和测试。 它提供了一个快速和准确的新途径，从散乱数据重建图像的重要问题。 在国家研究理事会（新兴生物医学成像的数学和物理学）最近发表的报告中，呼吁开发新的有效医学成像算法。 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，这使得人们对通过“光学”乳房X光检查开发早期检测方法的兴趣增加。 困难在于，与基本上以直线传播的x射线不同，光是高度散射的，这使得重建困难。 研究人员开发了解决这个问题的新方法，该方法使用偏微分方程的数值解方法来实现快速准确的重建。 除其他应用外，这种方法具有很好的潜力，可以导致X射线乳房摄影检查的光学替代方案，或者减少由假阳性导致的活检数量。

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.