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.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Michael H. Steuerwalt
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University of North Carolina at Charlotte
United States
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