Collaborative Research: Inversion of the Broken-Ray Radon Transform and Applications

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This project assembles a team of applied and computational mathematicians and physicists to develop, analyze and implement reconstruction algorithms for the broken-ray Radon transform (BRT) and its generalizations. The BRT describes the propagation of single-scattered particles or waves. This situation is typical of x-ray imaging at clinical energies or optical imaging of nearly transparent tissues and model organisms. The intent of the proposed research is to provide theoretically, numerically justified and practically applicable reconstruction algorithms for the BRT with applications to both x-ray computed tomography and optical tomography in the weak-scattering regime. In particular, the investigators propose to derive and analyze BRT-based scanning protocols and corresponding inversion techniques to reconstruct the absorption and scattering coefficients. Associated scanning protocols, which provide the optimum balance between spatial resolution and stability to noise, are to be developed. In addition, questions of uniqueness and stability (in the scale of Sobolev spaces) are a concern. Techniques of microlocal analysis may be used to characterize the propagation of singularities. Efficient numerical algorithms for inverting the BRT are to be implemented and tested using data derived from radiative transport forward solvers that account for both single- and multiple-scattering, hence connecting the research to the experimental regime. One of the grand challenges in imaging is to address the problem of scattering. It is generally believed that only unscattered particles or waves carry useful information about the medium through which they have traveled. The Investigators aim to show that this is not the case. By making use of mathematical methods and computational approaches that exploit the presence of scattering, they seek to transform a variety of biomedical and security-related x-ray and optical imaging technologies. This research is a collaboration between applied and computational mathematicians and physicists and their work with three graduate students. Broad dissemination of the results of the research is anticipated through publications and generation of publicly available software.