Collaborative Research: Direct Reconstruction Methods for Optical Tomography and Related Inverse Problems

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The objective of this proposal is to develop direct image reconstruction methods for optical tomography. Optical tomography is an emerging biomedical imaging modality that uses multiply-scattered light as a probe of tissue structure and function. The project will involve three components. (i) We will develop direct reconstruction methods, based on the inverse Born series, for recovering the absorption and diffusion coefficients of the radiative transport equation and the diffusion equation from boundary measurements. We will characterize the convergence, approximation error and stability of the proposed methods. (ii) We will develop fast reconstruction algorithms based on the inverse Born series. The algorithms will be designed to be used with data sets as large as 10^8 measurements. The algorithms will be tested using numerically simulated forward data and experimental data obtained from a noncontact optical tomography system. (iii) We will explore direct inversion methods for related inverse problems that share a common mathematical structure with optical tomography. In particular, we will develop and analyze reconstruction methods, based on inversion of the Born series, for electrical impedance tomography and for the Maxwell equations in the near-field.One of the grand challenges in biomedical imaging is to develop high-resolution methods for imaging of physiological and molecular function. The long-term goal of this research is to contribute to the solution to this problem by developing the necessary mathematical tools. The ability to detect and characterize disease at much earlier stages than is currently possible would be transformative and is critically linked to the development of robust and accurate image reconstruction algorithms for functional imaging. Two graduate students and an undergraduate will be trained to function in this interdisciplinary environment. The results of the research will be broadly disseminated and computer codes will be made publicly available for noncommercial use.