Inverse Problems in Optical Imaging

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Advances in imaging technologies such as computed tomography (CT) and magnetic resonance imaging (MRI) have transformed the practice of clinical medicine and basic biomedical research. Although the development of such technologies is well known to depend upon progress in physics and engineering, it is less well known that applied and computational mathematics has also played an essential role. This research project studies mathematical questions that arise in new medical imaging modalities that make use of near-infrared light either alone, or in combination with ultrasound. The research will study novel mathematical algorithms that will lead to improvements in optical imaging both with respect to resolution (visualizing structures at smaller scales) and computational speed. In particular, the project aims to devise robust and accurate image reconstruction algorithms that may lead to the detection and characterization of disease at much earlier stages than is currently possible.The objective of this research is to investigate inverse problems that arise in biomedical optical imaging. The project involves two components. (i) Development of mathematically-justified image reconstruction algorithms for optical tomography with diffuse light. The analysis will be carried out within the framework of radiative transport theory. The project aims to develop reconstruction methods based on topological reduction of the inverse Born series, for recovering the absorption and scattering coefficients of the radiative transport equation from boundary measurements, and to characterize the convergence and approximation error of these methods. It is planned to employ these results to develop and test fast reconstruction algorithms suitable for use with a noncontact optical tomography system. (ii) Study of inverse problems that arise in acousto-optic imaging. This recently developed imaging modality combines the high-resolution of ultrasound imaging with the physiologically important contrast of optical imaging. The work includes theoretical analysis of related inverse scattering problems, together with numerical tests of associated image reconstruction algorithms. In particular, the project aims to develop reconstruction methods for recovering the absorption and scattering coefficients of the radiative transport equation from acousto-optic measurements.