Inverse Problems with Internal Data

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Advances in imaging technologies such as computed tomography (CT), magnetic resonance imaging (MRI) and super resolution microscopy 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 biomedical imaging modalities in which new novel measurements play a key role. 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 principal investigator has research interests in applied mathematics and theoretical physics. He is also a physician. Graduate students will be trained to function in this interdisciplinary environment.The objective of this project is to investigate inverse problems with internal data that arise in biomedical optical imaging. Two classes of problems will be considered. (i) Teh PI will develop mathematically-justified methods for imaging below the diffraction limit of resolution, also known as superresolution imaging. The proposed work includes both analysis of the inverse scattering problem with internal sources and the development of reconstruction algorithms. The algorithms will be tested and characterized using data from physically realistic numerical simulations. (ii) The PI will study inverse problems that arise in acousto-optic imaging. The research will focus on the regime of coherent multiple scattering which leads to considerable mathematical simplifications compared to incoherent imaging. In particular, the PI will develop reconstruction methods for recovering the absorption and scattering coefficients of the radiative transport equation from coherent acousto-optic measurements. Finally, the role of improvements in modeling of the acousto-optic effect on image reconstruction will be investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.