Inverse Problems in Medical Imaging: Electromagnetic and Mechanical Property Imaging

Last two decades, much researches in biomedical imaging area deal with electromagnetic or mechanical property imaging instead of anatomical imaging since they manifest structural and pathological conditions of the tissue providing valuable diagnostic information. The corresponding objectives are in visualizing cross-sectional image reconstructions of admittivity, susceptibility, shear modulus distributions inside the human body. These electromagnetic or mechanical property imaging modalities belong to interdisciplinary research areas incorporating mathematical theories (PDEs, harmonic analysis, micro-local analysis), inverse problems and image reconstruction algorithms, image processing (denoising, segmentation, compressed sensing), numerical analysis and error estimates, MR physics and experimental techniques. To achieve these objectives, we need to set up a mathematical framework using the constitutive relation, related physical principles (electromagnetism, elasticity), and available measurement techniques, and it must be taken into account the well-posedness (uniqueness, existence, stability) and uncertainties in boundary conditions and material characterizations. It is often necessary to make various simplifications of reality with sacrificing physical details so that the simplified model is manageable and holds key information to be aimed. In this talk, we discuss these subjects from its mathematical framework to most recent outcomes. We present future directions and challenging issues in mathematics.