Coupled-physics imaging methods and geodesic X-ray transforms

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Improving feature detection is a constant challenge in medical and geophysical imaging, with tremendous benefits to society such as the adequate monitoring of medical conditions or natural resources, and the imaging of previously "invisible" features. This project presents two approaches leading to such improvements. A first approach is the design and theoretical analysis of imaging methods exploiting physical phenomena in new ways, leading to imaging strategies with both improved contrast and resolution, and providing access to new features such as anisotropic properties of muscle fibers. A second approach is to provide distortion-free reconstructions and accurately located inclusions in human bodies or the Earth's crust. This is achieved by using more realistic descriptions (specifically, spatially varying) of the propagation speed of the waves used to probe the medium of interest. The model considered becomes augmented with new technicalities, where additional phenomena (e.g., caustics) can occur and open mathematical questions abound. The present project provides theoretical and practical imaging answers to some of these new problems. This project focuses on the theoretical and mathematical understanding of: (i) coupled-physics inverse problems, which often consist of parameter identification problems in partial differential equations (PDE) using internal measurements; and (ii) geodesic X-ray transforms (a generalization of the famous Radon transform, in two dimensions), with applications to X-Ray Tomography in media with variable index of refraction. In the former topic, methods from analysis and PDEs are used to derive reconstruction algorithms for various models of increasing complexity, including additional constitutive parameters relevant for medical imaging, and including a transition from scalar to systems of PDEs. In the latter topic, the investigator uses tools of integral geometry, Fourier analysis, and PDEs to provide statements about injectivity/stability (or lack thereof), reconstruction formulas and their numerical validation, for the tensor tomography problem on Riemannian surfaces. Generalizations to higher dimensions, attenuated transforms and other types of flows (e.g., magnetic and thermostat flows) are also considered.