Taming unstable inverse problems: Mathematical routes toward high-resolution medical imaging modalities

This thesis explores two mathematical routes that make the transition from some severely ill-posed parameter reconstruction problems to better-posed versions of them. The general introduction starts by defining what we mean by an inverse problem and its theoretical analysis. We then provide motivations that come from the field of medical imaging. The first part consists in the analysis of an inverse problem involving the Boltzmann transport equation, with applications in Optical Tomography. There we investigate the reconstruction of the spatially-dependent part of the scattering kernel, from knowledge of angularly averaged outgoing traces of transport solutions and isotropic boundary sources. We study this problem in the stationary regime first, then in the time-harmonic regime. In particular we show, using techniques from functional analysis and stationary phase, that this inverse problem is severely ill-posed in the former setting, whereas it is mildly ill-posed in the latter. In this case, we deduce that making the measurements depend on modulation frequency allows to improve the stability of reconstructions. In the second part, we investigate the inverse problem of reconstructing a tensor-valued conductivity (or diffusion) coefficient in a second-order elliptic partial differential equation, from knowledge of internal measurements of power density type. This problem finds applications in the medical imaging modalities of Electrical Impedance Tomography and Optical Tomography, and the fact that one considers power densities is justified in practice by assuming a coupling of this physical model with ultrasound waves, a coupling assumption that is characteristic of so-called hybrid medical imaging methods. Starting from the famous Calderon's problem (i.e. the same parameter reconstruction problem from knowledge of boundary fluxes of solutions), and recalling its lack of injectivity and severe instability, we show how going from Dirichlet-to-Neumann data to considering the power density operator leads to reconstruction of the full conductivity tensor via explicit inversion formulas. Moreover, such reconstruction algorithms only require the loss of either zero or one derivative from the power density functionals to the unknown, depending on what part of the tensor one wants to reconstruct. The inversion formulas are worked out with the help of linear algebra and differential geometry, in particular calculus with the Euclidean connection. The practical pay-off of such theoretical improvements in injectivity and stability is twofold: (i) the lack of injectivity of Calderà³n's problem, no longer existing when using power density measurements, implies that future medical imaging modalities such as hybrid methods may make anisotropic properties of human tissues more accessible; (ii) the improvements in stability for both problems in transport and conductivity may yield practical improvements in the resolution of images of the reconstructed coefficients