Divergent X-ray tomography reconstruction and optimisation

In diverse fields such as medical imaging, astrophysics, geophysics, or material study, a common challenge exists: reconstructing the internal volume of an object using only physical measurements taken from its exterior or surface. This scientific approach is called tomography, and is the foundational concept that motivates this work. Specifically, the focus of my project is divergent (or cone beam) X-ray tomography, a technique used in medical imaging and material mechanics. Solving a tomography problems amounts to solving an inverse problem, where the ill-conditioning of the measurement operator, specifically the X-ray transform operator, and the inherent ill-posedness of inverse problems, in the sense of Hadamard, necessitate the integration of advanced optimisation techniques. Tackling these optimisation problems is highly challenging in practice due to the implementation of the X-ray transform operator and its mathematical adjoint, the backprojection operator: combination of High Performance Computing (HPC) and exact adjoint match between forward and backward operators has not been achieved in the currently available state-of-the-art packages. During this project, the impact of an adjoint mismatch on the convergence of optimisation algorithms is addressed. Additionally, the implementation of forward and matched adjoint operators in both 2D and 3D is explored. Given that adjoint mismatches are common in many state-of-the-art tomographic libraries, we explore whether it is worthwhile to approximate the adjoint instead of ensuring an exact match with the forward process for the sake of computational speed. Upon implementation, reconstruction techniques are investigated, from Bayesian formulations and prior hypotheses, to their corresponding representation problems in optimisation. Both synthetic and real data, in 2D and 3D, will be reconstructed using advanced techniques. Furthermore, we briefly explore novel applications unlocked with having consistent forward/adjoint codes, namely uncertainty quantification in scenarios with log-concave posterior distributions p(x|y) of the reconstructed image x, given the measurements y. This exploration becomes particularly relevant when using a Maximum-A-Posteriori (MAP) estimate.