Randomized Edge-preserving Methods for Inverse Problems

Inverse problems arise in numerous applications, including image deblurring, super-resolution imaging, and X-ray tomography. Since direct inversion methods are often dominated by noise and errors in the data, regularization techniques are required to obtain stable solutions. In particular, regularization with L1/L2 norms has recently been applied to X-ray tomography and solved within the Alternating Direction Method of Multipliers (ADMM) framework. This approach promotes sparsity, making it well-suited for signal reconstruction tasks where edge preservation is crucial. Unlike Total Variation (L1) minimization, L1/L2 minimization is non-convex and, therefore, more challenging to optimize.

Computational efficiency and storage constraints become critical concerns when datasets are large. Row-action methods, such as the Kaczmarz method, enable iterative solutions using subsets of the data, reducing memory requirements. In this dissertation, a randomized approach to edge-preserving regularization in inverse problems is developed, leveraging Kaczmarz-type methods to solve large-scale problems iteratively. More specifically, the Sampled Tikhonov framework is extended to generalized Tikhonov regularization and used within the ADMM solver. Furthermore, the performance of L1 and  L1/L2 minimization techniques implemented via ADMM and the Variable Projected Augmented Lagrangian (VPAL) method is investigated. Additionally, these methods using various inner L2 solvers, including the Sampled Generalized Tikhonov method, are also studied.