Feasibility-based Fixed Point Networks
This work addresses the need for more effective and stable signal recovery in inverse problems, such as medical imaging, by combining theoretical guarantees with data-driven methods, though it is incremental in nature.
The paper tackles the problem of recovering signals from noisy measurements in inverse problems by fusing data-driven regularization with convex feasibility, resulting in performance improvements over standard TV-based methods and comparable neural network approaches in CT reconstruction.
Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable recovery with respect to data perturbations. Hand-chosen analytic regularization can yield desirable theoretical guarantees, but such approaches have limited effectiveness recovering signals due to their inability to leverage large amounts of available data. To this end, this work fuses data-driven regularization and convex feasibility in a theoretically sound manner. This is accomplished using feasibility-based fixed point networks (F-FPNs). Each F-FPN defines a collection of nonexpansive operators, each of which is the composition of a projection-based operator and a data-driven regularization operator. Fixed point iteration is used to compute fixed points of these operators, and weights of the operators are tuned so that the fixed points closely represent available data. Numerical examples demonstrate performance increases by F-FPNs when compared to standard TV-based recovery methods for CT reconstruction and a comparable neural network based on algorithm unrolling.