The restoration of degraded images is a fundamental and challenging problem in image processing. This problem is ill-posed. The total-variation regularization and its variants are commonly used to convert to a well-posed problem. The resulting regularized model usually has a non-differentiable objective functional, which together with the large dimension of the underlying image makes the minimization theoretically and numerically difficult. Typical numerical treatments for this minimization are indirect in the sense that the methods are developed for a smoothed or dual model of the original model. With this project, the principal investigators use tools from convex analysis to find the solution of the image restoration models directly under a unified framework. The PIs address more general mathematical challenges and computational difficulties associated with the obtained fixed-point formulation. This project provides a fixed-point characterization for the solutions of models with least squares and max norm fidelity terms combined with the total variation regularization term. The study considers images corrupted by Gaussian noise, impulsive Gaussian noise and Poisson noise, which are all of relevance for different applications.
Restoring images from available data is required in a variety of applications including computer tomography; natural resources and pollution control via satellite imaging in environmental sciences; and fingerprint and face recognition in security identification. Advanced mathematical models and efficient computational algorithms for solving this problem are essential. The developed numerical schemes support improved automatic image restoration for these applications. Furthermore, interdisciplinary approaches resulting from the projects enrich upper level undergraduate and graduate curriculum development and teaching activities.