A Primal-Dual Active-Set Method for Non-Negativity Constrained Total Variation Deblurring Problems

This paper studies image deblurring problems using a total variation-based model, with a non-negativity constraint. The addition of the non-negativity constraint improves the quality of the solutions, but makes the solution process a difficult one. The contribution of our work is a fast and robust numerical algorithm to solve the non-negatively constrained problem. To overcome the nondifferentiability of the total variation norm, we formulate the constrained deblurring problem as a primal-dual program which is a variant of the formulation proposed by Chan, Golub, and Mulet for unconstrained problems. Here, dual refers to a combination of the Lagrangian and Fenchel duals. To solve the constrained primal-dual program, we use a semi-smooth Newton's method. We exploit the relationship between the semi-smooth Newton's method and the primal-dual active set method to achieve considerable simplification of the computations. The main advantages of our proposed scheme are: no parameters need significant adjustment, a standard inverse preconditioner works very well, quadratic rate of local convergence (theoretical and numerical), numerical evidence of global convergence, and high accuracy of solving the optimality system. The scheme shows robustness of performance over a wide range of parameters. A comprehensive set of numerical comparisons are provided against other methods to solve the same problem which show the speed and accuracy advantages of our scheme.