Fast numerical algorithms for total variation based image restoration

• Authors: Tony F. Chan;Mingqiang Zhu
• Venue: Fast numerical algorithms for total variation based image restoration
• Year: 2008

Image restoration models based on total variation (TV) have been popular and successful since their introduction by Rudin, Osher, and Fatemi (ROF) in 1992. The nonsmooth TV seminorm allows them to preserve sharp discontinuities (edges) in an image while removing noise and other unwanted fine scale detail. On the other hand, the TV term, which is the L1 norm of the gradient vector, poses computational challenge in solving those models efficiently. Furthermore, the global coupling of the gradient operator makes the problem extra harder than other L1 minimization problems where the variables under the L 1 norm are separable. In this paper we propose several new algorithms to tackle these difficulties from different perspectives. Numerical experiments show that they are competitive with the existing popular methods and some of them are significantly faster despite of their simplicity. The first algorithm we introduce is a primal-dual hybrid gradient descent method that alternates between the primal and the dual updates. It utilizes the information from both the primal and dual variables and therefore is able to converges faster than the pure primal or pure dual method in the same category. We then proposed gradient projection (GP) methods to solve the dual problem of the ROF model based on the special structure of the dual constraints. We also test variants of GP algorithms with different step selection strategies, including techniques based on the Barzilai-Borwein method. In this same line, a block co-ordinate descent method is proposed for solving the dual ROF problem. The subproblem at each single block can be solved exactly using different techniques. We also propose a basic multilevel optimization framework for the dual formulation, aiming to speedup our solution process for large scale problems. Finally, we study the connections between some existing methods and give an improvement to CGM method based on the primal-dual interior-point algorithms.