Duality-based algorithms for total-variation-regularized image restoration

• Published in: Computational optimization and applications 2010-11, Vol.47 (3), p.377-400
• Main Authors: Zhu, Mingqiang, Wright, Stephen J., Chan, Tony F.
• Format: Article
• Language: English
• Subjects: Algorithms; Applied mathematics; Convex and Discrete Geometry; Image processing systems; Lagrange multiplier; Management Science; Mathematics; Mathematics and Statistics; Operations Research; Operations Research/Decision Theory; Optimization; Optimization algorithms; Regularization methods; Statistics; Studies
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-008-9225-2

Image restoration models based on total variation (TV) have become popular since their introduction by Rudin, Osher, and Fatemi (ROF) in 1992. The dual formulation of this model has a quadratic objective with separable constraints, making projections onto the feasible set easy to compute. This paper proposes application of gradient projection (GP) algorithms to the dual formulation. We test variants of GP with different step length selection and line search strategies, including techniques based on the Barzilai-Borwein method. Global convergence can in some cases be proved by appealing to existing theory. We also propose a sequential quadratic programming (SQP) approach that takes account of the curvature of the boundary of the dual feasible set. Computational experiments show that the proposed approaches perform well in a wide range of applications and that some are significantly faster than previously proposed methods, particularly when only modest accuracy in the solution is required.