A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration

Abstract

We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondi erentiability of the quantity jruj in the de nition of the TV-norm before we apply a linearization technique such as Newton's method. This is accomplished by introducing an additional variable for the ux quantity appearing in the gradient of the objective function, which can be interpreted as the normal vector to the level sets of the image u. Our method can be viewed as a primal-dual method as proposed by Conn and Overton 9] and Andersen 3] for the minimization of a sum of Euclidean norms. In addition to possessing local quadratic convergence, experimental results show that the new method seems to be globally convergent.

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

“…By splitting the variables into suitable "active" and "inactive" sets, the Newton's equation is greatly simplified and can be effectively preconditioned by the FBIP preconditioners. Our algorithm inherits the advantages of the primal-dual method by Chan et al [8]. Compared to primal-only methods, the singularity of the term ∇u/|∇u| is overcome by introducing a dual variable.…”

“…An alternative way studied by Chan et al [8], Carter [4] and Chambolle [5] is to introduce a Fenchel dual variable p of u. This is done using the Fenchel transform to obtain…”

“…As a result, the matrix B is ill-conditioned for reasonable values of α. Therefore, we adopt the primal-dual approach proposed by Chan et al in [8] for solving unconstrained problems.…”

“…Compared to the optimality conditions of the primal problem (2), the singularity in ∇u/|∇u| is removed [8].…”

“…Chan et al [8] proposed to symmetrize the matrix so that CG could be applied. Following this suggestion, we symmetrize the system as:…”