A General Framework for a Class of First Order Primal-dual Algorithms for Tv Minimization

We generalize the primal-dual hybrid gradient (PDHG) algorithm proposed by Zhu and Chan in [M. Zhu, and T. F. Chan, An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration, UCLA CAM Report [08-34], May 2008], draw connections to similar methods and discuss convergence of several special cases and modifications. In particular, we point out a convergence result for a modified version of PDHG that has a similarly good empirical convergence rate for total variation (TV) minimization problems. Its convergence follows from interpreting it as an inexact Uzawa method. We also prove a convergence result for PDHG applied to TV denoising with some restrictions on the PDHG step size parameters. It is shown how to interpret this special case as a projected averaged gradient method applied to the dual functional. We discuss the range of parameters for which the inexact Uzawa method and the projected averaged gradient method can be shown to converge. We also present some numerical comparisons of these algorithms applied to TV denoising, TV deblurring and constrained l1 minimization problems.