An Upwind Finite-Difference Method for Total Variation–Based Image Smoothing

Abstract

Abstract. In this paper we study finite-difference approximations to the variational problem using the BV smoothness penalty that was introduced in an image smoothing context by Rudin, Osher, and Fatemi. We give a dual formulation for an upwind finite-difference approximation for the BV seminorm; this formulation is in the same spirit as one popularized by the first author for a simpler, less isotropic, finite-difference approximation to the (isotropic) BV seminorm. We introduce a multiscale method for speeding the approximation of both Chambolle's original method and of the new formulation of the upwind scheme. We demonstrate numerically that the multiscale method is effective, and we provide numerical examples that illustrate both the qualitative and quantitative behavior of the solutions of the numerical formulations.AMS subject classifications. 15A15, 15A09, 15A231. Introduction. In an influential paper, Rudin, Osher, and Fatemi [28] suggested using the bounded variation seminorm to smooth images. The functional proposed in their work has since found use in a wide array of problems (see, e.g., [9]), both in image processing and other applications. The novelty of the work was to introduce a method that preserves discontinuities while removing noise and other artifacts.In the continuous setting, the behavior of the solutions of the model proposed in [28] is well understood (see, e.g., [11], [12], and [27]). The qualitative properties of solutions of its discrete versions are not, perhaps, as well known or understood. In this work we study the behavior of solutions of the discrete approach used in, e.g., [13] as well as a new "upwind" variant of this model that better preserves edges and "isotropic" features. We also introduce a multiscale method for improving the initial guess of certain iterative methods for solving discrete variational problems based on the BV variational model.We begin by giving some background on the work in [28]. Working on the unit square I = [0,1] 2 , where the (isotropic) bounded variation seminorm is defined as

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