Galerkin method with splines for total variation minimization

Abstract

Total variation smoothing methods have been proven to be very efficient at discriminating between structures (edges and textures) and noise in images. Recently, it was shown that such methods do not create new discontinuities and preserve the modulus of continuity of functions. In this paper, we propose a Galerkin–Ritz method to solve the Rudin–Osher–Fatemi image denoising model where smooth bivariate spline functions on triangulations are used as approximating spaces. Using the extension property of functions of bounded variation on Lipschitz domains, we construct a minimizing sequence of continuous bivariate spline functions of arbitrary degree, d, for the TV- L2 energy functional and prove the convergence of the finite element solutions to the solution of the Rudin, Osher, and Fatemi model. Moreover, an iterative algorithm for computing spline minimizers is developed and the convergence of the algorithm is proved.