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-L 2 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.

Crouzeix–Raviart Approximation of the Total Variation on Simplicial Meshes

“…This leads again to an exaggeration of the total variation of characteristic functions, and as a result, to a smoothing of the discrete variational solutions, unless adaption is (well) implemented. It is difficult to expect this to improve using higher order approximations [37], as difficulties precisely arise when one needs to approximate discontinuous functions. One direction to improve this is suggested in [44], which proposes to use discontinuous P1 finite elements.…”

Possibility of using local spline functions for estimating displacement values when observing deformations of buildings

“…Capability to improve accuracy lies in the use of a more complex mathematical apparatus. For example, it seems promising to use the Galerkin-Ritz method or spline minimizers ( [10]).…”

A symmetric alternating minimization algorithm for total variation minimization

“…The first inequality follows from the function h is Lipschitz continuous with parameter L h = 1, the second inequality is obtained from ( 25) and (26). Using the fact that DW −1 D T 2 ≤ 1 together with the inequality (27), we know (24) holds. This completes the proof.…”

“…Although more modern and excellent techniques which are specifically tailored to image processing have been developed, total variation (TV) regularization is still a widely used method in the community of applied mathematics and engineering, for its good properties for preserving contours and sharp edges in objects with spatial structures; see for instance [26,27,32,39,41,42,47,63,64]. In this paper, we focus on solving a broad class of TV regularization problems, where the problem shall be reformulated as an unconstrained problem using the penalty decomposition approach [19,34] and then be efficiently solved in the framework of alternating minimization (AM) [11,14,25,37,40,43,45,51,60].…”