Discrete Total Variation with Finite Elements and Applications to Imaging

Abstract

The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TV-seminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart-Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV-L 2 and TV-L 1 , can be implemented in low and higher-order finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids.1. Introduction. The total-variation (TV)-seminorm | · | T V is ubiquitous as a regularizing functional in image analysis and related applications; see for instance [50,28,17,14]. When Ω ⊂ R 2 is a bounded domain, this seminorm is defined aswhere s ∈ [1, ∞], s * = s s−1 denotes the conjugate of s and | · | s * is the usual s * -norm of vectors in R 2 . Frequent choices include s = 2 (the isotropic case) and s = 1, see Figure 1.1.It has been observed in [21] that "the rigorous definition of the TV for discrete images has received little attention." In this paper we propose and analyze a discrete analogue of (1.1) for functions u belonging to a space DG r (Ω) or CG r (Ω) of globally discontinuous or continuous finite element functions of polynomial degree 1 0 ≤ r ≤ 4 on a geometrically conforming, simplicial triangulation of Ω, consisting of triangles T and interior edges E. 2 * This work was supported by DFG grants HE 6077/10-1 and SCHM 3248/2-1 within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization), which is gratefully acknowledged.

Additive Schwarz Methods for Convex Optimization as Gradient Methods

“…where n e denotes the unit outer normal to e. In [11,13], the following discretization of (4.12) constructed by replacing the solution space and the constraint set by S h (Ω) and K h , respectively, was proposed:…”

Total Variation Isoperimetric Profiles

“…in the case of geographic data accounting for the curvature and topography of the earth. To compute TV profiles in the presence of curvature, we easily could use a finite element formulation on triangulated surfaces, as suggested in [28]. • Our discussion focuses on total variation as an objective function, but we could attempt to generalize the construction of our profile by considering higher-order measurements popular in mathematical imaging like total generalized variation (TGV) [11].…”

A Finite Element Nonoverlapping Domain Decomposition Method with Lagrange Multipliers for the Dual Total Variation Minimizations

“…It means that the space X h may not be considered as a correct space for solving (1.2). To avoid this uncomfortable situation, one may use higher order Raviart-Thomas elements to discretize the dual formulation [1,15]. Nevertheless, thanks to Proposition 3.4, one can construct a sequence accumulating at u * in the L 1 (Ω)-topology from {u h * } h>0 .…”