Analysis of total variation flow and its finite element approximations

Abstract

Abstract. We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as h, k → 0, and to the total variation gradient flow problem as h, k, ε → 0 in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the mesh relation k = O(h 2 ). In particular, it is shown that all error bounds depend on 1 ε only in some lower polynomial order for small ε.

Numerical aspects of TV flow

“…The equivalence to TV regularisation, for instance, can no longer be expected to be true. Feng and Prohl have shown that given a domain Ω ⊂ R N , one can establish, for almost every time t in a given interval (0, T ), strong convergence of a weak solution u β (t) to u(t) in the L p -norm for any p ∈ [1, N/(N − 1)) [8].…”

“…It requires no additional parameters, it is well-posed [3,5,8], scaleinvariant [2], it preserves the shape of some objects [5], and it leads to constant signals in finite time [4]. In [21] it has further been shown that in the space-discrete 1-D case, TV flow with initial data f is equivalent to the so-called TV regularisation [1,6,17], where one minimises the energy functional…”

“…Sophisticated numerical ways of dealing with such nonlinear singular diffusion equations have been developed in the past, see e.g. [7,8,21]. They are either based on regularizing strategies [8] or on the simulation of the analytic behaviour of TV flow [7,21].…”

“…[7,8,21]. They are either based on regularizing strategies [8] or on the simulation of the analytic behaviour of TV flow [7,21]. The goal of the present paper is to discuss a number of numerical problems that can arise with such discretisations, and to investigate if similar problems arise in the context of regularized versions of (1).…”

Convection and total variation flow

“…Unfortunately, finite volumes are not suitable for the approximation of the total variation flow: indeed, if a sequence (u k ) k∈N of piecewise constant functions converges to u in L 1 , the total variation of u k does not converge in general to the total variation of u (see Bělík & Luskin (2003) for an example). The total variation flow must be approximated in W 1,1 -conforming discrete spaces, such as P 1 finite element spaces (Bartels, 2012;Feng & Prohl, 2003;Feng et al, 2005). Numerical schemes combining finite volumes and finite element schemes have already been considered for scalar conservation laws with a diffusion term (Feistauer et al, 1999) and for degenerate parabolic equations (Eymard et al, 2006).…”