Approximation of viscosity solution by morphological filters

Abstract

Abstract. We consider in R 2 all curvature equation ∂u ∂t = |Du|G(curv(u)) where G is a nondecreasing function and curv(u) is the curvature of the level line passing by x. These equations are invariant with respect to any contrast change u → g(u), with g nondecreasing. Consider the contrast invariant operator Tt : uo → u(t). A Matheron theorem asserts that all contrast invariant operator T can be put in a form (T u)(x) = infB∈B sup y∈B u(x + y). We show the asymptotic equivalence of both formulations. More precisely, we show that all curvature equations can be obtained as the iteration of Matheron operators T n h where h → 0 and n → ∞ with nh = t.Résumé. Nous considérons dans R 2 leséquations de courbure ∂u ∂t = |Du|G(curv(u)) où G est une fonction croissante et curv(u) représente la courbure de la ligne de niveau passant par le point x. Ceś equations sont invariantes pour tout changement de contraste u → g(u), avec g croissante. D'autre part, Matheron a prouvé que tout opérateur invariant par changement de contraste Tt : u0 → u(t) peut s'exprimer comme un schéma inf-sup (T u)(x) = infB∈B sup y∈B u(x + y). Nous démontrons l'équivalence asymtotique de ces deux approches. Plus précisément, nous prouvons que la solution de viscosité de touteéquation de courbure est la limite d'opérateurs de Matheron itérés T n h lorsque h → 0 et n → ∞ avec nh = t.

On the approximation of front propagation problems with nonlocal terms

“…More general schemes are given in Crandall and Lions [15], and their convergence is proved by the same method. To approach the viscosity solutions, we can also use inf-sup schemes, see [11] (for the mean curvature motion), [10,19] (for affine invariant scale space motion), [24] (for general motion depending on the curvature). In image processing, these inf-sup schemes are quite important because they have some invariance properties, see [19].…”

A Convolution-Thresholding Approximation of Generalized Curvature Flows

“…In what follows, we also use another result by Ishii and Souganidis [22] concerning locally uniform perturbations of the right-hand side of the equation. One can restate this result in the case of (1.2) as follows (see [28] …”

“…The result by Ishii and Souganidis presented in [22] can be restated in terms of the level-set equation (see [28] …”

“…In order to base the proof of our main result on this theorem, we follow Pasquignon [28] and restate it in terms of (1.2).…”

Second-order PDE’s and deterministic games

“…Similar semi-discrete schemes had been considered in the literature on computer vision (e.g. [2,13,14]), and in work on numerical schemes for computing viscosity solutions of second-order PDE's [5].…”