Abstract.We investigate the approximation of the evolution of compact hypersurfaces of R N depending, not only on terms of curvature of the surface, but also on non local terms such as the measure of the set enclosed by the surface.Mathematics Subject Classification. 65M12, 35K22.
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.
The problem of computation of a "good" skeleton is still open because several somehow opposite requirements must be satisfied: the skeleton must represent the connected components of the shape ( connectivity requirement). the skeleton must be noise insensitive. 0 the computation must be as independent as possible of the grid effects.We discuss several classical "thinning" algorithms and show that they can be reinterpreted as partial differential equations governing the shape evolution. We propose the best adapted partial differential equation to the computation of the skeleton, and define a reliable numerical scheme to compute it. Experiments and comparison of methods close the paper.
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