# Approximation of viscosity solution by morphological filters

• Volume: 4, page 335-359
• ISSN: 1292-8119

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## Abstract

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We consider in ${ℝ}^{2}$ all curvature equation $\frac{\partial u}{\partial t}=|Du|G\left(\mathrm{curv}\left(u\right)\right)$ 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 ${T}_{t}:{u}_{o}\to u\left(t\right)$. A Matheron theorem asserts that all contrast invariant operator T can be put in a form $\left(Tu\right)\left(𝐱\right)={inf}_{B\in ℬ}{sup}_{𝐲\in B}u\left(𝐱+𝐲\right)$. 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 ${T}_{h}^{n}$ where h → 0 and n → ∞ with nh=t.

## How to cite

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Pasquignon, Denis. "Approximation of viscosity solution by morphological filters." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 335-359. <http://eudml.org/doc/197342>.

@article{Pasquignon2010,
abstract = { We consider in $\mathbb\{R\}^2$ all curvature equation $\frac\{\partial u\}\{\partial t\}=|Du|G(\{\rm curv\}(u))$ 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 $T_t: u_o\to u(t)$. A Matheron theorem asserts that all contrast invariant operator T can be put in a form $(Tu)(\{\bf x\}) = \inf_\{B\in \{\cal B\}\}\sup_\{\{\bf y\}\in B\} u(\{\bf x\}+\{\bf y\})$. 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 $T_h^n$ where h → 0 and n → ∞ with nh=t. },
author = {Pasquignon, Denis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Viscosity solutions; inf-sup scheme; morphological filter.; morphological filters; viscosity solution; curvature equation; Matheron operators},
language = {eng},
month = {3},
pages = {335-359},
publisher = {EDP Sciences},
title = {Approximation of viscosity solution by morphological filters},
url = {http://eudml.org/doc/197342},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Pasquignon, Denis
TI - Approximation of viscosity solution by morphological filters
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 335
EP - 359
AB - We consider in $\mathbb{R}^2$ all curvature equation $\frac{\partial u}{\partial t}=|Du|G({\rm curv}(u))$ 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 $T_t: u_o\to u(t)$. A Matheron theorem asserts that all contrast invariant operator T can be put in a form $(Tu)({\bf x}) = \inf_{B\in {\cal B}}\sup_{{\bf y}\in B} u({\bf x}+{\bf y})$. 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 $T_h^n$ where h → 0 and n → ∞ with nh=t.
LA - eng
KW - Viscosity solutions; inf-sup scheme; morphological filter.; morphological filters; viscosity solution; curvature equation; Matheron operators
UR - http://eudml.org/doc/197342
ER -

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