# Approximation of viscosity solution by morphological filters

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

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topPasquignon, 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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