# Finite-differences discretizations of the mumford-shah functional

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 33, Issue: 2, page 261-288
- ISSN: 0764-583X

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topChambolle, Antonin. "Finite-differences discretizations of the mumford-shah functional." ESAIM: Mathematical Modelling and Numerical Analysis 33.2 (2010): 261-288. <http://eudml.org/doc/197581>.

@article{Chambolle2010,

abstract = {
About two years ago, Gobbino [21]
gave a proof of a De Giorgi's conjecture
on the approximation of the Mumford-Shah energy by means of
finite-differences based non-local functionals.
In this work, we introduce a discretized version of De Giorgi's
approximation, that may be seen as a generalization of
Blake and Zisserman's “weak membrane” energy
(first introduced in the image segmentation framework).
A simple adaptation of Gobbino's results allows us to
compute the Γ-limit of this discrete functional as
the discretization step goes to zero; this generalizes a previous
work by the author on the “weak membrane” model [10].
We deduce how to design in a systematic way discrete
image segmentation functionals with “less anisotropy” than
Blake and Zisserman's original energy, and we show in
some numerical experiments how it improves the method.
},

author = {Chambolle, Antonin},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Free discontinuity problems; Γ-convergence;
special bounded variation (SBV) functions;
finite differences; image processing.; finite-differences discretizations; convergence; Mumford and Shah's functional; Blake and Zisserman's weak membrane energy; discrete functional; segmentations of images},

language = {eng},

month = {3},

number = {2},

pages = {261-288},

publisher = {EDP Sciences},

title = {Finite-differences discretizations of the mumford-shah functional},

url = {http://eudml.org/doc/197581},

volume = {33},

year = {2010},

}

TY - JOUR

AU - Chambolle, Antonin

TI - Finite-differences discretizations of the mumford-shah functional

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 33

IS - 2

SP - 261

EP - 288

AB -
About two years ago, Gobbino [21]
gave a proof of a De Giorgi's conjecture
on the approximation of the Mumford-Shah energy by means of
finite-differences based non-local functionals.
In this work, we introduce a discretized version of De Giorgi's
approximation, that may be seen as a generalization of
Blake and Zisserman's “weak membrane” energy
(first introduced in the image segmentation framework).
A simple adaptation of Gobbino's results allows us to
compute the Γ-limit of this discrete functional as
the discretization step goes to zero; this generalizes a previous
work by the author on the “weak membrane” model [10].
We deduce how to design in a systematic way discrete
image segmentation functionals with “less anisotropy” than
Blake and Zisserman's original energy, and we show in
some numerical experiments how it improves the method.

LA - eng

KW - Free discontinuity problems; Γ-convergence;
special bounded variation (SBV) functions;
finite differences; image processing.; finite-differences discretizations; convergence; Mumford and Shah's functional; Blake and Zisserman's weak membrane energy; discrete functional; segmentations of images

UR - http://eudml.org/doc/197581

ER -

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