Finite-differences discretizations of the mumford-shah functional

Antonin Chambolle

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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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.

How to cite

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Chambolle, 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 -

Citations in EuDML Documents

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  1. Blaise Bourdin, Antonin Chambolle, Design-dependent loads in topology optimization
  2. Blaise Bourdin, Antonin Chambolle, Design-dependent loads in topology optimization
  3. Andrea Braides, Anneliese Defranceschi, Enrico Vitali, A compactness result for a second-order variational discrete model
  4. Andrea Braides, Anneliese Defranceschi, Enrico Vitali, A compactness result for a second-order variational discrete model
  5. Andrea Braides, Giuseppe Riey, A variational model in image processing with focal points

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