Numerical study of a new global minimizer for the Mumford-Shah functional in R3

Benoît Merlet

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 3, page 553-569
  • ISSN: 1292-8119

Abstract

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In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in 𝐑 3 . The singular set of such a new minimizer belongs to a three parameters family of sets ( 0 < δ 1 , δ 2 , δ 3 < π ) . We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of 𝐒 2 with three reentrant corners. The necessary conditions are constraints on the eigenvalue and on the ratios between the three singular coefficients of the associated eigenvector. We use numerical methods (Singular Functions Method and Moussaoui's extraction formula) to compute the eigenvalues and the singular coefficients. We conclude that there is no ( δ 1 , δ 2 , δ 3 ) for which the necessary conditions are satisfied and this shows that the hypothesis was wrong.

How to cite

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Merlet, Benoît. "Numerical study of a new global minimizer for the Mumford-Shah functional in R3." ESAIM: Control, Optimisation and Calculus of Variations 13.3 (2007): 553-569. <http://eudml.org/doc/250007>.

@article{Merlet2007,
abstract = { In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in $\mathbf\{R\}^3$. The singular set of such a new minimizer belongs to a three parameters family of sets $(0<\delta_1,\delta_2,\delta_3<\pi)$. We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of $\mathbf\{S\}^2$ with three reentrant corners. The necessary conditions are constraints on the eigenvalue and on the ratios between the three singular coefficients of the associated eigenvector. We use numerical methods (Singular Functions Method and Moussaoui's extraction formula) to compute the eigenvalues and the singular coefficients. We conclude that there is no $(\delta_1,\delta_2,\delta_3)$ for which the necessary conditions are satisfied and this shows that the hypothesis was wrong. },
author = {Merlet, Benoît},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Mumford-Shah functional; numerical analysis; boundary value problems for second-order; elliptic equations in domains with corners},
language = {eng},
month = {6},
number = {3},
pages = {553-569},
publisher = {EDP Sciences},
title = {Numerical study of a new global minimizer for the Mumford-Shah functional in R3},
url = {http://eudml.org/doc/250007},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Merlet, Benoît
TI - Numerical study of a new global minimizer for the Mumford-Shah functional in R3
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/6//
PB - EDP Sciences
VL - 13
IS - 3
SP - 553
EP - 569
AB - In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in $\mathbf{R}^3$. The singular set of such a new minimizer belongs to a three parameters family of sets $(0<\delta_1,\delta_2,\delta_3<\pi)$. We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of $\mathbf{S}^2$ with three reentrant corners. The necessary conditions are constraints on the eigenvalue and on the ratios between the three singular coefficients of the associated eigenvector. We use numerical methods (Singular Functions Method and Moussaoui's extraction formula) to compute the eigenvalues and the singular coefficients. We conclude that there is no $(\delta_1,\delta_2,\delta_3)$ for which the necessary conditions are satisfied and this shows that the hypothesis was wrong.
LA - eng
KW - Mumford-Shah functional; numerical analysis; boundary value problems for second-order; elliptic equations in domains with corners
UR - http://eudml.org/doc/250007
ER -

References

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  11. P. Grisvard, Singularities in boundary value problems, Recherches Math. Appl. 22. Masson, Paris (1992).  Zbl0766.35001
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  14. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math.42 (1989) 577–685.  Zbl0691.49036

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