This paper is concerned with the computation of the topological gradient associated to a fourth order Kirchhoff type partial differential equation and to a second order cost function. This computation is motivated by fine structure detection in image analysis. The study of the topological sensitivity is performed both in the cases of a circular inclusion and a crack.
Keywords: Topological gradient, fourth order PDE, fine structures, 2D imaging
Aubert, Gilles 1 ; Drogoul, Audric 1
@article{COCV_2015__21_4_1120_0,
author = {Aubert, Gilles and Drogoul, Audric},
title = {Topological gradient for a fourth order operator used in image analysis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1120--1149},
year = {2015},
publisher = {EDP Sciences},
volume = {21},
number = {4},
doi = {10.1051/cocv/2014061},
mrnumber = {3395758},
zbl = {1325.49050},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2014061/}
}
TY - JOUR AU - Aubert, Gilles AU - Drogoul, Audric TI - Topological gradient for a fourth order operator used in image analysis JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 1120 EP - 1149 VL - 21 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2014061/ DO - 10.1051/cocv/2014061 LA - en ID - COCV_2015__21_4_1120_0 ER -
%0 Journal Article %A Aubert, Gilles %A Drogoul, Audric %T Topological gradient for a fourth order operator used in image analysis %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 1120-1149 %V 21 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2014061/ %R 10.1051/cocv/2014061 %G en %F COCV_2015__21_4_1120_0
Aubert, Gilles; Drogoul, Audric. Topological gradient for a fourth order operator used in image analysis. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1120-1149. doi: 10.1051/cocv/2014061
R.A. Adams, Sobolev spaces. Pure Appl. Math. Academic Press, New York (1978). | Zbl
, Sensitivity analysis with respect to a local perturbation of the material property. Asymp. Anal. 49 (2006) 87–108. | MR | Zbl
, The topological asymptotic for the Navier-Stokes equations. ESAIM: COCV 11 (2005) 401–425. | MR | Zbl | Numdam
and , Topological asymptotic analysis of the Kirchhoff plate bending problem. ESAIM: COCV 17 (2011) 705–721. | MR | Zbl | Numdam
and , Topological gradient for fourth-order PDE and application to the detection of fine structures in 2D images. C.R. Math. 352 (2014) 609–613. | MR | Zbl | DOI
, From restoration by topological gradient to medical image segmentation via an asymptotic expansion. Math. Comput. Model. 49 (2009) 2191–2205. | MR | Zbl | DOI
D. Auroux, M. Masmoudi and L. Jaafar Belaid, Image restoration and classification by topological asymptotic expansion, in Variational Formulations in Mechanics: Theory and Applications. Edited by E. Taroco, E.A. de Souza Neto and A.A. Novotny. CIMNE, Barcelona, Spain (2007) 23–42.
, , , and , Application of the topological gradient to image restoration and edge detection. Eng. Anal. Boundary Elements 32 (2008) 891–899. | Zbl | DOI
G. Chen and J. Zhou, Boundary Element Methods with Applications to Nonlinear Problems. Atlantis Stud. Math. Eng. Sci. (1992). | MR | Zbl
, Numerical analysis of the topological gradient method for fourth order models and applications to the detection of fine structures in imaging. SIAM J. Imaging Sci. 7 (2014) 2700–2731. | MR | Zbl | DOI
A. Drogoul, Topological gradient method applied to the detection of edges and fine structures in imaging. Ph.D. thesis, University of Nice Sophia Antipolis (2014).
, Exact solution of a hypersingular integral equation. J. Integral Equations Appl. 4 (1992) 197–204. | MR | Zbl | DOI
M. Masmoudi, The topological asymptotic. In Computational Methods for Control Applications. Vol. 16 of GAKUTO Internat. Ser. Math. Appl. Tokyo, Japan (2001). | Zbl
J.-C. Nédélec, Acoustic and electromagnetic equations: integral representations for harmonic problems. Appl. Math. Sci. Springer, New York (2001). | MR | Zbl
, On elliptic partial differential equations. Annali della Scuola Normale Superiore di Pisa – Classe di Scienze 13 (1959) 115–162. | MR | Zbl | Numdam
D. Ruiz, A note on the uniformity of the constant in the Poincaré inequality. Preprint arXiv:1208.6045 (2012). | MR | Zbl
and , On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251–1272. | MR | Zbl | DOI
, An unbiased detector of curvilinear structures. IEEE Trans. Pattern Anal. Mach. Intell. 20 (1998) 113–125. | DOI
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