The aim of this article is to propose a new method for the grey-level image classification problem. We first present the classical variational approach without and with a regularization term in order to smooth the contours of the classified image. Then we present the general topological asymptotic analysis, and we finally introduce its application to the grey-level image classification problem.
Mots-clés : image classification, topological asymptotic expansion, image restoration
@article{M2AN_2007__41_3_607_0,
author = {Auroux, Didier and Belaid, Lamia Jaafar and Masmoudi, Mohamed},
title = {A topological asymptotic analysis for the regularized grey-level image classification problem},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {607--625},
publisher = {EDP-Sciences},
volume = {41},
number = {3},
year = {2007},
doi = {10.1051/m2an:2007027},
mrnumber = {2355713},
zbl = {1138.68622},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007027/}
}
TY - JOUR AU - Auroux, Didier AU - Belaid, Lamia Jaafar AU - Masmoudi, Mohamed TI - A topological asymptotic analysis for the regularized grey-level image classification problem JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 607 EP - 625 VL - 41 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007027/ DO - 10.1051/m2an:2007027 LA - en ID - M2AN_2007__41_3_607_0 ER -
%0 Journal Article %A Auroux, Didier %A Belaid, Lamia Jaafar %A Masmoudi, Mohamed %T A topological asymptotic analysis for the regularized grey-level image classification problem %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 607-625 %V 41 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007027/ %R 10.1051/m2an:2007027 %G en %F M2AN_2007__41_3_607_0
Auroux, Didier; Belaid, Lamia Jaafar; Masmoudi, Mohamed. A topological asymptotic analysis for the regularized grey-level image classification problem. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 3, pp. 607-625. doi : 10.1051/m2an:2007027. http://www.numdam.org/articles/10.1051/m2an:2007027/
[1] , Shape optimization by the homogenization method. Applied Mathematical Sciences 146, Springer (2002). | MR | Zbl
[2] and, Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Eur. J. Mech. A Solids 12 (1993) 839-878. | Zbl
[3] , and, A level-set method for shape optimization. C. R. Acad. Sci. Sér. I 334 (2002) 1125-1130. | Zbl
[4] ,, and, Structural optimization using topological and shape sensitivity via a level set method 34 (2005) 59-80. | MR
[5] , and, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II - The full Maxwell equations. J. Math. Pures Appl. 80 (2001) 769-814. | Zbl
[6] , and, Crack detection by the topological gradient method. Control Cybern. 34 (2005) 119-138. | MR
[7] and, Optimal partitions, regularized solutions, and application to image classification. Appl. Anal. 84 (2005) 15-35.
[8] and, Mathematical problems in image processing. Applied Mathematical Sciences 147, Springer-Verlag, New York (2002). | MR | Zbl
[9] , and, Wavelet-based level set evolution for classification of textured images. IEEE Trans. Image Process. 12 (2003) 1634-1641.
[10] , Optimal topology design of continuum structure: an introduction. Technical report, Department of Mathematics, Technical University of Denmark, Lyngby, Denmark (1996).
[11] ,, and, Bayesian image classification using Markov random fields. Image Vision Comput. 14 (1996) 285-293.
[12] and, A multiscale random field model for Bayesian image segmentation. IEEE Trans. Image Process. 3 (1994) 162-177.
[13] , Finite Element Method for Elliptic Problems. North Holland (2002). | MR | Zbl
[14] , and, Surface reconstruction using active contour models. SPIE Int. Symp. Optics, Imaging and Instrumentation, San Diego California USA (July 1993).
[15] and, Analyse mathématique et calcul numérique pour les sciences et les techniques. Collection CEA, Masson, Paris (1987). | MR | Zbl
[16] , and, Some improvements to Bayesian image segmentation - Part one: modelling. Traitement du signal 14 (1997) 373-382. | Zbl
[17] , and, Some improvements to Bayesian image segmentation - Part two: classification. Traitement du signal 14 (1997) 383-395. | Zbl
[18] and, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem of continuous dependance. Arch. Rational Mech. Anal. 105 (1989) 299-326. | Zbl
[19] , and, The topological asymptotic for PDE systems: The elasticity case. SIAM J. Control Optim. 39 (1991) 17-49. | Zbl
[20] ,, and, Image restoration and edge detection by topological asymptotic expansion. C. R. Acad. Sci. Paris. Ser. I Math. 342 (2006) 313-318. | Zbl
[21] , Modélisations markoviennes multirésolutions en vision par ordinateur - Application à la segmentation d'images SPOT. Ph.D. thesis, INRIA, Sophia Antipolis, France (1994).
[22] , The topological asymptotic, in Computational Methods for Control Applications, R. Glowinski, H. Karawada and J. Periaux Eds., GAKUTO Internat. Ser. Math. Sci. Appl. 16, Tokyo, Japan (2001) 53-72. | Zbl
[23] and, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577-685. | Zbl
[24] and, Geodesic active regions and level set methods for supervised texture segmentation. Int. Jour. Computer Vision 46 (2002) 223-247. | Zbl
[25] and, Integrating region growing and edge detection. IEEE Trans. Pattern Anal. Machine Intelligence 12 (1990) 225-233.
[26] and, Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intelligence 12 (1990) 629-638.
[27] , and, The topological asymptotic for the Helmholtz equation. SIAM J. Control Optim. 42 (2003) 1523-1544. | Zbl
[28] ,, and, A level set method for image classification. Int. J. Comput. Vision 40 (2000) 187-197. | Zbl
[29] ,, and, A variational model for image classification and restauration. IEEE Trans. Pattern Anal. Machine Intelligence 22 (2000) 460-472.
[30] , Level set methods evolving interfaces in geometry, fluid mechanics, computer vision, and materials science. Cambride University Press (1996). | MR | Zbl
[31] and, Topological derivatives of shape functionals for elasticity systems. Int. Ser. Numer. Math. 139 (2002) 231-244. | Zbl
[32] and, Variational methods in image segmentation. Birkhauser (1995). | MR
[33] and, Reduced Non-Convex Functional Approximations for Image Restoration and Segmentation. UCLA CAM Report 97-56 (1997).
[34] , and, A level set method for structural topology optimization. Comput. Methods Appl. Mech. Engrg. 192 (2003) 227-246. | Zbl
[35] , Efficient image segmentation using partial differential equations and morphology. Pattern Recogn. 34 (2001) 1813-1824. | Zbl
Cité par Sources :
