Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-18T09:08:15.183Z Has data issue: false hasContentIssue false

A topological asymptotic analysis for the regularized grey-level image classification problem

Published online by Cambridge University Press:  02 August 2007

Didier Auroux
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier Toulouse 3, 31062 Toulouse cedex 9, France
Lamia Jaafar Belaid
Affiliation:
ENIT-LAMSIN, BP37, 1002 Tunis Belvédère, Tunisia
Mohamed Masmoudi
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier Toulouse 3, 31062 Toulouse cedex 9, France
Get access

Abstract

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

G. Allaire, Shape optimization by the homogenization method. Applied Mathematical Sciences 146, Springer (2002).
Allaire, G. and Kohn, R., Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Eur. J. Mech. A Solids 12 (1993) 839878.
Allaire, G., Jouve, F. and Toader, A.-M., A level-set method for shape optimization. C. R. Acad. Sci. Sér. I 334 (2002) 11251130.
Allaire, G., de Gournay, F., Jouve, F. and Toader, A.-M., Structural optimization using topological and shape sensitivity via a level set method, Internal report, n° 555, CMAP, École polytechnique. Control Cybern. 34 (2005) 59-80.
Ammari, H., Vogelius, M.S. and Volkov, D., 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) 769814. CrossRef
Amstutz, S., Horchani, I. and Masmoudi, M., Crack detection by the topological gradient method. Control Cybern. 34 (2005) 119-138.
Aubert, G. and Aujol, J.-F., Optimal partitions, regularized solutions, and application to image classification. Appl. Anal. 84 (2005) 1535. CrossRef
G. Aubert and P. Kornprobst, Mathematical problems in image processing. Applied Mathematical Sciences 147, Springer-Verlag, New York (2002).
Aujol, J.-F., Aubert, G. and Blanc-Féraud, L., Wavelet-based level set evolution for classification of textured images. IEEE Trans. Image Process. 12 (2003) 16341641. CrossRef
M. Bendsoe, Optimal topology design of continuum structure: an introduction. Technical report, Department of Mathematics, Technical University of Denmark, Lyngby, Denmark (1996).
Berthod, M., Kato, Z., Yu, S. and Zerubia, J., Bayesian image classification using Markov random fields. Image Vision Comput. 14 (1996) 285293. CrossRef
Bouman, C.A. and Shapiro, M., A multiscale random field model for Bayesian image segmentation. IEEE Trans. Image Process. 3 (1994) 162177. CrossRef
P.G. Ciarlet, Finite Element Method for Elliptic Problems. North Holland (2002).
L. Cohen, E. Bardinet and N. Ayache, Surface reconstruction using active contour models. SPIE Int. Symp. Optics, Imaging and Instrumentation, San Diego California USA (July 1993).
R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Collection CEA, Masson, Paris (1987).
Descombes, X., Morris, R. and Zerubia, J., Some improvements to Bayesian image segmentation – Part one: modelling. Traitement du signal 14 (1997) 373382.
Descombes, X., Morris, R. and Zerubia, J., Some improvements to Bayesian image segmentation – Part two: classification. Traitement du signal 14 (1997) 383395.
Friedman, A. and Vogelius, M.S., Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem of continuous dependance. Arch. Rational Mech. Anal. 105 (1989) 299326. CrossRef
Garreau, S., Guillaume, P. and Masmoudi, M., The topological asymptotic for PDE systems: The elasticity case. SIAM J. Control Optim. 39 (1991) 1749.
Jaafar Belaid, L., Jaoua, M., Masmoudi, M. and Siala, L., Image restoration and edge detection by topological asymptotic expansion. C. R. Acad. Sci. Paris. Ser. I Math. 342 (2006) 313318. CrossRef
Z. Kato, Modélisations markoviennes multirésolutions en vision par ordinateur - Application à la segmentation d'images SPOT. Ph.D. thesis, INRIA, Sophia Antipolis, France (1994).
M. Masmoudi, 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.
Mumford, D. and Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577685. CrossRef
Paragios, N. and Deriche, R., Geodesic active regions and level set methods for supervised texture segmentation. Int. Jour. Computer Vision 46 (2002) 223247. CrossRef
Pavlidis, T. and Liow, Y.-T., Integrating region growing and edge detection. IEEE Trans. Pattern Anal. Machine Intelligence 12 (1990) 225233. CrossRef
Perona, P. and Malik, J., Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intelligence 12 (1990) 629638. CrossRef
Samet, B., Amstutz, S. and Masmoudi, M., The topological asymptotic for the Helmholtz equation. SIAM J. Control Optim. 42 (2003) 15231544. CrossRef
Samson, C., Blanc-Féraud, L., Aubert, G. and Zerubia, J., A level set method for image classification. Int. J. Comput. Vision 40 (2000) 187197. CrossRef
Samson, C., Blanc-Féraud, L., Aubert, G. and Zerubia, J., A variational model for image classification and restauration. IEEE Trans. Pattern Anal. Machine Intelligence 22 (2000) 460472. CrossRef
J.A. Sethian, Level set methods evolving interfaces in geometry, fluid mechanics, computer vision, and materials science. Cambride University Press (1996).
Sokolowski, J. and Zochowski, A., Topological derivatives of shape functionals for elasticity systems. Int. Ser. Numer. Math. 139 (2002) 231244.
S. Solimini and J.M. Morel, Variational methods in image segmentation. Birkhauser (1995).
L. Vese and T. Chan, Reduced Non-Convex Functional Approximations for Image Restoration and Segmentation. UCLA CAM Report 97–56 (1997).
Wang, M.Y., Wang, D. and Guo, A., A level set method for structural topology optimization. Comput. Methods Appl. Mech. Engrg. 192 (2003) 227246. CrossRef
Weickert, J., Efficient image segmentation using partial differential equations and morphology. Pattern Recogn. 34 (2001) 18131824. CrossRef