Active Contours on Manifolds & Global Minimization of the Active Contour Model

Xavier Bresson
Tuesday, 30.5.2006, 11:30
Taub 337

In this talk, I will present two image segmentation models based on the active contour method. The first model introduces an evolution equation for active contours embedded on parametric Riemannian manifolds. A first application of this equation, called "multiscale active contours" [1], allows to segment multiscale structures in general scale spaces s.a. the linear scale space and also the curvature or the Beltrami scale spaces. A second application leads to the segmentation of structures on omnidirectional images defined on spherical, hyperbolic and parabolic manifolds [2]. The current work attempts to generalize the previous models to non-parametric manifolds. The second model I will present concerns the global minimization of the active contour model [7]. Inspired by the work of Chan-Esedoglu-Nikolova [3] and using the weighted total variation norm, we propose a global minimization framework for the active contour model based on the Rudin-Osher-Fatemi model [4] and the Mumford-Shah functional [5]. Besides, the global minimization process is speed up thanks to the work of Chambolles [6]. The current work consists of generalizing this model to natural images containing smooth regions and textures using the image decomposition idea.

[1] X. Bresson, P. Vandergheynst and J. Thiran, "Multiscale Active Contours",
International Journal of Computer Vision, In press.
[2] I. Bogdanova, X. Bresson, J. Thiran and P. Vandergheynst, "Laplacian
Operator, Diffusion Flow and Active Contour on non-Euclidean Images",
TR-ITS-2005.20, August 2005.
[3] T. Chan, S. Esedoglu, and M. Nikolova, "Algorithms for Finding Global
Minimizers of Image Segmentation and Denoising Models, UCLA CAM Report 04-54," 2
[4] L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear Total Variation Based Noise
Removal Algorithms," Physica D, vol. 60(1-4), pp. 259-268, 1992.
[5] D. Mumford and J. Shah, "Optimal Approximations of Piecewise Smooth
Functions and Associated Variational Problems," Communications on Pure and
Applied Mathematics, vol. 42, pp. 577-685, 1989.
[6] A. Chambolle, "An Algorithm for Total Variation Minimization and
Applications," Journal of Mathematical Imaging and Vision, vol. 20(1-2), pp.
89-97, 2004.
[7] X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, "Fast
Global Minimization of the Active Contour/Snake Model", Journal of Mathematical
Imaging and Vision, Submitted.

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