Skip to the content.

KAUST-IAMCS Workshop on Modeling and Simulation of Wave Propagation and Applications

Rajae Aboulaich, Mohammed V University at Agdal (Morocco)
A Nash-Game Approach for Image Restoration and Segmentation


  • Rajae Aboulaich
  • Abderrahmane Habbal
  • Moez Kallel
  • Maher Moakher


In this work, we propose a game theory approach to simultaneously restore and segment noisy images. We define two players: one is restoration, with the image intensity as strategy, and the other is segmentation, with contours as strategy. Cost functions are the classical relevant ones for restoration and segmentation, respectively.

The two players play a static game with complete information, and we consider as solution to the game the so-called Nash Equilibrium. For the computation of this equilibrium, we present an iterative method with relaxation. The results of numerical experiments performed on some real images show the relevance and efficiency of the proposed algorithm.

Image segmentation, which is the process of extracting objects from an image, is one of the most important problems in image processing. It has several applications ranging from object recognition and motion detection to medical image analysis. In general, the segmentation of images is a very difficult problem in image processing. For the last few decades, this problem has been formulated as a variational problem leading to partial differential equations.

Most often, the image to be segmented is polluted with noise whose origin can be attributed to the acquisition devices, transmitting channels, random variations in luminosity or temperature during acquisition, etc. It is therefore essential to remove or reduce the noise before segmenting the image. Similar to the image segmentation problem, the restoration of images can be performed using variational methods. We can see [3,4,5,6] for a survey and analysis of the variational methods used in the image restoration and segmentation problems. In what follows, we briefly recall some classical variational models used for the restoration, segmentation, and joint restoration and segmentation of noisy images.


  1. R. Aboulaich, S. Boujena, E. El Guarmah, Sur un modèle non-linéaire pour le débruitage de l'image, CRAS, EDP, .
  2. R. Aboulaich, S. Boujena, E. El Guarmah, A non linear model for image processing, Math. Model. Nat. Phenom., Vol 3, Num. 1, .
  3. R. Aboulaich, D. Meskine, and A. Souissi, New diffusion models in image processing, Computers and Mathematics with Applications, .
  4. A. Chambolle and P-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., (76)(2), , 167-188.
  5. Y. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM Journal of Applied Mathematics, 66(4) 1383-1406, .
  6. P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (), 629-639.
  7. L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, (60)(), 259-268.
  8. Sapiro, G., Geometric partial differential equations and image analysis, Cambridge University Press ().
  9. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, F. Lenzen, Variational methods in imaging, Applied Mathematical Sciences, vol. 167. Springer, New York ().