A New Relaxation Method for a Discrete Image Restoration Problem

Maïtine Bergounioux; Mounir Haddou

doi:10.68381/jca17055

Abstract

The method we present in this paper has been motivated by a restoration problem in a tomography context. We are interested in blurred and noised binary images restoration. We consider the discrete version of a minimization problem settled in the space of bounded variation functions. We give a general abstract formulation of the (discrete) optimization problem with binary constraints and provide approximate and penalized formulations. Convergence results are given and we present numerical tests.

Author information

Contact details are reproduced from the original publication and may be historical.

Maïtine Bergounioux

UFR Sciences - Mathématiques, Université d'Orléans, Route de Chartres, 45067 Orléans, France
and: Labo. MAPMO, UMR 6628

maitine.bergounioux@univ-orleans.fr

Mounir Haddou

UFR Sciences - Mathématiques, Université d'Orléans, Route de Chartres, 45067 Orléans, France
and: Labo. MAPMO, UMR 6628

mounir@haddou@univ-orleans.fr

Keywords

Tomography
optimization
penalization

References

I. Abraham, R. Abraham, M. Bergounioux: An active curve approach for tomographic reconstruction of binary radially symmetric objects, in: Numerical Mathematics and Advanced Applications (Graz, 2007), K. Kunisch, G. Of, O. Steinbach (eds.), Springer, Berlin (2008) 663–670.https://doi.org/10.1007/978-3-540-69777-0_79 Google Scholar
R. Abraham, M. Bergounioux, E. Trélat: A penalization approach for tomographic reconstruction of binary axially symmetric objects, Appl. Math. Optim. 58 (2008) 345–371.https://doi.org/10.1007/s00245-008-9039-8 Google Scholar
L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, Oxford (2000).https://doi.org/10.1093/oso/9780198502456.001.0001 Google Scholar
G. Aubert, P. Kornprobst: Mathematical Problems in Image Processing, Applied Mathematical Sciences 147, Springer, New York (2006).https://doi.org/10.1007/978-0-387-44588-5 Google Scholar
T. D. Capricelli, P. L. Combettes: A convex programming algorithm for noisy discrete tomography, in: Advances in Discrete Tomography and its Applications, G. T. Herman, A. Kuba (eds.), Birkhäuser, Boston (2007) 207–226.https://doi.org/10.1007/978-0-8176-4543-4_10 Google Scholar
A. Chambolle: An algorithm for total variation minimization and applications, J. Math. Imaging Vis. 20 (2004) 89–97.https://doi.org/10.1023/B:JMIV.0000011325.36760.1e Google Scholar
F. H. Clarke: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics 5, SIAM, Philadelphia (1990).https://doi.org/10.1137/1.9781611971309 Google Scholar
M. Haddou: A new class of smoothing methods for mathematical programs with equilibrium constraints, Pac. J. Optim. 5 (2009) 87–95.Google Scholar
R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1996).Google Scholar
J. A. Sethian: Theory, algorithms and applications of level set methods for propagating interfaces, in: Acta Numerica Vol. 5, A. Iserles (ed.), Cambridge University Press, Cambridge (1996) 309–395.https://doi.org/10.1017/S0962492900002671 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.

[1] I. Abraham, R. Abraham, M. Bergounioux: An active curve approach for tomographic reconstruction of binary radially symmetric objects, in: Numerical Mathematics and Advanced Applications (Graz, 2007), K. Kunisch, G. Of, O. Steinbach (eds.), Springer, Berlin (2008) 663–670.https://doi.org/10.1007/978-3-540-69777-0_79 Google Scholar

[2] R. Abraham, M. Bergounioux, E. Trélat: A penalization approach for tomographic reconstruction of binary axially symmetric objects, Appl. Math. Optim. 58 (2008) 345–371.https://doi.org/10.1007/s00245-008-9039-8 Google Scholar

[3] L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, Oxford (2000).https://doi.org/10.1093/oso/9780198502456.001.0001 Google Scholar

[4] G. Aubert, P. Kornprobst: Mathematical Problems in Image Processing, Applied Mathematical Sciences 147, Springer, New York (2006).https://doi.org/10.1007/978-0-387-44588-5 Google Scholar

[5] T. D. Capricelli, P. L. Combettes: A convex programming algorithm for noisy discrete tomography, in: Advances in Discrete Tomography and its Applications, G. T. Herman, A. Kuba (eds.), Birkhäuser, Boston (2007) 207–226.https://doi.org/10.1007/978-0-8176-4543-4_10 Google Scholar

[6] A. Chambolle: An algorithm for total variation minimization and applications, J. Math. Imaging Vis. 20 (2004) 89–97.https://doi.org/10.1023/B:JMIV.0000011325.36760.1e Google Scholar

[7] F. H. Clarke: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics 5, SIAM, Philadelphia (1990).https://doi.org/10.1137/1.9781611971309 Google Scholar

[8] M. Haddou: A new class of smoothing methods for mathematical programs with equilibrium constraints, Pac. J. Optim. 5 (2009) 87–95.Google Scholar

[9] R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1996).Google Scholar

[10] J. A. Sethian: Theory, algorithms and applications of level set methods for propagating interfaces, in: Acta Numerica Vol. 5, A. Iserles (ed.), Cambridge University Press, Cambridge (1996) 309–395.https://doi.org/10.1017/S0962492900002671 Google Scholar