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Regularization of inverse problems in image processing

Abstract : Inverse problems are to recover the data that has been processed or corrupted. Since they are ill-posed they require a regularization. In image processing, the total variation as a regularization tool has the advantage of preserving the discontinuities while creating smooth regions. These results are established in this thesis in a continuous setting for general energies. In addition, we propose and examine a variant of the total variation. We establish a dual formulation that allows us to prove that this variant coincides with the total variation for sets of finite perimeter. Nowadays, non-local methods exploiting the self-similarities of images is particularly successful. We adapt this approach to the problem of spectrum completion, which has applications for general inverse problems. The final part is devoted to the algorithmic aspects inherent to the optimization of the convex energies we considered. We study the convergence and the complexity of the recently developed Primal-Dual algorithms.
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https://pastel.archives-ouvertes.fr/pastel-00787790
Contributor : Khalid Jalalzai <>
Submitted on : Tuesday, February 12, 2013 - 11:02:38 PM
Last modification on : Friday, October 23, 2020 - 4:38:17 PM
Long-term archiving on: : Monday, May 13, 2013 - 4:12:41 AM

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  • HAL Id : pastel-00787790, version 1

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Khalid Jalalzai. Regularization of inverse problems in image processing. Functional Analysis [math.FA]. Ecole Polytechnique X, 2012. English. ⟨pastel-00787790⟩

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