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Contribution à l'analyse de textures en traitement d'images par méthodes variationnelles et équations aux dérivées partielles

Jean-François Aujol 1, 2
2 ARIANA - Inverse problems in earth monitoring
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - SIS - Signal, Images et Systèmes
Abstract : This Ph.D. thesis is a work in applied mathematics. It deals with image processing problems, and uses specific mathematical tools.

The aim of the two first parts is to propose a model for decomposing an image f into three components : f=u+v+w. The first one, u, contains the geometrical information. It can be considered as a sketch of the original image f. The second one, v, contains the textures. The third one, w, contains the noise of the original image. Our approach relies on the use of mathematical spaces
adapted to each component : the space BV of functions with bounded variations for u, a space G close to the dual space of BV for v, and a negative Besov space E for w. We carry out the complete mathematical analysis of the different models we propose.
We illustrate our approach with many numerical examples, and we give two applications: a first one in SAR image restoration, and a second one in image compression.

In the third and last part, we only deal with the texture component of an image. We propose a supervised classification algorithm for textured images. Our approach relies on the use of the active contours method. We build a data term specific to textures thanks to a packet wavelet transform. We thus get a functional, whose minimum corresponds to the classification we look for. We numerically solve this problem with a coupled PDEs system. We illustrate our approach with many numerical examples. We also carry out the theoretical study of the classification functional.
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Theses
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https://tel.archives-ouvertes.fr/tel-00006303
Contributor : Jean-François Aujol <>
Submitted on : Tuesday, June 22, 2004 - 2:08:32 PM
Last modification on : Monday, October 12, 2020 - 10:30:13 AM
Long-term archiving on: : Friday, April 2, 2010 - 9:14:48 PM

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  • HAL Id : tel-00006303, version 1

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Jean-François Aujol. Contribution à l'analyse de textures en traitement d'images par méthodes variationnelles et équations aux dérivées partielles. Mathématiques [math]. Université Nice Sophia Antipolis, 2004. Français. ⟨tel-00006303⟩

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