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Equations aux dérivées partielles appliquées à la restauration et à l'agrandissement des images

Abstract : This thesis consists in two parts in which we study two problems of computer vision : image restoration in the first part and image zooming in the second part. In the introduction of our first part we present the main mathematical models that have been proposed for the restoration (denoising and edge enhancement) of digital images. Then we discuss edge-detection theory and the Malik and Perona model. We introduce various variants that have been proposed to stabilized the (ill-posed) Malik and Perona equation and in particular the model which we will study in the following chapters. For this model in all dimensions, we prove in small time existence and uniqueness of a classical solution. We then construct (in one and two dimensions) a numerical approximation and prove its convergence to a weak solution. As conclusion of this part, we present some experiments. We begin the second part by presenting existing image zooming methods. Using geometrical arguments, we propose a new approach based on a partial differential equation. Next we prove the well posedness of the model using the theory of the viscosity solutions. We then discuss the discretisation of the model, and finally present some experiments.
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Contributor : Abdelmounim Belahmidi <>
Submitted on : Wednesday, June 25, 2003 - 12:42:39 PM
Last modification on : Wednesday, September 23, 2020 - 4:31:13 AM
Long-term archiving on: : Friday, April 2, 2010 - 7:17:44 PM


  • HAL Id : tel-00003022, version 1



Abdelmounim Belahmidi. Equations aux dérivées partielles appliquées à la restauration et à l'agrandissement des images. Mathématiques [math]. Université Paris Dauphine - Paris IX, 2003. Français. ⟨tel-00003022⟩



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