Equations aux différences partielles définies sur des graphes pour le traitement d'images et de données

Vinh Thong Ta 1
1 Equipe Image - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
Abstract : This works deals with images and non uniform data processing by partial difference equations (PdE) over weighted graphs. Transcription and adaption of continuous models to discrete formulations are considered within this PdEs-based framework. The considered continuous models (from image processing domain) are defined as variational models or approaches based on partial differential equations. The continuous models considered in this work are: regularization models, mathematical morphology, and the eikonal equation. To adapt these latter models within a discrete setting, we introduce a large family of discrete differential operators defined on weighted graphs: weighted differences, discrete gradients and $p$-Laplacian. These operators enable the transcription and the adaption of continuous models and provide a general formulation for considering numerous applications for images and arbitrary data. Potentialities of our discrete regularization, mathematical morphology and eikonal equation models are shown in applications such as image and data filtering, simplification, segmentation, clustering and classification. Our formulation also unifies local and non local patch-based processing. We have intensively used this latter configuration and shown the superiority of such a scheme in the context of image processing. Our approach is based on weighted graphs. This point provides a natural extension of continuous models for the processing of arbitrary data that can be represented by a weighted graph (for instance$:$ images, manifolds, data sets, data bases, etc.). Finally, this work opens new insights for image processing and new possible applications in machine learning.
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Vinh Thong Ta. Equations aux différences partielles définies sur des graphes pour le traitement d'images et de données. Interface homme-machine [cs.HC]. Université de Caen, 2009. Français. ⟨tel-00472041⟩

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