Mathematical Morphology and Eikonal Equations on Graphs for Nonlocal Image and Data Processing

Vinh Thong Ta (1) , Abderrahim Elmoataz (1) , Olivier Lézoray (1)

1 Equipe Image - Laboratoire GREYC - UMR6072

Mathematical morphology (MM) offers a wide range of operators to address various image processing problems. These operators can be defined in terms of algebraic (discrete) sets or as partial differential equations (PDEs). In this paper, we introduce a novel formulation of MM formalized as a framework of partial difference equations (PdEs) over weighted graphs of arbitrary topology. Then, we present and analyze a new family of morphological operators defined on weighted graphs. The proposed framework recovers local algebraic and PDEs-based formulations of MM. It also introduces nonlocal configurations for morphological image processing and extends PDEs-based methods to process any discrete data that can be described by a graph such as high dimensional data defined on irregular domains. Moreover, based on same ideas, we propose an adaptation of the eikonal equation on weighted graphs. Our formulation of the eikonal leads to novel applications of this equation such as nonlocal image segmentation and weighted distances or data clustering.

Mots clés: partial difference equations (PdEs); weighted graphs; Mathematical morphology; eikonal equation; arbitrary discrete data sets; nonlocal image segmentation; data clustering

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement des images [eess.IV]; Mathématique discrète [cs.DM]; Equations aux dérivées partielles [math.AP]; Traitement du signal et de l'image [eess.SP]

Vinh Thong Ta, Abderrahim Elmoataz, Olivier Lézoray. Mathematical Morphology and Eikonal Equations on Graphs for Nonlocal Image and Data Processing. 2009. ⟨hal-00365431⟩