Non-Local Morphological PDEs and p-Laplacian Equation on Graphs with applications in image processing and machine learning

Abderrahim Elmoataz (1) , Xavier Desquesnes (1) , Olivier Lezoray (1)

1 Equipe Image - Laboratoire GREYC - UMR6072

In this paper, we introduce a new class of nonlocal p-Laplacian operators that interpolate between non-local Laplacian and infinity Laplacian. These operators are discrete analogous of the game p-laplacian operators on Euclidean spaces, and involve discrete morphological gradient on graphs. We study the Dirichlet problem associated with the new p-Laplacian equation and prove existence and uniqueness of it's solution. We also consider non-local diffusion on graphs involving these operators. Finally, we propose to use these operators as a unified framework for solution of many inverse problems in image processing and machine learning.

Mots clés: p-Laplacian; PDEs-based morphology on graphs; image processing; machine learning; Tug-of-war games

Domaines: Traitement des images [eess.IV]

Abderrahim Elmoataz, Xavier Desquesnes, Olivier Lezoray. Non-Local Morphological PDEs and p-Laplacian Equation on Graphs with applications in image processing and machine learning. IEEE Journal of Selected Topics in Signal Processing, 2012, 6 (7), pp.764-779. ⟨10.1109/JSTSP.2012.2216504⟩. ⟨hal-00813009⟩