Nonlocal perimeters and curvature flows on graphs with applications in image processing and high-dimensional data classification

Imad El Bouchairi (1, 2, 3) , Abderrahim Elmoataz (1, 2, 3) , Jalal M. Fadili (1, 2, 4)

1 Equipe Image - Laboratoire GREYC - UMR6072
2 GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen
3 UNICAEN - Université de Caen Normandie
4 ENSICAEN - École Nationale Supérieure d'Ingénieurs de Caen

In this paper, we revisit the notion of perimeter on graphs, introduced in [19], and we extend it to so-called inner and outer perimeters. We will also extend the notion of total variation on graphs. Thanks to the co-area formula, we show that discrete total variations can be expressed through these perimeters. Then, we propose a novel class of curvature operators on graphs that unifies both local and nonlocal mean curvature on an Euclidean domain. This leads us to translate and adapt the notion of the mean curvature flow on graphs as well as the level set mean curvature, which can be seen as approximate schemes. Finally, we exemplify the usefulness of these methods in image processing, 3D point cloud processing, and high dimensional data classification.

Mots clés: Perimeter; total variation; graph cuts; mean curvature flow; image processing; data clustering; PdE on graph

Domaines: Traitement du signal et de l'image [eess.SP]; Mathématique discrète [cs.DM]; Algorithme et structure de données [cs.DS]; Théorie de l'information [cs.IT]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Imad El Bouchairi, Abderrahim Elmoataz, Jalal M. Fadili. Nonlocal perimeters and curvature flows on graphs with applications in image processing and high-dimensional data classification. SIAM Journal on Imaging Sciences, inPress, 16 (1), ⟨10.1137/22M148598X⟩. ⟨hal-03631338v3⟩