Partial differential equations and variational methods for geometric processing of images

Buet, Blanche; Mirebeau, Jean-Marie; Gennip, Yves van; Desquilbet, François; Dréo, Johann; Barbaresco, Frédéric; Leonardi, Gian Paolo; Masnou, Simon; Schönlieb, Carola‐Bibiane

doi:10.5802/smai-jcm.55

Cited by 1 publication

(1 citation statement)

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“…Finite graphs and networks have been widely and successfully used in a variety of fields such as machine learning, data mining, image analysis and social sciences where one is facing analysis and modelling of high dimensional unstructured datasets. In this context, extending the models and methods from variational methods and PDEs to solve problems on graphs is an active research area; see [40,8,21,22] and references therein. Many of these problems, such as classification, clustering or segmentation, can be often formulated in terms of minimizing a graph perimeter (graph cut) or a related functional (normalized cut, ratio cut, balanced cut, etc.).…”

Section: Introductionmentioning

confidence: 99%

Nonlocal Perimeters and Curvature Flows on Graphs with Applications in Image Processing and High-Dimensional Data Classification

Bouchairi

Elmoataz

Fadili

2023

SIAM J. Imaging Sci.

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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.

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Section: Introductionmentioning

confidence: 99%