Partial difference Equations on Graphs and Networks: from images to graph signal processing.
Résumé
Partial differential equations (PDEs) play a key role for mathematical modeling throughout
applied and natural sciences. In this context, many PDEs have been studied to describe importantprocesses, e.g., in physics, biology, economy, image processing, computer vision.
In particular they have been applied successfully in image and signal processing to a broad varietyof applications, e.g., isotropic and anisotropic filterings, non local filtering, image regularization and inpainting, or image segmentation (to name a few).
Recently, there is high interest in adapting and solving PDEs on data which is given by
arbitrary graphs and networks. The demand for such methods is motivated by existing and potential future applications, such as in machine learning and mathematical image processing. Indeed, any kind of data can be represented by a graph in an abstract form in which the vertices are associated to the data and the edges correspond to relationships within the data. In order to translate and solve PDEs on graphs, different discrete vector calculus have been proposed in the literature in recent years. One simple discrete calculus on graphs is based on discrete partial differences, which enables one to solve PDEs on both regular as well as irregular data domains in a unified and simple manner. This mimetic approach consists of replacing continuous partial differential operators, e.g., gradient or divergence, by a reasonable discrete analogue, which makes it possible to transfer many important tools and results from the continuous setting.
This tutorial aims at proposing a comprehensive introduction to the field of Partial
difference Equations (PdE) on graphs and their applications in image, signal and data processing, and machine learning. It lies at the interface of the following topics: local and non-local continuous PDE, PDE on graphs, Tug-of-War games, signal processing on graphs and local and non local manifold processing.