Location: MIC-Arena, Maria-Goeppert-Str. 3, 23562 Lübeck
Title: Partial Difference Equations on Graphs with Applications in Image and Data Processing
Abstract: Graph-based methods have emerged as a promising tool for many applications in machine learning and data processing. One key feature of these methods is the possibility to incorporate nonlocal relationships in the data rather than using only local neighborhoods. The recent trend in the literature is to translate well-studied variational problems and PDEs to the graph setting and overcome hereby drawbacks of classical approaches.
In this talk we give a short introduction to the concept of partial difference equations on graphs and show that many classical numerical discretization schemes can be embedded in a graph setting and thus be interpreted as special cases in a more general framework. As one example we discuss a recently proposed family of partial difference operators on graphs and study equations involving these operators. This family covers local and nonlocal variational p-Laplacian and ∞-Laplacian as well as gradient operators used in morphology. We analyze a corresponding parabolic equation involving these operators which enables us to interpolate adaptively between p-Laplacian diffusion-based filtering and morphological filtering, i.e., erosion and dilation. Furthermore, in the case p = ∞ we investigate a connection to a stochastic game known as Tug-of-War and give a possibility to introduce nonlocality into this setting.
Finally, we demonstrate the advantages of graph-based methods for different tasks in image and point cloud processing, such as filtering, segmentation, clustering, and inpainting.
Joint work with A. Elmoataz, Université de Caen, France
