Due to recent technical advances in the development of 3D sensors and new application fields there is a growing interest in processing of 3D point cloud data. Furthermore, point clouds can be found as representations of manifolds in high-dimensional spaces, as for example in computational biology and machine learning. One challenging problem in current research is to translate well-established methods from Euclidean spaces, e.g., variational and PDE-based methods, to surfaces represented by discrete point cloud data.
In this talk we discuss how to construct a weighted graph from point cloud data based on geometry and associated features of the given points. We introduce a discrete calculus which enables us to mimic partial differential operators in this setting. Based on these operators we formulate different partial difference equations on weighted graphs. By solving the latter we are able to tackle challenging problems in point cloud data processing, e.g., filtering, colorization, or inpainting. Finally, we discuss how to use the proposed weighted graph framework for machine learning applications.
This talk is based on previous works of A. Elmoataz (Université de Caen Normandie), F. Lozes (Université de Caen Normandie), and recent collaborative works.