Solving PDEs on point clouds with applications to shape analysis

March 12, 2024
Using point clouds is the most natural and ubiquitous way of representing geometry and data in 3D and higher. In this talk, I will present a framework of solving geometric PDEs directly on point clouds based on local tangent space parametrization. Then I will talk about some applications in shape analysis for point clouds. Unlike images, which have a canonical form of representation as functions defined on a uniform grid on a rectangular domain, surfaces and manifolds in 3D and higher are geometric objects that do not have a canonical or natural form of representation or global parametrization. Moreover, their embeddings in the ambient space are not intrinsic. We show how geometric PDEs can be used to “connect the dots” and extract intrinsic geometric information for the underlying point clouds for shape analysis.