Solving Geometric PDEs on Manifolds Represented as Point Clouds and Applications

Rongjie Lai
Rensselaer Polytechnic Institute

Analyzing and inferring the underlying global intrinsic structures of data from its local information are critical in many fields. In practice, coherent structures of data allow us to model data as low dimensional manifolds, represented as point clouds, in a possible high dimensional space. It is a challenge to extract global geometric information hidden in the point clouds due to the lack of global connectivity. In our recent work, systematical numerical methods are proposed to solving PDEs on manifolds sampled as point clouds. These methods can achieve high order accuracy and enjoy flexibility of solving different type of equations on manifolds with possible high co-dimesion. We use the proposed methods to consider special designed geometric PDEs on point clouds, which provides us a bridge to link local and global information. Based on this method, I will discuss a few applications to geometric understanding for point clouds, including computation of LB eigen-systems for point clouds, extraction of global skeletons structure from point clouds, extraction of conformal structures from point clouds, and intrinsic comparisons among point clouds etc. In addition, our methods can also be extended to solve PDEs on manifolds only represented as incomplete distance information. I will also demonstrate our preliminary results of this method for reconstructing and understanding distance data based on solutions of Laplace-Beltrame equations.

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