Constructing Laplace operator from point clouds in Rd

Authors:
Mikhail Belkin;Jian Sun;Yusu Wang
Affiliations:
The Ohio State University, Columbus, OH;Stanford University, Palo Alto, CA;The Ohio State University, Columbus, OH
Venue:
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Year:
2009

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Abstract

We present an algorithm for approximating the Laplace-Beltrami operator from an arbitrary point cloud obtained from a k-dimensional manifold embedded in the d-dimensional space. We show that this PCD Laplace (Point-Cloud Data Laplace) operator converges to the Laplace-Beltrami operator on the underlying manifold as the point cloud becomes denser. Unlike the previous work, we do not assume that the data samples are independent identically distributed from a probability distribution and do not require a global mesh. The resulting algorithm is easy to implement. We present experimental results indicating that even for point sets sampled from a uniform distribution, PCD Laplace converges faster than the weighted graph Laplacian. We also show that using our PCD Laplacian we can directly estimate certain geometric invariants, such as manifold area.