Order-randomized Laplacian mesh smoothing

Yang, Ying; Rushmeier, Holly; Ivrissimtzis, Ioannis; Floater, Michael S.; Lyche, Tom; Mazure, Marie-Laurence; Mørken, Knut; Schumaker, Larry L.

Abstract

In this paper we compare three variants of the graph Laplacian smoothing. The first is the standard synchronous implementation, corresponding to multiplication by the graph Laplacian matrix. The second is a voter process inspired asynchronous implementation, assuming that every vertex is equipped with an independent exponential clock. The third is in-between the first two, with the vertices updated according to a random permutation of them. We review some well-known results on spectral graph theory and on voter processes, and we show that while the convergence of the synchronous Laplacian is graph dependent and, generally, does not converge on bipartite graphs, the asynchronous converges with high probability on all graphs. The differences in the properties of these three approaches are illustrated with examples including both regular grids and irregular meshes.

Citation

Yang, Y., Rushmeier, H., Ivrissimtzis, I., Floater, M. S., Lyche, T., Mazure, M., …Schumaker, L. L. (2017). Order-randomized Laplacian mesh smoothing. In Mathematical methods for curves and surfaces : 9th International Conference, MMCS 2016, Tønsberg, Norway, June 23 - June 28, 2016. Revised selected papers (312-323). https://doi.org/10.1007/978-3-319-67885-6_17

• Conference Name: 9th International Conference on Mathematical Methods for Curves and Surfaces
• Conference Location: Tønsberg, Norway
• Start Date: Jun 23, 2016
• End Date: Jun 28, 2016
• Acceptance Date: Mar 7, 2017
• Online Publication Date: Oct 18, 2017
• Publication Date: Oct 18, 2017
• Deposit Date: Aug 15, 2017
• Publicly Available Date: Oct 18, 2018
• Pages: 312-323
• Series Title: Lecture notes in computer science
• Series Number: 10521
• Series ISSN: 0302-9743,1611-3349
• Book Title: Mathematical methods for curves and surfaces : 9th International Conference, MMCS 2016, Tønsberg, Norway, June 23 - June 28, 2016. Revised selected papers.
• ISBN: 9783319678849
• DOI: https://doi.org/10.1007/978-3-319-67885-6_17