Geometry, Imaging and Computing

Volume 3 (2016)

Number 1-2

Dynamic unified surface Ricci flow

Pages: 31 – 56

DOI: http://dx.doi.org/10.4310/GIC.2016.v3.n1.a2

Authors

Wei Chen (School of Software and Technology, Dalian University of Technology, Dalian, Liaoning, China)

Min Zhang (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.; and Brigham and Women’s Hospital, Harvard Medical School, Boston, Mass., U.S.A.)

Na Lei (DUT-RU International School of Information Science & Engineering, Dalian University of Technology, Dalian, Liaoning, China; and Key Laboratory for Ubiquitous Network and Service Software of Liaoning Province, China)

David Xianfeng Gu (Department of Computer Science , Stony Brook University, Stony Brook, New York, U.S.A.)

Abstract

Surface parameterization plays a fundamental role in geometric modeling and processing. Surface Ricci flow deforms the Riemannian metric proportional to the curvature, such that the curvature evolves according to a diffusion-reaction process, and converges to the target curvature. Surface Ricci flow is a powerful tool to design Riemannian metrics from user-prescribed curvatures. In discrete setting, there are several schemes, which can be unified to a coherent framework.

Conventional discrete surface Ricci flow method is vulnerable to mesh quality. For a given target curvature and a low quality mesh, the method may encounter degeneracy. In general, it is difficult to analyze the existence of the solution to the conventional unified Ricci flow. This greatly prevents the unified Ricci flow from largescale real applications.

In the current work, in order to conquer this problem, we propose the dynamic unified Ricci flow method. The novel method updates the triangulation during the flow, such that the triangulation is always power Delaunay. In theory, dynamic Ricci flow guarantees the existence of solutions to the flow with target curvatures satisfying Gauss–Bonnet condition; in practice, the dynamic Ricci flow is much more robust than conventional method. Our experimental results demonstrate the efficiency, efficacy and robustness of the dynamic Ricci flow method.

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This project has been partially supported by NSFC 11271156, AFOSR FA9550-14-1-0193, NSF DMS-1418255, NSFC 61303078.

Received 3 January 2017

Published 19 April 2018