research-article

On centroidal voronoi tessellation—energy smoothness and fast computation

Authors:

Chenglei YangAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 28, Issue 4

Article No.: 101, Pages 1 - 17

https://doi.org/10.1145/1559755.1559758

Published: 08 September 2009 Publication History

Abstract

Centroidal Voronoi tessellation (CVT) is a particular type of Voronoi tessellation that has many applications in computational sciences and engineering, including computer graphics. The prevailing method for computing CVT is Lloyd's method, which has linear convergence and is inefficient in practice. We develop new efficient methods for CVT computation and demonstrate the fast convergence of these methods. Specifically, we show that the CVT energy function has 2nd order smoothness for convex domains with smooth density, as well as in most situations encountered in optimization. Due to the 2nd order smoothness, it is possible to minimize the CVT energy functions using Newton-like optimization methods and expect fast convergence. We propose a quasi-Newton method to compute CVT and demonstrate its faster convergence than Lloyd's method with various numerical examples. It is also significantly faster and more robust than the Lloyd-Newton method, a previous attempt to accelerate CVT. We also demonstrate surface remeshing as a possible application.

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On centroidal voronoi tessellation—energy smoothness and fast computation

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cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 28, Issue 4

August 2009

116 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/1559755

Issue’s Table of Contents

Copyright © 2009 ACM.

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Publication History

Published: 08 September 2009

Accepted: 01 May 2009

Received: 01 December 2008

Published in TOG Volume 28, Issue 4

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Ministry of Science and Technology of the People's Republic of China
European Research Council
National Natural Research Council of China
Research Grants Council, University Grants Committee, Hong Kong

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