Isotropic Mesh Simplification by Evolving the Geodesic Delaunay Triangulation Shi-Qing Xin Shuang-Min Chen Ying He School of Computer Engineering Nanyang Technological University Singapore Email: {sqxin|smchen|yhe}@ntu.edu.sg Guo-Jin Wang State Key Laboratory of CAD & CG Department of Mathematics Zhejiang University, China Email: wanggj@zju.edu.cn Xianfeng Gu Hong Qin Department of Computer Science Stony Brook University New York, USA Email: {gu|qin}@cs.sunysb.edu Abstract—In this paper, we present an intrinsic algorithm for isotropic mesh simplification. Starting with a set of unevenly distributed samples on the surface, our method computes the geodesic Delaunay triangulation with regard to the sample set and iteratively evolves the Delaunay triangulation such that the Delaunay edges become almost equal in length. Finally, our method outputs the simplified mesh by replacing each curved Delaunay edge with a line segment. We conduct experiments on numerous real-world models of complicated geometry and topology. The promising experimental results demonstrate that the proposed method is intrinsic and insensitive to initial mesh triangulation. I. I NTRODUCTION Mesh simplification is the process of reducing the number of faces used in the surface while keeping the overall shape, volume and boundaries preserved as much as possible. Typi- cally, mesh simplification is used to improve rendering speed or to minimize data size or compression requirements [1]. Topology-preserving and feature-preserving schemes are of- ten desired. Mesh simplification has been extensively studied in the past two decades. There exists a rich body of literatures in mesh simplification. The representative works include progressive meshes [2] and quadric error metrics scheme [3]. We refer the readers to [4], [5] for a complete survey. In this paper, we present a new method for isotropic mesh simplification. Our method is different than the existing approaches in that geodesic Delaunay triangulation is em- ployed. Therefore, the proposed method is intrinsic and independent of the embedding space. Starting with a set of initial sample points that are arbitrarily distributed on the input mesh, our method computes the geodesic Delaunay triangulation with regard to the sample set and iteratively evolves the Delaunay triangulation such that the Delaunay edges become almost equal in length. Our evolving step iteratively invokes the following operations: 1) Compute the geodesic Delaunay triangulation based on the sample set; 2) For each sample point, compute a standard deviation of the first order incident Delaunay edge lengths and predict a better sample point for substitute; 3) Update the sample point set. The iterative algorithm terminates when the maximum stan- dard deviation is less than a prescribed tolerance. Finally, our method outputs the simplified mesh by replacing each curved Delaunay edge with a line segment. Compared with existing methods, our Delaunay evolving algorithm takes advantage of the geodesic Voronoi diagram, and therefore it is intrinsic, robust and insensitive to initial mesh triangulation. Furthermore, our algorithm refines the Delaunay edges to be as equal as possible in length, which leads to a simplified mesh of high triangulation quality. In addition, the evolving is only a local operation and thus can be parallelized. We apply our approach to real-world models with non-trivial topology and geometry. The experimental results demonstrate the efficacy of our algorithm. Figure 1 shows an example of our mesh simplification algorithm. The remaining of the paper is organized as follows: Sec- tion II reviews the related work on mesh simplification, geodesic Delaunay triangulation, discrete geodesics and sur- face sampling. After that, Section III presents our algorithm for isotropic mesh simplification followed by the experi- mental results in Section IV. Finally, Section V draws the conclusion and discusses the future work. II. RELATED WORK Our work is closely related to mesh simplification, geodesic Voronoi diagram/Delaunay triangulation, discrete geodesics, and surface sampling. Mesh simplification The existing mesh simplification algorithms can be classified into the following categories: • Volumetric approach [6] converts the input polygonal mesh into a volumetric description (e.g., voxels), then simplifies the mesh by 3D morphological operators. This approach is robust and flexible in that it can handle mesh degeneracies. • Simplification envelopes [7] use a geometric construc- tion to control the simplification, i.e., building a new surface inside the space formed by the two offsetting surfaces with regard to the user-specified threshold. 2011 Eighth International Symposium on Voronoi Diagrams in Science and Engineering 978-0-7695-4483-0/11 $26.00 © 2011 IEEE DOI 10.1109/ISVD.2011.14 39