Abstract
Voronoi diagrams on triangulated surfaces based on the geodesic metric play a key role in many applications of computer graphics. Previous methods of constructing such Voronoi diagrams generally depended on having an exact geodesic metric. However, exact geodesic computation is time-consuming and has high memory usage, limiting wider application of geodesic Voronoi diagrams (GVDs). In order to overcome this issue, instead of using exact methods, we reformulate a graph method based on Steiner point insertion, as an effective way to obtain geodesic distances. Further, since a bisector comprises hyperbolic and line segments, we utilize Apollonius diagrams to encode complicated structures, enabling Voronoi diagrams to encode a medial-axis surface for a dense set of boundary samples. Based on these strategies, we present an approximation algorithm for efficient Voronoi diagram construction on triangulated surfaces. We also suggest a measure for evaluating similarity of our results to the exact GVD. Although our GVD results are constructed using approximate geodesic distances, we can get GVD results similar to exact results by inserting Steiner points on triangle edges. Experimental results on many 3D models indicate the improved speed and memory requirements compared to previous leading methods.
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The authors would like to thank the reviewers for their valuable suggestions. This work was supported in part by the Youth Teacher Development Foundation of Harbin Institute of Technology (IDGA10002143), the National Natural Science Foundation of China (62072139, 62272277, 62072284), the National Key R&D Program of China (2021YFB1715900), and the Joint Funds of the National Natural Science Foundation of China (U22A2033).
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Wenlong Meng received his Ph.D. degree from Shandong University, in 2022. He is currently a lecturer with the School of Computer Science and Technology, Harbin Institute of Technology, Weihai. His research interests include computational geometry, computer graphics, and computer-aided design.
Pengbo Bo received his B.S. and M.E. degrees in computer science at the School of Computer Science, Shandong University, Jinan, in 2001 and 2004, respectively, and his Ph.D. degree in computer science from the University of Hong Kong, in 2010. He is a professor and doctoral supervisor at the School of Computer Science and Technology, Harbin Institute of Technology, Weihai. His research interests include computer graphics, 3D visual computing, and CNC machining.
Xiaodong Zhang received his M.S. degree in Harbin Institute of Technology. He is currently a lecturer with the School of Computer Science and Technology, Harbin Institute of Technology, Weihai. His research interests include design and analysis of algorithms.
Jixiang Hong is an undergraduate student at Taishan College, Shandong University. He entered the university in 2019. He majors in computer science and technology. His interests are focused in the area of computer graphics.
Shiqing Xin received his Ph.D. degree from Zhejiang University, in 2009. He was a research fellow with Nangyang Technological University, Singapore, for three years. He is currently an associate professor with the School of Computer Science, Shandong University. He has authored or coauthored more than 60 papers in top journals and conferences, including IEEE Transactions on Visualization and Computer Graphics and ACM Transactions on Graphics. His research interests include various geometry processing algorithms, especially geodesic computation approaches and Voronoi/power tessellation methods. He was the recipient of three best paper awards and many other academic awards.
Changhe Tu received his B.Sc., M.Eng., and Ph.D. degrees from Shandong University, in 1990, 1993, and 2003, respectively. He is currently a professor with the School of Computer Science and Technology, Shandong University. He currently leads the CG-VIS Group, Shandong University. He has authored or coauthored more than 100 papers in international journals and conferences. His research interests include computer graphics, 3D vision, and computer-aided geometric design.
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Meng, W., Bo, P., Zhang, X. et al. An efficient algorithm for approximate Voronoi diagram construction on triangulated surfaces. Comp. Visual Media 9, 443–459 (2023). https://doi.org/10.1007/s41095-022-0326-0
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DOI: https://doi.org/10.1007/s41095-022-0326-0