ABSTRACT
We propose the first GPU solution to compute the 2D constrained Delaunay triangulation (CDT) of a planar straight line graph (PSLG) consisting of points and edges. There are many CPU algorithms developed to solve the CDT problem in computational geometry, yet there has been no known prior approach using the parallel computing power of the GPU to solve this problem efficiently. For the special case of the CDT problem with a PSLG consisting of just points, which is the normal Delaunay triangulation problem, a hybrid approach has already been presented that uses the GPU together with the CPU to partially speed up the computation. Our work, on the other hand, accelerates the whole computation by the GPU. Our implementation using the CUDA programming model on NVIDIA GPUs is numerically robust with good speedup, of up to an order of magnitude, compared to the best sequential implementations on the CPU. This result is reflected in our experiment with both randomly generated PSLGs and real world GIS data with millions of points and edges.
Supplemental Material
- Aurenhammer, F. 1991. Voronoi diagrams -- a survey of a fundamental geometric data structure. ACM Computing Surveys 23, 3, 345--405. Google ScholarDigital Library
- Boissonnat, J.-D. 1988. Shape reconstruction from planar cross sections. Computer Vision, Graphics, and Image Processing 44, 1, 1--29. Google ScholarDigital Library
- Cao, T.-T., Edelsbrunner, H., and Tan, T.-S. 2010. Proof of correctness of the digital Delaunay triangulation algorithm. http://www.comp.nus.edu.sg/~tants/delaunay2DDownload_files/notes-30-april-2011.pdf.Google Scholar
- Cao, T.-T., Tang, K., Mohamed, A., and Tan, T.-S. 2010. Parallel banding algorithm to compute exact distance transform with the GPU. In I3D '10: Proc. ACM Symp. Interactive 3D Graphics and Games, ACM, New York, NY, USA, 83--90. Google ScholarDigital Library
- CGAL, 2011. CGAL, Computational Geometry Algorithms Library. http://www.cgal.org.Google Scholar
- Chew, L. P. 1989. Constrained Delaunay triangulations. Algorithmica 4, 97--108.Google ScholarDigital Library
- Dwyer, R. 1987. A faster divide-and-conquer algorithm for constructing delaunay triangulations. Algorithmica 2, 137--151.Google ScholarDigital Library
- Fischer, I., and Gotsman, C. 2006. Fast approximation of high-order voronoi diagrams and distance transforms on the GPU. J. Graphics Tools 11, 4, 39--60.Google ScholarCross Ref
- Fortune, S. 1987. A sweepline algorithm for Voronoi diagrams. Algorithmica 2, 153--174.Google ScholarDigital Library
- Fortune, S. 1997. Handbook of discrete and computational geometry. CRC Press, Inc., Boca Raton, FL, USA, ch. Voronoi diagrams and Delaunay triangulations, 377--388. Google ScholarDigital Library
- Guibas, L., Knuth, D., and Sharir, M. 1992. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica 7, 381--413.Google ScholarDigital Library
- Highnam, P. T. 1982. The ears of a polygon. In Information Processing Letters, 196--198.Google Scholar
- Hoff, III, K. E., Keyser, J., Lin, M., Manocha, D., and Culver, T. 1999. Fast computation of generalized Voronoi diagrams using graphics hardware. In Proc. ACM SIGGRAPH '99, ACM Press/Addison-Wesley Publishing Co., New York, NY, USA, 277--286. Google ScholarDigital Library
- Kallmann, M., Bieri, H., and Thalmann, D. 2003. Fully dynamic constrained Delaunay triangulations. In Geometric Modelling for Scientific Visualization, G. Brunnett, B. Hamann, and H. Mueller, Eds. Springer-Verlag.Google Scholar
- Lee, D., and Lin, A. 1986. Generalized Delaunay triangulation for planar graphs. Discrete and Computational Geometry 1, 201--217.Google ScholarDigital Library
- NVIDIA, 2011. NVIDIA CUDA C Programming Guide, Version 4.0. http://developer.download.nvidia.com/compute/DevZone/docs/html/C/doc/CUDA_C_Programming_Guide.pdf.Google Scholar
- Prasad, L., and Skourikhine, A. N. 2006. Vectorized image segmentation via trixel agglomeration. Pattern Recognition 39 (April), 501--514. Google ScholarDigital Library
- Rong, G., Tan, T.-S., Cao, T.-T., and Stephanus. 2008. Computing two-dimensional Delaunay triangulation using graphics hardware. In I3D '08: Proc. Symp. Interactive 3D Graphics and Games, ACM, New York, NY, USA, 89--97. Google ScholarDigital Library
- Shewchuk, J. 1996. Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. In Applied Computational Geometry Towards Geometric Engineering, M. Lin and D. Manocha, Eds., vol. 1148 of Lecture Notes in Computer Science. Springer Berlin/Heidelberg, 203--222. Google ScholarDigital Library
- Shewchuk, J. R. 1996. Robust adaptive floating-point geometric predicates. ACM, New York, NY, USA, SoCG '96, 141--150. Google ScholarDigital Library
- Su, P., and Scot Drysdale, R. L. 1997. A comparison of sequential Delaunay triangulation algorithms. Computational Geometry: Theory and Applications 7 (April), 361--385. Google ScholarDigital Library
Index Terms
- Computing 2D constrained Delaunay triangulation using the GPU
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