The Community for Technology Leaders
RSS Icon
Issue No.05 - May (2013 vol.19)
pp: 736-748
Meng Qi , GR/ST/21-256C, Nat. Univ. of Singapore, Singapore, Singapore
Thanh-Tung Cao , Nat. Univ. of Singapore, Singapore, Singapore
Tiow-Seng Tan , Sch. of Comput., Nat. Univ. of Singapore, Singapore, Singapore
We propose the first graphics processing unit (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 existing CPU algorithms to solve the CDT problem in computational geometry, yet there has been no prior approach to solve this problem efficiently using the parallel computing power of the GPU. For the special case of the CDT problem where the PSLG consists of just points, which is simply the normal Delaunay triangulation (DT) problem, a hybrid approach using the GPU together with the CPU to partially speed up the computation has already been presented in the literature. Our work, on the other hand, accelerates the entire computation on the GPU. Our implementation using the CUDA programming model on NVIDIA GPUs is numerically robust, and runs up to an order of magnitude faster than the best sequential implementations on the CPU. This result is reflected in our experiment with both randomly generated PSLGs and real-world GIS data having millions of points and edges.
Graphics processing units, Instruction sets, Arrays, Strips, Standards, Color,image vectorization, GPGPU, parallel computation, computational geometry, Voronoi diagram
Meng Qi, Thanh-Tung Cao, Tiow-Seng Tan, "Computing 2D Constrained Delaunay Triangulation Using the GPU", IEEE Transactions on Visualization & Computer Graphics, vol.19, no. 5, pp. 736-748, May 2013, doi:10.1109/TVCG.2012.307
[1] F. Aurenhammer, "Voronoi Diagrams - A Survey of a Fundamental Geometric Data Structure," ACM Computing Surveys, vol. 23, no. 3, pp. 345-405, 1991.
[2] J. Bernal, "Inserting Line Segments into Triangulations and Tetrahedralizations," Technical Report 5596, Nat'l Inst. of Standards and Tech nology, 1995.
[3] J.-D. Boissonnat, "Shape Reconstruction from Planar Cross Sections," Computer Vision, Graphics, and Image Processing, vol. 44, no. 1, pp. 1-29, 1988.
[4] T.-T. Cao, H. Edelsbrunner, and T.-S. Tan, "Proof of Correctness of the Digital Delaunay Triangulation Algorithm," http://www. notes-30-april-2011.pdf . 2011.
[5] T.-T. Cao, K. Tang, A. Mohamed, and T.-S. Tan, "Parallel Banding Algorithm to Compute Exact Distance Transform with the GPU," Proc. ACM Symp. Interactive 3D Graphics and Games (I3D '10), pp. 83-90, 2010.
[6] CGAL, "CGAL, Computational Geometry Algorithms Library," http:/, 2011.
[7] L.P. Chew, "Constrained Delaunay Triangulations," Algorithmica, vol. 4, pp. 97-108, 1989.
[8] V. Domiter, "Constrained Delaunay Triangulation Using Plane Subdivision," Proc. Eighth Central European Seminar on Computer Graphics, pp. 105-110, 2004.
[9] R. Dwyer, "A Faster Divide-and-Conquer Algorithm for Constructing Delaunay Triangulations," Algorithmica, vol. 2, pp. 137-151, 1987.
[10] I. Fischer and C. Gotsman, "Fast Approximation of High-Order Voronoi Diagrams and Distance Transforms on the GPU," J. Graphics Tools, vol. 11, no. 4, pp. 39-60, 2006.
[11] S. Fortune, "A Sweepline Algorithm for Voronoi Diagrams," Algorithmica, vol. 2, pp. 153-174, 1987.
[12] S. Fortune, "Voronoi diagrams and Delaunay triangulations," Handbook of Discrete and Computational Geometry. B. Raton, ed., pp. 377-388, CRC Press, Inc., 1997.
[13] C.M. Gold, "A Review of Potential Applications of Voronoi Methods in Geomatics," Proc. Canadian Conf. GIS, pp. 1647-1656, 1994.
[14] R.L. Graham, "An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set," Information Processing Letters, vol. 1, no. 4, pp. 132-133, 1972.
[15] L. Guibas, D. Knuth, and M. Sharir, "Randomized Incremental Construction of Delaunay and Voronoi Diagrams," Algorithmica, vol. 7, pp. 381-413, 1992.
[16] P.T. Highnam, "The Ears of a Polygon," Information Processing Letters, pp. 196-198, 1982.
[17] K.E. HoffIII, J. Keyser, M. Lin, D. Manocha, and T. Culver, "Fast Computation of Generalized Voronoi Diagrams Using Graphics Hardware," Proc. ACM SIGGRAPH '99, pp. 277-286, 1999.
[18] K.H. Huebner, D.L. Dewhirst, D.E. Smith, and T.G. Byrom, The Finite Element Method for Engineers. Wiley, 2001.
[19] M. Kallmann, "Shortest Paths with Arbitrary Clearance from Navigation Meshes," Proc. ACM SIGGRAPH, pp. 159-168, 2010.
[20] D. Lee and A. Lin, "Generalized Delaunay Triangulation for Planar Graphs," Discrete and Computational Geometry, vol. 1, pp. 201-217, 1986.
[21] NVIDIA, "NVIDIA CUDA C Programming Guide," DevZone/docs/html/C/docCUDA_C_Programming_Guide.pdf , 2012.
[22] L. Prasad and A.N. Skourikhine, "Vectorized Image Segmentation via Trixel Agglomeration," Pattern Recognition, vol. 39, pp. 501-514, Apr. 2006.
[23] F.P. Preparata and M.I. Shamos, Computational Geometry: An Introduction, Springer-Verlag, Inc., 1985.
[24] M. Qi, T.-T. Cao, and T.-S. Tan, "Computing 2D Constrained Delaunay Triangulation Using the GPU," Proc. ACM SIGGRAPH, pp. 39-46, 2012.
[25] G. Rong, T.-S. Tan, T.-T. Cao, and Stephanus, "Computing Two-Dimensional Delaunay Triangulation using Graphics Hardware," Proc. Symp. Interactive 3D Graphics and Games (I3D '08), pp. 89-97, 2008.
[26] M.I. Shamos and D. Hoey, "Closest-Point Problems," Proc. 16th Ann. Symp. Foundations of Computer Science (FOCS '75), pp. 151-162, 1975.
[27] J. Shewchuk, "Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator," Applied Computational Geometry Towards Geometric Eng., vol. 1148, pp. 203-222, 1996.
[28] J.R. Shewchuk, "Robust Adaptive Floating-Point Geometric Predicates," Proc. 12th Ann. Symp. Computational Geometry (SoCG '96), pp. 141-150, 1996.
[29] P. Su and R.L. Scot Drysdale, "A Comparison of Sequential Delaunay Triangulation Algorithms," Computational Geometry: Theory and Applications, vol. 7, pp. 361-385, Apr. 1997.
[30] L.A. Treinish, "Visualization of Scattered Meteorological Data," IEEE Computer Graphics and Applications, vol. 15, no. 4, pp. 20-26, July 1995.
40 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool