The Visual Computer

, Volume 28, Issue 11, pp 1115–1125 | Cite as

Analytical solutions for sketch-based convolution surface modeling on the GPU

  • Xiaoqiang Zhu
  • Xiaogang JinEmail author
  • Shengjun Liu
  • Hanli Zhao
Original Article


Convolution surfaces are attractive for modeling objects of complex evolving topology. This paper presents some novel analytical convolution solutions for planar polygon skeletons with both finite-support and infinite-support kernel functions. We convert the double integral over a planar polygon into a simple integral along the contour of the polygon based on Green’s theorem, which reduces the computational cost and allows for efficient parallel computation on the GPU. For finite support kernel functions, a skeleton clipping algorithm is presented to compute the valid skeletons. The analytical solutions are integrated into a prototype modeling system on the GPU (Graphics Processing Unit). Our modeling system supports point, polyline and planar polygon skeletons. Complex objects with arbitrary genus can be modeled easily in an interactive way. Resulting convolution surfaces with high quality are rendered with interactive ray casting.


Convolution surface Closed-form solution Planar polygon skeleton Sketch-based modeling CUDA 



Xiaogang Jin was supported by the National Key Basic Research Foundation of China (Grant No. 2009CB320801), the NSFC-MSRA Joint Funding (Grant No. 60970159), the National Natural Science Foundation of China (Grant No. 60933007), and the Zhejiang Provincial Natural Science Foundation of China (Grant No. Z1110154). Shengjun Liu was supported by the National Natural Science Foundation of China (Grant No. 61173119).

Supplementary material

(MP4 13.3 MB)


  1. 1.
    Alexe, A., Gaildrat, V., Barthe, L.: Interactive modelling from sketches using spherical implicit functions. In: Proceedings of the 3rd International Conference on Computer Graphics, Virtual Reality, Visualisation and Interaction in Africa, AFRIGRAPH ’04, pp. 25–34. ACM Press, New York (2004) CrossRefGoogle Scholar
  2. 2.
    Alexe, A., Barthe, L., Cani, M., Gaildrat, V.: Shape modeling by sketching using convolution surfaces. In: Pacific Graphics, Short paper (2005) Google Scholar
  3. 3.
    Angelidis, A., Cani, M.P.: Adaptive implicit modeling using subdivision curves and surfaces as skeletons. In: Proceedings of the 7th ACM symposium on Solid Modeling and Applications, SMA’02, pp. 45–52. ACM Press, New York (2002) CrossRefGoogle Scholar
  4. 4.
    Bernhardt, A., Pihuit, A., Cani, M.P., Barthe, L.: Matisse: painting 2d regions for modeling free-form shapes. In: EUROGRAPHICS Workshop on Sketch-Based Interfaces and Modeling, SBIM’08, pp. 57–64. Eurographics Association, Annecy (2008) Google Scholar
  5. 5.
    Bloomenthal, J., Shoemake, K.: Convolution surfaces. Comput. Graph. 25(4), 251–256 (1991) CrossRefGoogle Scholar
  6. 6.
    David, F.: Procedural Elements for Computer Graphics, 2nd edn. McGraw-Hill, New York (1997) Google Scholar
  7. 7.
    Eyiyurekli, M., Grimm, C., Breen, D.: Editing level-set models with sketched curves. In: Proceedings of the 6th Eurographics Symposium on Sketch-Based Interfaces and Modeling, SBIM’09, p. 45–52. Eurographics Association, New Orleans (2009) CrossRefGoogle Scholar
  8. 8.
    Gourmel, O., Pajot, A., Paulin, M., Barthe, L., Poulin, P.: Fitted bvh for fast raytracing of metaballs. Comput. Graph. Forum 29(2), 281–288 (2010) CrossRefGoogle Scholar
  9. 9.
  10. 10.
    Hubert, E.: Convolution surfaces based on polygons for infinite and compact support kernels. Graph. Models 74(1) (2012). doi: 10.1016/j.gmod.2011.07.001
  11. 11.
    Igarashi, T., Matsuoka, S., Tanaka, H.: Teddy: a sketching interface for 3d freeform design. In: SIGGRAPH, pp. 409–416. ACM Press, New York (1999) Google Scholar
  12. 12.
    Jin, X., Tai, C.: Analytical methods for polynomial weighted convolution surfaces with various kernels. Comput. Graph. 26(3), 437–447 (2002) CrossRefGoogle Scholar
  13. 13.
    Jin, X., Tai, C.: Convolution surfaces for arcs and quadratic curves with a varying kernel. Vis. Comput. 18(8), 530–546 (2002) CrossRefGoogle Scholar
  14. 14.
    Jin, X., Tai, C., Feng, J., Peng, Q.: An analytical convolution surface model for line skeletons with polynomial weighted distributions. J. Graph. Tools 6(3), 1–12 (2001) CrossRefGoogle Scholar
  15. 15.
    Jin, X., Tai, C., Zhang, H.: Implicit modeling from polygon soup using convolution. Vis. Comput. 25(3), 279–288 (2009) CrossRefGoogle Scholar
  16. 16.
    Kanamori, Y., Szego, Z., Nishita, T.: GPU-based fast ray casting for a large number of metaballs. Comput. Graph. Forum 27(3), 351–360 (2008) CrossRefGoogle Scholar
  17. 17.
    Karpenko, O., Hughes, J.: Smoothsketch: 3d free-form shapes from complex sketches. ACM Trans. Graph. 25(3), 589–598 (2006) CrossRefGoogle Scholar
  18. 18.
    Karpenko, O., Hughes, J., Raskar, R.: Free-from sketching with variational implicit surfaces. Comput. Graph. Forum 21(3), 585–594 (2002) CrossRefGoogle Scholar
  19. 19.
    Knoll, A., Hijazi, Y., Kensler, A., Scjptt, M.: Fast ray tracing of arbitrary implicit surfaces with interval and affine arithmetic. Comput. Graph. Forum 28(1), 26–40 (2007) CrossRefGoogle Scholar
  20. 20.
    Kravtsov, D., Fryazinov, O., Adzhiev, V., Pasko, A., Comninos, P.: Embedded implicit stand-ins for animated meshes: a case of hybrid modelling. Comput. Graph. Forum 29(1), 128–140 (2010) CrossRefGoogle Scholar
  21. 21.
    Lorensen, W., Cline, H.: Marching cubes: a high resolution 3d surface construction algorithm. Comput. Graph. 21(4), 163–169 (1987) CrossRefGoogle Scholar
  22. 22.
    McCormack, J., Sherstyuk, A.: Creating and rendering convolution surfaces. Comput. Graph. Forum 17(2), 113–120 (1998) CrossRefGoogle Scholar
  23. 23.
    Schmidt, R., Wyvill, B., Sousa, M., Jorge, J.: Shapeshop: sketch-based solid modeling with blobtrees. In: EUROGRAPHICS Workshop on Sketch-Based Interfaces and Modeling, SBIM’05, pp. 53–62. Eurographics Association, Dublin (2005) Google Scholar
  24. 24.
    Schmidt, R., Wyvill, B., Sousa, M., Jorge, J.: Sketch-based modeling with the blobtree. In: SIGGRAPH, Technical Sketch. ACM Press, New York (2005) Google Scholar
  25. 25.
    Sherstyuk, A.: Fast ray tracing of implicit surfaces. Comput. Graph. Forum 18(2), 139–147 (1999) CrossRefGoogle Scholar
  26. 26.
    Sherstyuk, A.: Kernel functions in convolution surfaces: a comparative analysis. Vis. Comput. 15(4), 171–182 (1999) zbMATHCrossRefGoogle Scholar
  27. 27.
    Tai, C., Zhang, H., Fong, C.: Prototype modeling from sketched silhouettes based on convolution surfaces. Comput. Graph. Forum 23(4), 71–83 (2004) CrossRefGoogle Scholar
  28. 28.
    Wilfred, K.: Advanced Calculus, 5th edn. Addison-Wesley Longman, Boston (2002) Google Scholar
  29. 29.
    Wyvill, B., Overveld, K.: Tilling techniques for implicit skeletal models. In: SIGGRAPH courses. ACM Press, New York (1996) Google Scholar
  30. 30.
    Wyvill, B., Guy, A., Galin, E.: Extending the csg tree: warping, blending and boolean operations in an implicit surface modeling system. Comput. Graph. Forum 18(2), 149–158 (1999) CrossRefGoogle Scholar
  31. 31.
    Wyvill, B., Foster, K., Jepp, P., Schmidt, R., Sousa, M., Jorge, J.: Sketch based construction and rendering of implicit models. In: EUROGRAPHICS, Workshop on Computational Aesthetics in Graphics, Visualization and Image, pp. 67–74. Eurographics Association, Girona (2005) Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Xiaoqiang Zhu
    • 1
  • Xiaogang Jin
    • 1
    Email author
  • Shengjun Liu
    • 2
  • Hanli Zhao
    • 3
  1. 1.State Key Lab of CAD&CGZhejiang UniversityHangzhouP.R. China
  2. 2.School of Mathematical Science and Computing TechnologyCentral South UniversityChangshaP.R. China
  3. 3.College of Physics & Electronic Information EngineeringWenzhou UniversityWenzhouP.R. China

Personalised recommendations