The Visual Computer

, Volume 31, Issue 10, pp 1351–1363 | Cite as

Interpolation and parallel adjustment of center-sampled trees with new balancing constraints

  • Byungmoon Kim
  • Panagiotis Tsiotras
  • Jeong-Mo Hong
  • Oh-young Song
Original Article


We present a novel tree balancing constraint that is slightly stronger than the well-known 2-to-1 balancing constraint used in octree data structures (Tu and O’hallaron, Balanced refinement of massive linear octrees. Tech. Rep. CMU-CS-04-129. Carnegie Mellon School of Computer Science, Pennsylvania, 2004). The new balancing produces a limited number of local cell connectivity types (stencils): 5 for a quadtree and 21 for an octree. Using this constraint, we interpolate the data sampled at cell centers using weights pre-computed by interpolation or by generating interpolation codes for each stencil. In addition, we develop a parallel tree adjustment algorithm, and show that the imposed balancing constraint is satisfied even when the tree is adjusted in parallel. We also show that the adjustment has high parallelization performance. We finally apply the new balancing scheme to level set image segmentation and smoke simulation problems.


Octree Quadtree Balanced tree  Interpolation Parallelization Smoke simulation Segmentation 



The second author was supported by the US National Science Foundation, award CMMI-0856565. The third author was supported by National Research Foundation of Korea (NRF) (Grant NRF-2011-0023134).

Supplementary material (1.9 mb)
Supplementary material 1 (mov 1975 KB)


  1. 1.
    Adalsteinsson, D., Sethian, J.: Fast level set method for propagating interfaces. J. Comput. Phys. 118, 269–277 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bai, Y., Han, X., Prince, J.L.: Octree grid topology preserving geometric deformable model for three-dimensional medical image segmentation. In: Information Processing in Medical Imaging (IPMI 2007), pp. 20:556–68 (2007)Google Scholar
  3. 3.
    Benson, D., Davis, J.: Octree textures. ACM Transactions on Graphics. In: Proc. of SIGGRAPH, 21, pp. 785–790 (2002)Google Scholar
  4. 4.
    Brox, T., Weickert, J.: Level set segmentation with multiple regions. IEEE Trans. Image Process. 15(10), 3213–3218 (2006)CrossRefGoogle Scholar
  5. 5.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. In: International Conference of Computer Vision (ICCV), pp. 694–699 (1995)Google Scholar
  6. 6.
    Chan, T.F., Sandberg, B.Y., Vese, L.A.: Active contours without edges for vector-valued images. J. Vis. Commun. Image Represent. 11(2), 130–141 (2000)CrossRefGoogle Scholar
  7. 7.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (1999)CrossRefGoogle Scholar
  8. 8.
    Chen, H., Min, C.H., Gibou, F.: A supra-convergent finite difference scheme for the poisson and heat equations on irregular domains and non-graded adaptive cartesian grids. J. Sci. Comput. 31, 19–60 (2007)Google Scholar
  9. 9.
    Cremers, D.: A variational framework for image segmentation combining motion estimation and shape regularization. In: IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), pp. 53–58 (2003)Google Scholar
  10. 10.
    DeBry, D., Gibbs, J., Petty, D.D., Robins, N.: Painting and rendering textures on unparameterized models. ACM Transactions on Graphics. In: Proc. of SIGGRAPH, 21, pp. 763–768 (2002)Google Scholar
  11. 11.
    Foster, N., Fedkiw, R.: Practical animation of liquids. In: ACM SIGGRAPH, pp. 15–22 (2001)Google Scholar
  12. 12.
    Frisken, S.F., Perry, R.N., Rockwood, A.P., Jones, T.R.: Adaptively sampled distance fields: a general representation of shape for computer graphics. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’00, pp. 249–254. ACM Press/Addison-Wesley Publishing Co., New York, NY, USA (2000). doi: 10.1145/344779.344899
  13. 13.
    Gibou, F., Min, C.H., Ceniceros, H.: Finite difference schemes for incompressible flows on non-graded adaptive cartesian grids. Fluid Dyn. Mater. Process. 154, 199–208 (2007)MathSciNetGoogle Scholar
  14. 14.
    Ju, T., Losasso, F., Schaefer, S., Warren, J.: Dual contouring of hermite data. ACM Transactions on Graphics. In: Proc. of SIGGRAPH, 21(3), pp. 339–346 (2002)Google Scholar
  15. 15.
    Kim, B., Tsiotras, P.: Image segmentation on cell-center sampled quadtree and octree grids. In: Proceedings of SPIE Electronic Imaging / Wavelet Applications in Industrial Processing VI, pp. 265–278 (2009)Google Scholar
  16. 16.
    Losasso, F., Gibou, F., Fedkiw, R.: Simulating water and smoke with an octree data structure. In: ACM SIGGRAPH, pp. 457–462 (2004)Google Scholar
  17. 17.
    Malladi, R., Sethian, J.A., Vemuri, B.C.: Evolutionary fronts for topology-independent shape modeling and recovery. In: Proceedings of the third European conference on Computer vision, pp. 1–13 (1994)Google Scholar
  18. 18.
    Milne, B.: Adaptive Level Set Methods Interfaces. PhD thesis, Dept. of Mathematics, University of California, Berkeley, CA (1995)Google Scholar
  19. 19.
    Min, C.H., Gibou, F.: A second order accurate projection method for the incompressible navier-stokes equations on fully adaptive grids. J. Comput. Phys. 219, 912–929 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Min, C.H., Gibou, F.: A second order accurate level set method on non-graded adaptive grids. J. Comput. Phys. 225, 300–321 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Min, C.H., Gibou, F., Ceniceros, H.: A supra-convergent finite difference scheme for the variable coefficient poisson equation on fully adaptive grids. J. Comput. Phys. 202, 577–601 (2006)MathSciNetGoogle Scholar
  22. 22.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on hamilton-jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Osher, S.J., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin. ISBN 0-387-95482-1 (2002)Google Scholar
  24. 24.
    Parashar, M., Browne, J.C.: Distributed dynamic data-structures for parallel adaptive mesh-refinement. In: Proceedings of the international conference for high performance computing (1995)Google Scholar
  25. 25.
    Plewa, T., Linde, T., Weirs (Editors), V.G.: Adaptive Mesh Refinement - Theory and Applications. In: Proceedings of the Chicago Workshop on Adaptive Mesh Refinement Methods, Sept. 3–5, 2003. Lecture Notes in Computational Science and Engineering, Vol. 41. Springer, Berlin (2003)Google Scholar
  26. 26.
    Sagan, H.: Space-Filling Curves. Springer, Berlin (1994)Google Scholar
  27. 27.
    Schaefer, S., Warren, J.: Dual marching cubes: primal contouring of dual grids. In: Proceedings of Pacific Graphics, pp. 70–76 (2004)Google Scholar
  28. 28.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge. ISBN 0-521-64557-3 (1999)Google Scholar
  29. 29.
    Stam, J.: Stable fluids. In: ACM SIGGRAPH, pp. 121–128 (1999)Google Scholar
  30. 30.
    Strain, J.: Fast tree-based redistancing for level set computations. J. Comput. Phys. 152(2), 648–666 (1999)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Tu, T., O’hallaron, D.R.: Balanced refinement of massive linear octrees. Tech. Rep. CMU-CS-04-129, Carnegie Mellon School of Computer Science, Pennsylvania (2004)Google Scholar
  32. 32.
    Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the mumford and shah model. IEEE Trans. Image Process. 50(3), 271–293 (2002)zbMATHGoogle Scholar
  33. 33.
    Westermann, R., Kobbelt, L., Ertl, T.: Real-time exploration of regular volume data by adaptive reconstruction of isosurfaces. Vis. Comput. 15(2), 100–111 (1999)CrossRefGoogle Scholar
  34. 34.
    Wilhelms, J., Gelder, A.V.: Octrees for faster isosurface generation. ACM Trans. Graph. 11(3), 201–227 (1991)CrossRefGoogle Scholar
  35. 35.
    Yerry, M.A., Shephard, M.S.: Automatic three-dimensional mesh generation by the modified-octree technique. Int. J. Numer. Methods Eng. 20(11), 1965–1990 (1984)zbMATHCrossRefGoogle Scholar
  36. 36.
    Zhou, K., Gong, M., Huang, X., Guo, B.: Data-parallel octrees for surface reconstruction. IEEE Trans. Vis. Comput. Graph. 177(5), 681–699 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Byungmoon Kim
    • 1
  • Panagiotis Tsiotras
    • 2
  • Jeong-Mo Hong
    • 3
  • Oh-young Song
    • 4
  1. 1.Adobe SystemsSan JoseUSA
  2. 2.Georgia Institute of TechnologyAtlantaUSA
  3. 3.Dongguk UniversitySeoulSouth Korea
  4. 4.Sejong UniversitySeoulSouth Korea

Personalised recommendations