Journal of Scientific Computing

, Volume 46, Issue 2, pp 243–264 | Cite as

A Memory and Computation Efficient Sparse Level-Set Method

  • Wladimir J. van der Laan
  • Andrei C. Jalba
  • Jos B. T. M. RoerdinkEmail author
Open Access


Since its introduction, the level set method has become the favorite technique for capturing and tracking moving interfaces, and found applications in a wide variety of scientific fields. In this paper we present efficient data structures and algorithms for tracking dynamic interfaces through the level set method. Several approaches which address both computational and memory requirements have been very recently introduced. We show that our method is up to 8.5 times faster than these recent approaches. More importantly, our algorithm can greatly benefit from both fine- and coarse-grain parallelization by leveraging SIMD and/or multi-core parallel architectures.


Level sets Sparse-grid method Tile management 


  1. 1.
    Adalsteinsson, D., Sethian, J.A.: A fast level set method for propagating interfaces. J. Comput. Phys. 118(2), 269–277 (1995). zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bridson, R.E.: Computational aspects of dynamic surfaces. Ph.D. thesis, Stanford University, Stanford, CA, USA (2003). Adviser-Ronald Fedkiw Google Scholar
  3. 3.
    Chopp, D.L.: Computing minimal surfaces via level set curvature flow. J. Comput. Phys. 106(1), 77–91 (1993). zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Droske, M., Meyer, B., Rumpf, M., Schaller, C.: An adaptive level set method for medical image segmentation. In: IPMI ’01: Proceedings of the 17th International Conference on Information Processing in Medical Imaging, pp. 416–422. Springer, London (2001) Google Scholar
  5. 5.
    Enright, D., Fedkiw, R., Ferziger, J., Mitchell, I.: A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183, 83–116 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Enright, D., Losasso, F., Fedkiw, R.: A fast and accurate semi-Lagrangian particle level set method. Comput. Struct. 83(6–7), 479–490 (2005) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Fournier, M., Dischler, J.M., Bechmann, D.: 3D distance transform adaptive filtering for smoothing and denoising triangle meshes. In: GRAPHITE ’06: Proceedings of the 4th International Conference on Computer Graphics and Interactive Techniques in Australasia and Southeast Asia, pp. 407–416. ACM, New York (2006). CrossRefGoogle Scholar
  8. 8.
    Houston, B., Nielsen, M.B., Batty, C., Nilsson, O., Museth, K.: Hierarchical RLE level set: a compact and versatile deformable surface representation. ACM Trans. Graph. 25(1), 151–175 (2006) CrossRefGoogle Scholar
  9. 9.
    Lefohn, A.E., Kniss, J.M., Hansen, C.D., Whitaker, R.T.: A streaming narrow-band algorithm: interactive computation and visualization of level sets. IEEE Trans. Vis. Comput. Graph. 10(4), 422–433 (2004) CrossRefGoogle Scholar
  10. 10.
    Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994). zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Losasso, F., Gibou, F., Fedkiw, R.: Simulating water and smoke with an octree data structure. In: SIGGRAPH ’04: ACM SIGGRAPH 2004 Papers, pp. 457–462. ACM, New York (2004). CrossRefGoogle Scholar
  12. 12.
    Min, C.: Local level set method in high dimension and codimension. J. Comput. Phys. 200(1), 368–382 (2004). zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Nielsen, M.B., Museth, K.: Dynamic Tubular Grid: An efficient data structure and algorithms for high resolution level sets. J. Sci. Comput. 26(3), 261–299 (2006). zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Osher, S., Fedkiw, R.: Level set methods: an overview and some recent results. J. Comput. Phys. 169(2), 463–502 (2001). zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Osher, S., Fedkiw, R.: Level-Set Methods and Dynamic Implicit Surfaces. Springer, New York (2002) Google Scholar
  16. 16.
    Osher, S., Chan, T., dong Liu, X., dong Liu, X.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDE-based fast local level set method. J. Comput. Phys. 155(2), 410–438 (1999). zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, New York (1982) zbMATHGoogle Scholar
  20. 20.
    Sethian, J.A.: Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1996) zbMATHGoogle Scholar
  21. 21.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
  22. 22.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988). zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Strain, J.: Tree methods for moving interfaces. J. Comput. Phys. 151(2), 616–648 (1999). zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Tsai, R., Osher, S.: Level set methods and their applications in image science. Commun. Math. Sci. 1(4), 623–656 (2003) zbMATHMathSciNetGoogle Scholar
  25. 25.
    Whitaker, R.T.: A level-set approach to 3D reconstruction from range data. Int. J. Comput. Vis. 29(3), 203–231 (1998). CrossRefGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  • Wladimir J. van der Laan
    • 1
  • Andrei C. Jalba
    • 2
  • Jos B. T. M. Roerdink
    • 1
    Email author
  1. 1.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations