Arbitrary 3D Resolution Discrete Ray Tracing of Implicit Surfaces

  • Nilo Stolte
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3429)


A new approach to ray tracing implicit surfaces based on recursive space subdivision is presented in this paper. Interval arithmetic, already used to calculate intersections in ray tracing and ray casting (numerically or subdividing 1D or 2D spaces), is now used here to implement a ray tracing based on reliable rays traversals into a potentially infinite octree-like subdivided space, eliminating explicit intersections. Novel, robust and efficient algorithms for ray voxelization and BSP octant ordering are used to recursively traverse rays through the space. Implicit surfaces are robustly voxelized and hierarchically stored into an octree to a certain given level. During rendering, the subdivision based voxelization of surfaces and rays continues further down until a resolution near the discrete domain of the floating point numbers is acquired. To guarantee robustness of the ray voxelization, interval arithmetic with calculations performed under appropriate rounding modes in Pentium-4 x87 and SSE2 FPUs respectively is applied. The major advantage is that the traversal algorithm is guaranteed to find reliable intersections between the rays and the scene without any explicit intersection calculation, solving a known precision problem of the ray traversal in a previous approach, used here for comparison. The precision of the traversal can be arbitrarily increased within the limitation of the floating point representation.


Computer Graphic Interval Arithmetic Implicit Surface Traversal Algorithm Subdivision Level 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Whitted, T.: An Improved Ilumination Model for Shaded Display. Communications of the ACM 23, 343–349 (1980)CrossRefGoogle Scholar
  2. 2.
    Rubin, S.M., Whitted, T.: A 3-Dimensional Representation for Fast Rendering Complex Scenes. Computer Graphics 14, 110–116 (1980)CrossRefGoogle Scholar
  3. 3.
    Kay, T., Kajiya, J.: Ray Tracing Complex Scenes. Computer Graphics 20, 269–278 (1986)CrossRefGoogle Scholar
  4. 4.
    Glassner, A.S.: Space Subdivision for Fast Ray Tracing. IEEE - CGA 10, 15–22 (1984)Google Scholar
  5. 5.
    Jevans, D., Wyvill, B.: Adaptative Voxel Subdivision for Ray Tracing. In: Proceedings of Graphics Interface 1989, Toronto, Ontario, June 1989, pp. 164–172. Canadian Information Processing Society (1989)Google Scholar
  6. 6.
    Gargantini, I.: Ray tracing an Octree: Numerical Evaluation of the First Intersection. Computer Graphics forum 12, 199–210 (1993)CrossRefGoogle Scholar
  7. 7.
    Endl, R., Sommer, M.: Classification of Ray-Generators in Uniform Subdivisions and Octrees for Ray Tracing. protect Computer Graphics forum 13, 3–19 (1994)CrossRefGoogle Scholar
  8. 8.
    Fujimoto, A., Tanaka, T., Iwata, K.: ARTS: Accelerated Ray Tracing System. IEEE - CGA 6, 16–26 (1986)Google Scholar
  9. 9.
    Stolte, N., Caubet, R.: Discrete Ray-Tracing of Huge Voxel Spaces. Computer Graphics Forum 14, 383–394 (1995)CrossRefGoogle Scholar
  10. 10.
    Yagel, R., Cohen, D., Kaufman, A.: Discrete Ray Tracing. IEEE - CGA 12, 19–28 (1992)Google Scholar
  11. 11.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ (1966)zbMATHGoogle Scholar
  12. 12.
    Moore, R.E.: Methods and Application of Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia (1979)Google Scholar
  13. 13.
    Snyder, J.M.: Interval Analysis For Computer Graphics. Computer Graphics 26, 121–130 (1992)CrossRefGoogle Scholar
  14. 14.
    Duff, T.: Interval Arithmetic and Recursive Subdivision for Implicit Functions and Constructive Solid Geometry. Computer Graphics 26, 131–138 (1992)CrossRefGoogle Scholar
  15. 15.
    Goldberg, D.: What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys 23, 5–48 (1991)CrossRefGoogle Scholar
  16. 16.
    de Cusatis Junior, A., de Figueiredo, L.H., Gattass, M.: Interval Methods for Ray Casting Implicit Surfaces with Affine Arithmetic. In: SIBGRAPHI, pp. 17–20 (1999)Google Scholar
  17. 17.
    Kalra, D., Barr, A.: Guaranteed Ray Intersections with Implicit Surfaces. Computer Graphics 23, 297–306 (1989)CrossRefGoogle Scholar
  18. 18.
    Stolte, N.: High Resolution Discrete Spaces: A New Approach for Modeling and Realistic Rendering (Espaces Discrets de Haute Résolutions: Une Nouvelle Approche pour la Modelisation et le Rendu d’Images Réalistes). PhD thesis, Université Paul Sabatier - Toulouse - France (1996)Google Scholar
  19. 19.
    Kaufman, A.: An Algorithm for 3D Scan-Conversion of Polygons. In: Eurographics 1987, Amsterdam, North Holand, August 1987, pp. 197–208 (1987)Google Scholar
  20. 20.
    Kaufman, A.: Efficient Algorithms for 3D Scan-Conversion of Parametric Curves, Surfaces, and Volumes. Computer Graphics 21, 171–179 (1987)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Greene, N.: Voxel Space Automata: Modeling with Stochastic Growth Processes in Voxel Space. Computer Graphics 23, 175–184 (1989)CrossRefGoogle Scholar
  22. 22.
    Taubin, G.: Rasterizing Algebraic Curves and Surfaces. IEEE - CGA, 14–23 (1994)Google Scholar
  23. 23.
    Stolte, N., Caubet, R.: Comparison between different Rasterization Methods for Implicit Surfaces. In: Earnshaw, R., Vince, J.A., Jones, H. (eds.) Visualization and Modeling, pp. 191–201. Academic Press, London (1997)Google Scholar
  24. 24.
    Stolte, N., Kaufman, A.: Novel Techniques for Robust Voxelization and Visualization of Implicit Surfaces. Graphical Models 63, 387–412 (2001)zbMATHCrossRefGoogle Scholar
  25. 25.
    Bidasaria, H.B.: Defining and Rendering of Textured Objects through The Use of Exponential Functions. Graphical Models and Image Processing 54, 97–102 (1992)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Nilo Stolte
    • 1
  1. 1.École de Technologie SupérieureMontréalCanada

Personalised recommendations