Parallel Monte Carlo Sampling Scheme for Sphere and Hemisphere

  • Ivan T. Dimov
  • Anton A. Penzov
  • Stanislava S. Stoilova
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4310)


The sampling of certain solid angle is a fundamental operation in realistic image synthesis, where the rendering equation describing the light propagation in closed domains is solved. Monte Carlo methods for solving the rendering equation use sampling of the solid angle subtended by unit hemisphere or unit sphere in order to perform the numerical integration of the rendering equation.

In this work we consider the problem for generation of uniformly distributed random samples over hemisphere and sphere. Our aim is to construct and study the parallel sampling scheme for hemisphere and sphere. First we apply the symmetry property for partitioning of hemisphere and sphere. The domain of solid angle subtended by a hemisphere is divided into a number of equal sub-domains. Each sub-domain represents solid angle subtended by orthogonal spherical triangle with fixed vertices and computable parameters. Then we introduce two new algorithms for sampling of orthogonal spherical triangles.

Both algorithms are based on a transformation of the unit square. Similarly to the Arvo’s algorithm for sampling of arbitrary spherical triangle the suggested algorithms accommodate the stratified sampling. We derive the necessary transformations for the algorithms. The first sampling algorithm generates a sample by mapping of the unit square onto orthogonal spherical triangle. The second algorithm directly compute the unit radius vector of a sampling point inside to the orthogonal spherical triangle. The sampling of total hemisphere and sphere is performed in parallel for all sub-domains simultaneously by using the symmetry property of partitioning. The applicability of the corresponding parallel sampling scheme for Monte Carlo and Quasi-Monte Carlo solving of rendering equation is discussed.


Solid Angle Closed Domain Bulgarian Academy Global Illumination Spherical Triangle 
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  1. 1.
    Arvo, J.: Stratifed sampling of spherical triangles. In: Computer Graphics Proceedings, Annual Conference Series, ACM Siggraph, pp. 437–438. ACM Press, New York (1995)Google Scholar
  2. 2.
    Dutré, P.: Global Illumination Compendium, Script of September 29 2003,
  3. 3.
    Kajiya, J.T.: The Rendering Equation. Computer Graphics- Proceedings of SIGGRAPH‘86 20(4), 143–150 (1986)CrossRefGoogle Scholar
  4. 4.
    Keller, A.: Quasi-Monte Carlo Methods in Computer Graphics: The Global Illumination Problem. In: Lectures in Applied Mathematics, vol. 32, pp. 455–469 (1996)Google Scholar
  5. 5.
    Rogers, D.F.: Procedural Elements for Computer Graphics. McGraw-Hill, New York (1985)Google Scholar
  6. 6.
    Urena, C.: Computation of Irradiance from Triangles by Adaptive Sampling. Computer Graphics Forum 19(2), 165–171 (2000)zbMATHCrossRefGoogle Scholar
  7. 7.
    Yershova, A., LaValle, S.M.: Deterministic Sampling Methods for Spheres and SO(3). In: Robotics and Automation, 2004. IEEE Proceedings of ICRA’04, vol. 4, pp. 3974–3980. IEEE Computer Society Press, Los Alamitos (2004)CrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Ivan T. Dimov
    • 1
  • Anton A. Penzov
    • 2
  • Stanislava S. Stoilova
    • 3
  1. 1.Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 25 A, 1113 Sofia, Bulgaria and ACET Centre, University of Reading Whiteknights, PO Box 217, Reading, RG6 6AHUK
  2. 2.Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 25 A, 1113 SofiaBulgaria
  3. 3.Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 8, 1113 SofiaBulgaria

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