Parallel Monte Carlo Approach for Integration of the Rendering Equation

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


This paper is addressed to the numerical solving of the rendering equation in realistic image creation. The rendering equation is integral equation describing the light propagation in a scene accordingly to a given illumination model. The used illumination model determines the kernel of the equation under consideration. Nowadays, widely used are the Monte Carlo methods for solving the rendering equation in order to create photorealistic images.

In this work we consider the Monte Carlo solving of the rendering equation in the context of the parallel sampling scheme for hemisphere. Our aim is to apply this sampling scheme to stratified Monte Carlo integration method for parallel solving of the rendering equation. The domain for integration of the rendering equation is a hemisphere. We divide the hemispherical domain into a number of equal sub-domains of orthogonal spherical triangles. This domain partitioning allows to solve the rendering equation in parallel. It is known that the Neumann series represent the solution of the integral equation as a infinity sum of integrals. We approximate this sum with a desired truncation error (systematic error) receiving the fixed number of iteration. Then the rendering equation is solved iteratively using Monte Carlo approach. At each iteration we solve multi-dimensional integrals using uniform hemisphere partitioning scheme. An estimate of the rate of convergence is obtained using the stratified Monte Carlo method.

This domain partitioning allows easy parallel realization and leads to convergence improvement of the Monte Carlo method. The high performance and Grid computing of the corresponding Monte Carlo scheme are discussed.


Integration Domain Monte Carlo Approach Global Illumination Illumination Model Random Sampling Point 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dimov, I.T., Gurov, T.V., Penzov, A.A.: A Monte Carlo Approach for the Cook-Torrance Model. In: Li, Z., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2004. LNCS, vol. 3401, pp. 257–265. Springer, Heidelberg (2005)Google Scholar
  2. 2.
    Dimov, I., et al.: Parallel Importance Separation and Adaptive Monte Carlo Algorithms for Multiple Integrals. In: Dimov, I.T., et al. (eds.) NMA 2002. LNCS, vol. 2542, pp. 99–107. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Dimov, I.T., Penzov, A.A., Stoilova, S.S.: Parallel Monte Carlo Sampling Scheme for Sphere and Hemisphere. In: Boyanov, T., et al. (eds.) NMA 2006. LNCS, vol. 4310, pp. 148–155. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Dutré, P.: Global Illumination Compendium. Script of September 29 2003,
  5. 5.
    Georgieva, R., Ivanovska, S.: Importance Separation for Solving Integral Equations. In: Lirkov, I., et al. (eds.) LSSC 2003. LNCS, vol. 2907, pp. 144–152. Springer, Heidelberg (2004)Google Scholar
  6. 6.
    Ivanovska, S., Karaivanova, A.: Parallel Importance Separation for Multiple Integrals and Integral Equations. In: Bubak, M., et al. (eds.) ICCS 2004. LNCS, vol. 3039, pp. 499–506. Springer, Heidelberg (2004)Google Scholar
  7. 7.
    Karaivanova, A.: Adaptive Monte Carlo methods for numerical integration. Mathematica Balkanica 11, 201–213 (1997)MathSciNetGoogle Scholar
  8. 8.
    Karaivanova, A., Dimov, I.: Error analysis of an adaptive Monte Carlo method for numerical integration. Mathematics and Computers in Simulation 47, 391–406 (1998)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Kajiya, J.T.: The Rendering Equation, Computer Graphics, vol. Computer Graphics - Proceedings of SIGGRAPH‘86 20(4), 143–150 (1986)CrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Penzov, A.A.: Shading and Illumination Models in Computer Graphics - a literature survey, MTA SZTAKI, Research Report CG-4, Budapest (1992)Google Scholar
  12. 12.
    Sobol, I.: Monte Carlo Numerical Methods (in Russian). Nauka, Moscow (1975)Google Scholar
  13. 13.
    Szirmay-Kalos, L.: Monte-Carlo Methods in Global Illumination, Script in WS of, 1999/2000,
  14. 14.
    Urena, C.: Computation of Irradiance from Triangles by Adaptive Sampling. Computer Graphics Forum 19(2), 165–171 (2000)zbMATHCrossRefGoogle Scholar
  15. 15.
    Veach, E.: Robust Monte Carlo Methods for Light Transport Simulation, Ph.D. Dissertation, Stanford University (December 1997)Google 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

Personalised recommendations