Diaphony of Uniform Samples over Hemisphere and Sphere

  • I. T. Dimov
  • S. S. Stoilova
  • N. Mitev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)


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. In this work we consider the problem for generation of uniformly distributed random samples over hemisphere and sphere. Using two algorithms we obtain samples in orthogonal spherical triangle and spherical quadrangle. Our aim is to prove uniform distribution of the obtained samples. The importance of the uniformly distributed samples is determined by the effectiveness of the algorithms for numerical solution of the rendering equation. We use numerical characteristic for uniform distribution of points, called diaphony. The diaphony of these samples is calculated numerically. Analysis and comparison of the obtained results are made.


Sampling Point Solid Angle Light Propagation Uniform Sample Closed Domain 
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., Penzov, A.A., Stoilova, S.S.: Parallel Monte Carlo Sampling Scheme for Sphere and Hemisphere. In: Li, Z., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2004. LNCS, vol. 3401, pp. 148–155. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Dimov, I.T., Penzov, A.A., Stoilova, S.S.: Parallel Monte Carlo Approach for Intgration of the Rendering Equation. In: Li, Z., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2004. LNCS, vol. 3401, pp. 140–147. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Grozdanov, V., Stoilova, S.: On the Theory of b-adic Diaphony. Comp. Ren. Akad. Bul. Sci. 54(3), 31–34 (2001)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2, 84–90 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hellekalek, P., Leeb, H.: Dyadic Diaphony. Acta Arithmetica LXXX(2), 187–196Google Scholar
  6. 6.
    Keller, A.: Quasi-Monte Carlo Methods in Computer Graphics: The Global Illumination Problem. Lectures in Applied Mathematics 32, 455–469 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Roth, K.: On irregularities of distribution. Mathematika 1, 73–79 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Weyl, H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77(S), 313–352 (1916)CrossRefzbMATHGoogle Scholar
  9. 9.
    Zinterhof, P.: Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden. Sitzungsber. Osterr. Akad. Wiss. Math.-Naturwiss. abt II 185, 121–132 (1976)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • I. T. Dimov
    • 1
  • S. S. Stoilova
    • 2
  • N. Mitev
    • 1
  1. 1.Institute for Parallel ProcessingBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations