This Article 
 Bibliographic References 
 Add to: 
Optimal Sampling for Hemicubes
March 1995 (vol. 1 no. 1)
pp. 60-76

Abstract—The hemicube estimates of form factors are based on a finite set of sample directions. We obtain several optimal arrangements of sample directions, which minimize the variance of these estimates. They are based on changing the size or shape of the pixels or the shape of the hemicube, or using non-uniform pixel grids. The best reduces the variance by 43%.

The variance calculation is based on the assumption that the errors in the estimate are caused by the projections of single polygon edges, and that the positions and orientations of these edges are random. This replaces the infinite dimensional space of possible environments by the two dimensional space of great circles on the unit sphere, making the numerical variance minimization possible.

[1] C. Goral,K. Torrance,D. Greenberg,, and B. Battaile,“Modeling the interaction of light between diffuse surfaces,” Computer Graphics, vol. 8, no. 3, Siggraph’84 proc., pp. 213-222, 1984.
[2] M.F. Cohen and D.P. Greenberg, "The Hemi-Cube: A Radiosity Solution for Complex Environments," Computer Graphics (Proc. SIGGRAPH '85),San Francisco, Calif., vol. 19, no. 3, pp. 31-40, July 1985.
[3] M.F. Cohen, S.E. Chen, J.R. Wallace, and D.P. Greenberg, "A Progressive Refinement Approach to Fast Radiosity Image Generation," Proc. SIGGRAPH 88, pp. 75-84, 1988.
[4] R. Siegal and J. Howell,Thermal Radiation Heat Transfer, Third Edition, Hemisphere Publishing Corp., Washington, 1992.
[5] T. Nishita and E. Nakamae,“Continuous tone representation of three-dimensional objects taking into account of shadows andinterreflection,” Computer Graphics, vol. 19, no. 3, Siggraph’85 proc., pp. 23-30, 1985.
[6] D. Baum,H. Rushmeier,, and J. Winget,“Improving radiosity solutions through the use of analytically determined form-factors,” Computer Graphics, vol. 23, no. 3, Siggraph’89 proc., pp. 325-334, 1989.
[7] P. Schröder and P. Hanrahan,“On the form factor between two polygons,” Computer Graphics, Ann. Conf. Series, pp. 163-164, 1993.
[8] F. Sillion and C. Peuch,“A general two-pass method integrating specular and diffuse reflection,” Computer Graphics, vol. 23, no. 3, Siggraph’89 proc., pp. 335-344, 1989.
[9] R. Recker,D. George,, and D. Greenberg,“Acceleration techniques for progressive refinement radiosity,” Computer Graphics, vol. 24, no. 2, pp. 59-66, Mar. 1990.
[10] J. Wallace,K. Elmquist,, and E. Haines,“A ray tracing algorithm for progressive radiosity,” Computer Graphics, vol. 23, no. 3, Siggraph’89 proc. pp. 315-324, 1989.
[11] T. Whitted, “An Improved Illumination Model for Shaded Display,” Comm. ACM, vol. 23, no. 6, pp. 343-349, 1980.
[12] M. Lee,R. Redner,, and S. Uselton,“Statistically optimized sampling for distributed ray tracing,” Computer Graphics, vol. 19, no. 3, pp. 61-67, July 1985.
[13] D. Burnett,Finite Element Analysis, Addison Wesley, Reading Mass., 1988.
[14] H.R. Zatz, "Galerkin Radiosity: A Higher-Order Solution Methods for Global Illumination," Computer Graphics Proc., Ann. Conf. Series: SIGGRAPH '93,Anaheim, Calif., pp. 213-220. ACM, Aug. 1993.
[15] H.S. Javitz and A. Valdes, “The Sri Ides Statistical Anomaly Detector,” Proc. IEEE Computer Society Symp. Security and Privacy, May 1991.
[16] R. Troutman and N.L. Max., "Radiosity Algorithms Using Higher Order Finite Element Methods," Computer Graphics Proc., Ann. Conf. Series: SIGGRAPH '93,Anaheim, Calif., pp. 209-212.New York: ACM SIGGRAPH, Aug. 1993.
[17] S.J. Gortler, P. Schroder, M.F. Cohen, and P. Hanrahan, "Wavelet Radiosity," Computer Graphics Proc., Ann. Conf. Series: SIGGRAPH '93,Anaheim, Calif., pp. 221-230, Aug. 1993.
[18] S. E. Chen,H. Rushmeier,G. Miller,, and D. Turner,“A progressive multi-pass method for global illumination,” Computer Graphics, vol. 25, no. 4, Siggraph’91 proc., pp. 165-174, 1991.
[19] D. Gay,“Algorithm 611, collected algorithms from the ACM,” ACM Trans. Mathematical Software, vol. 9, no. 4, pp. 503-524, 1983
[20] N. Max and R. Troutman,“Optimal hemicube sampling,” Proc. Fourth Eurographics Workshop on Rendering, École Normale Superieure, pp. 185-200 and pp. 348-351.
[21] J. Beran-Koehn and M. Pavicic,“A cubic tetrahedral adaptation of thehemi-cube algorithm,” Graphic Gems II, J. Arvo, ed., Academic Press, Boston, pp. 299-302, 1991.
[22] M.A.Z. Dippé and E.H. Wold, Antialiasing through Stochastic Sampling Computer Graphics (SIGGRAPH '85 Proc.), B.A. Barsky, ed., vol. 19, pp. 69-78, July 1985.

Index Terms:
Hemicube, radiosity, form factor, sampling, variance, optimization.
Nelson Max, "Optimal Sampling for Hemicubes," IEEE Transactions on Visualization and Computer Graphics, vol. 1, no. 1, pp. 60-76, March 1995, doi:10.1109/2945.468388
Usage of this product signifies your acceptance of the Terms of Use.