Affine double- and triple-product wavelet integrals for rendering

ABSTRACT

Many problems in computer graphics involve integrations of products of functions. Double- and triple-product integrals are commonly used in applications such as all-frequency relighting or importance sampling, but are limited to distant illumination. In contrast, near-field lighting from planar area lights involves an affine transform of the source radiance at different points in space. Our main contribution is a novel affine double- and triple-product integral theory; this generalization enables one of the product functions to be scaled and translated. We study the computational complexity in a number of bases, with particular attention to the common Haar wavelets. We show that while simple analytic formulae are not easily available, there is considerable sparsity that can be exploited computationally. We demonstrate a practical application to compute near-field lighting from planar area sources, that can be easily combined with most relighting algorithms. We also demonstrate initial results for wavelet importance sampling with near-field area lights, and image processing directly in the wavelet domain.

REFERENCES

• 1: Annen, T., Kautz, J., Durand, F., and Seidel, H.-P. 2004. Spherical harmonic gradients for mid-range illumination. In Eurographics Symposium on Rendering. 331--336.
• 2: Beylkin, G. 1992a. On the fast algorithm for multiplication of functions in the wavelet bases. In International Conference on Wavelets and Applications. 53--61.
• 3: G. Beylkin, On the representation of operators in bases of compactly supported wavelets, SIAM Journal on Numerical Analysis, v.29 n.6, p.1716-1740, Dec. 1992 [doi>10.1137/0729097]
• 4: Bracewell, R. N., Chang, K.-Y., Jha, A. K., and Wang, Y.-H. 1993. Affine theorem for two-dimensional fourier transform. Electron. Lett. 29, 3, 304.
• 5: Jin-Xiang Chai , Xin Tong , Shing-Chow Chan , Heung-Yeung Shum, Plenoptic sampling, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.307-318, July 2000 [doi>10.1145/344779.344932]
• 6: Petrik Clarberg , Wojciech Jarosz , Tomas Akenine-Möller , Henrik Wann Jensen, Wavelet importance sampling: efficiently evaluating products of complex functions, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005
• 7: Julie Dorsey , James Arvo , Donald Greenberg, Interactive Design of Complex Time-Dependent Lighting, IEEE Computer Graphics and Applications, v.15 n.2, p.26-36, March 1995 [doi>10.1109/38.365003]
• 8: Drori, I. and Lischinski, D. 2003. Fast multiresolution image operations in the wavelet domain. IEEE Trans. Visualiz. Comput. Graph. 9, 3, 395--411.
• 9: Frédo Durand , Nicolas Holzschuch , Cyril Soler , Eric Chan , François X. Sillion, A frequency analysis of light transport, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005
• 10: Anders Wang Kristensen , Tomas Akenine-Möller , Henrik Wann Jensen, Precomputed local radiance transfer for real-time lighting design, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005
• 11: Kutil, R. 2005. Approximating linear image operations in the wavelet domain. Tech. rep. 2005-08, Department of Scientific Computing, University of Salzburg, Austria.
• 12: Marc Levoy , Pat Hanrahan, Light field rendering, Proceedings of the 23rd annual conference on Computer graphics and interactive techniques, p.31-42, August 1996 [doi>10.1145/237170.237199]
• 13: Liu, X., Sloan, P.-P. J., Shum, H.-Y., and Snyder, J. 2004. All-Frequency precomputed radiance transfer for glossy objects. In EuroGraphics Symposium on Rendering'04. 337--344.
• 14: Mei, C., Shi, J., and Wu, F. 2004. Rendering with spherical radiance transport maps. In Eurographics'04. 281--290.
• 15: Ren Ng , Ravi Ramamoorthi , Pat Hanrahan, All-frequency shadows using non-linear wavelet lighting approximation, ACM Transactions on Graphics (TOG), v.22 n.3, July 2003
• 16: Ren Ng , Ravi Ramamoorthi , Pat Hanrahan, Triple product wavelet integrals for all-frequency relighting, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004
• 17: Nimeroff, J. S., Simoncelli, E., and Dorsey, J. 1994. Efficient re-rendering of naturally illuminated environments. In Eurographics Rendering Workshop, 359--373.
• 18: Ravi Ramamoorthi , Dhruv Mahajan , Peter Belhumeur, A first-order analysis of lighting, shading, and shadows, ACM Transactions on Graphics (TOG), v.26 n.1, p.2-es, January 2007 [doi>10.1145/1189762.1189764]
• 19: Zhong Ren , Rui Wang , John Snyder , Kun Zhou , Xinguo Liu , Bo Sun , Peter-Pike Sloan , Hujun Bao , Qunsheng Peng , Baining Guo, Real-time soft shadows in dynamic scenes using spherical harmonic exponentiation, ACM Transactions on Graphics (TOG), v.25 n.3, July 2006
• 20: Peter-Pike Sloan , Jan Kautz , John Snyder, Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments, ACM Transactions on Graphics (TOG), v.21 n.3, July 2002
• 21: Eric J. Stollnitz , Tony D. Derose , David H. Salesin, Wavelets for computer graphics: theory and applications, Morgan Kaufmann Publishers Inc., San Francisco, CA, 1996
• 22: G. Strang, Wavelets and dilation equations: a brief introduction, SIAM Review, v.31 n.4, p.614-627, Dec. 1989 [doi>10.1137/1031128]
• 23: Weifeng Sun , Amar Mukherjee, Generalized wavelet product integral for rendering dynamic glossy objects, ACM Transactions on Graphics (TOG), v.25 n.3, July 2006
• 24: Wang, R., Tran, J., and Luebke, D. 2004. All-Frequency relighting of non-diffuse objects using separable BRDF approximation. In Eurographics Symposium on Rendering, 345--354.
• 25: Jiaping Wang , Kun Xu , Kun Zhou , Stephen Lin , Shimin Hu , Baining Guo, Spherical harmonics scaling, The Visual Computer: International Journal of Computer Graphics, v.22 n.9, p.713-720, September 2006 [doi>10.1007/s00371-006-0057-8]
• 26: Wang, R., Ng, R., Luebke, D., and Humphreys, G. 2006b. Efficient wavelet rotation for environment map rendering. In Eurographics Symposium on Rendering. 173--182.
• 27: Kun Zhou , Yaohua Hu , Stephen Lin , Baining Guo , Heung-Yeung Shum, Precomputed shadow fields for dynamic scenes, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General

Additional Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.3 Picture/Image Generation
  I.4 IMAGE PROCESSING AND COMPUTER VISION
      I.4.0 General

General Terms:
Algorithms, Theory

Keywords:
Rendering, double and triple products, image processing, importance sampling, near-field illumination, relighting, wavelets