“Integrating Clipped Spherical Harmonics Expansions” by Belcour, Xie, Hery, Meyer, Jarosz, et al. …

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    Integrating Clipped Spherical Harmonics Expansions

Session/Category Title:   Smart Integration for Real-Time Rendering

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Abstract:


    Many applications in rendering rely on integrating functions over spherical polygons. We present a new numerical solution for computing the integral of spherical harmonics (SH) expansions clipped to polygonal domains. Our solution, based on zonal decompositions of spherical integrands and discrete contour integration, introduces an important numerical operating for SH expansions in rendering applications. Our method is simple, efficient, and scales linearly in the bandlimited integrand’s harmonic expansion. We apply our technique to problems in rendering, including surface and volume shading, hierarchical product importance sampling, and fast basis projection for interactive rendering. Moreover, we show how to handle general, nonpolynomial integrands in a Monte Carlo setting using control variates. Our technique computes the integral of bandlimited spherical functions with performance competitive to (or faster than) more general numerical integration methods for a broad class of problems, both in offline and interactive rendering contexts. Our implementation is simple, relying only on self-contained SH evaluation and discrete contour integration routines, and we release a full source CPU-only and shader-based implementations (<750 lines of commented code).

References:


    1. Thomas Annen, Jan Kautz, Frédo Durand, and Hans-Peter Seidel. 2004. Spherical harmonic gradients for mid-range illumination. In Proceedings of the 15th Eurographics Conference on Rendering Techniques (EGSR’04). 331–336. Google ScholarDigital Library
    2. James R. Arvo. 1995. Applications of irradiance tensors to the simulation of non-Lambertian phenomena. In Proceedings of the 1995 SIGGRAPH Annual Conference. Google ScholarDigital Library
    3. J. Beckmann, H. N. Mhaskar, and J. Prestin. 2012. Quadrature formulas for integration of multivariate trigonometric polynomials on spherical triangles. International Journal on Geomathematics 3, 1, 119–138.Google ScholarCross Ref
    4. J. Beckmann, H. N. Mhaskar, and J. Prestin. 2014. Local numerical integration on the sphere. International Journal on Geomathematics 5, 2, 143–162.Google ScholarCross Ref
    5. Benedikt Bitterli, Jan Novák, and Wojciech Jarosz. 2015. Portal-masked environment map sampling. In Proceedings of the 26th Eurographics Symposium on Rendering (EGSR’15). 13–19.Google ScholarCross Ref
    6. Min Chen and James Arvo. 2001. Simulating non-Lambertian phenomena involving linearly-varying luminaires. In Proceedings of the 12th Eurographics Conference on Rendering. 25–38. Google ScholarDigital Library
    7. Min Chen and James R. Arvo. 2000. A closed-form solution for the irradiance due to linearly-varying luminaires. In Rendering Techniques 2000. Eurographics. Springer, 137–148. Google ScholarDigital Library
    8. Petrik Clarberg and Tomas Akenine-Möller. 2008a. Exploiting visibility correlation in direct illumination. In Proceedings of the 19th Eurographics Conference on Rendering (EGSR’08). 1125–1136. Google ScholarDigital Library
    9. Petrik Clarberg and Tomas Akenine-Möller. 2008b. Practical product importance sampling for direct illumination. Computer Graphics Forum 27, 2, 681–690.Google ScholarCross Ref
    10. Petrik Clarberg, Wojciech Jarosz, Tomas Akenine-Möller, and Henrik Wann Jensen. 2005. Wavelet importance sampling: Efficiently evaluating products of complex functions. ACM Transactions on Graphics 24, 3, 1166–1175. Google ScholarDigital Library
    11. Ronald Cools and Philip Rabinowitz. 1993. Monomial cubature rules since “Stroud”: A compilation. Journal of Computational and Applied Mathematics 48, 3, 309–326. Google ScholarDigital Library
    12. Shaohua Fan, Stephen Chenney, Bo Hu, Kam-Wah Tsui, and Yu-Chi Lai. 2006. Optimizing control variate estimators for rendering. Computer Graphics Forum 25, 3, 351–358.Google ScholarCross Ref
    13. Q. T. Le Gia and H. N. Mhaskar. 2009. Localized linear polynomial operators and quadrature formulas on the sphere. SIAM Journal on Numerical Analysis 47, 1, 440–466. Google ScholarDigital Library
    14. Paul Green, Jan Kautz, Wojciech Matusik, and Frédo Durand. 2006. View-dependent precomputed light transport using nonlinear Gaussian function approximations. In Proceedings of the Symposium on Interactive 3D Graphics and Games. ACM, New York, NY. Google ScholarDigital Library
    15. Charles Han, Bo Sun, Ravi Ramamoorthi, and Eitan Grinspun. 2007. Frequency domain normal map filtering. ACM Transactions on Graphics 26, 3, Article No. 28. Google ScholarDigital Library
    16. Eric Heitz, Jonathan Dupuy, Stephen Hill, and David Neubelt. 2016. Real-time polygonal-light shading with linearly transformed cosines. ACM Transactions on Graphics 35, 4, Article No. 41. Google ScholarDigital Library
    17. Kerstin Hesse, Ian H. Sloan, and Robert S. Womersley. 2010. Numerical integration on the sphere. In Handbook of Geomathematics. Springer, 1–35.Google Scholar
    18. R. W. James. 1969. Transformation of spherical harmonics under change of reference frame. Geophysical Journal of the Royal Astronomical Society 17, 3, 305–316.Google ScholarCross Ref
    19. Wojciech Jarosz, Nathan A. Carr, and Henrik Wann Jensen. 2009. Importance sampling spherical harmonics. Computer Graphics Forum 28, 2, 577–586.Google ScholarCross Ref
    20. Boz Kempski. 1995. Extension of the Whittaker-Shannon Sampling Series Aided by Symbolic Computation. Ph.D. Dissertation. Anglia Polytechnic.Google Scholar
    21. Anders Wang Kristensen, Tomas Akenine-Möller, and Henrik Wann Jensen. 2005. Precomputed local radiance transfer for real-time lighting design. ACM Transactions on Graphics 24, 3, 1208–1215.. Google ScholarDigital Library
    22. Eric P. Lafortune and Yves D. Willems. 1994. The ambient term as a variance reducing technique for Monte Carlo ray tracing. In Photorealistic Rendering Techniques. Focus on Computer Graphics. Springer, 168–176.Google Scholar
    23. Pascal Lecocq, Arthur Dufay, Gaël Sourimant, and Jean-Eudes Marvie. 2016. Accurate analytic approximations for real-time specular area lighting. In Proceedings of the Symposium on Interactive 3D Graphics and Games. Google ScholarDigital Library
    24. Christian Lessig, Tyler de Witt, and Eugene Fiume. 2012. Efficient and accurate rotation of finite spherical harmonics expansions. Journal of Computational Physics 231, 2, 243–250. Google ScholarDigital Library
    25. Wojciech Matusik, Hanspeter Pfister, Matt Brand, and Leonard McMillan. 2003. A data-driven reflectance model. ACM Transactions on Graphics 22, 759–769. Google ScholarDigital Library
    26. Soham Mehta, Ravi Ramamoorthi, Mark Meyer, and Christophe Hery. 2012. Analytic tangent irradiance environment maps for anisotropic surfaces. Computer Graphics Forum 31, 4, 1501–1508. Google ScholarDigital Library
    27. Ren Ng, Ravi Ramamoorthi, and Pat Hanrahan. 2003. All-frequency shadows using non-linear wavelet lighting approximation. ACM Transactions on Graphics 22, 3, 376–381. Google ScholarDigital Library
    28. Ren Ng, Ravi Ramamoorthi, and Pat Hanrahan. 2004. Triple product wavelet integrals for all-frequency relighting. ACM Transactions on Graphics 23, 3, 477–487. Google ScholarDigital Library
    29. Jan Novák, Andrew Selle, and Wojciech Jarosz. 2014. Residual ratio tracking for estimating attenuation in participating media. ACM Transactions on Graphics 33, 6, Article No. 179. Google ScholarDigital Library
    30. Derek Nowrouzezahrai, Patricio Simari, and Eugene Fiume. 2012. Sparse zonal harmonic factorization for efficient SH rotation and shading. ACM Transactions on Graphics 31, 3, Article No. 23. Google ScholarDigital Library
    31. Vincent Pegoraro and Steven G. Parker. 2009. An analytical solution to single scattering in homogeneous participating media. Computer Graphics Forum 28, 2, 329–335.Google ScholarCross Ref
    32. Vincent Pegoraro, Mathias Schott, and Steven G. Parker. 2009. An analytical approach to single scattering for anisotropic media and light distributions. In Proceedings of the Graphics Interface Conference. Google ScholarDigital Library
    33. Matt Pharr, Wenzel Jakob, and Greg Humphreys. 2016. Physically Based Rendering: From Theory to Implementation (3rd ed.). Morgan Kaufmann, San Francisco, CA. Google ScholarDigital Library
    34. Ravi Ramamoorthi and Pat Hanrahan. 2001. An efficient representation for irradiance environment maps. In Proceedings of the SIGGRAPH Annual Conference. ACM, New York, NY, 497–500. Google ScholarDigital Library
    35. Ravi Ramamoorthi and Pat Hanrahan. 2002. Frequency space environment map rendering. ACM Transactions on Graphics 21, 3, 517–526. Google ScholarDigital Library
    36. Peter-Pike Sloan. 2008. Stupid spherical harmonics (SH) tricks. In Proceedings of the Game Developers Conference.Google Scholar
    37. Peter-Pike Sloan, Jan Kautz, and John Snyder. 2002. Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments. ACM Transactions on Graphics 21, 3, 527–536. Google ScholarDigital Library
    38. Peter-Pike Sloan, Ben Luna, and John Snyder. 2005. Local, deformable precomputed radiance transfer. ACM Transactions on Graphics 24, 3, 1216–1224. Google ScholarDigital Library
    39. Cyril Soler, Mahdi M. Bagher, and Derek Nowrouzezahrai. 2015. Efficient and accurate spherical kernel integrals using isotropic decomposition. ACM Transactions on Graphics 34, 6, Article No. 161. Google ScholarDigital Library
    40. Bo Sun, Ravi Ramamoorthi, Srinivasa G. Narasimhan, and Shree K. Nayar. 2005. A practical analytic single scattering model for real time rendering. ACM Transactions on Graphics 24, 3, 1040–1049. Google ScholarDigital Library
    41. Weifeng Sun and Amar Mukherjee. 2006. Generalized wavelet product integral for rendering dynamic glossy objects. ACM Transactions on Graphics 25, 3, 955–966. Google ScholarDigital Library
    42. Yu-Ting Tsai and Zen-Chung Shih. 2006. All-frequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation. ACM Transactions on Graphics 25, 3, 967–976. Google ScholarDigital Library
    43. Jiaping Wang, Peiran Ren, Minmin Gong, John Snyder, and Baining Guo. 2009. All-frequency rendering of dynamic, spatially-varying reflectance. ACM Transactions on Graphics 28, 5, Article No. 133. Google ScholarDigital Library

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