“Analytic spherical harmonic coefficients for polygonal area lights” by Zhang and Ramamoorthi

  • ©Jingwen Zhang and Ravi Ramamoorthi



Entry Number: 54


    Analytic spherical harmonic coefficients for polygonal area lights

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




    Spherical Harmonic (SH) lighting is widely used for real-time rendering within Precomputed Radiance Transfer (PRT) systems. SH coefficients are precomputed and stored at object vertices, and combined interactively with SH lighting coefficients to enable effects like soft shadows, interreflections, and glossy reflection. However, the most common PRT techniques assume distant, low-frequency environment lighting, for which SH lighting coefficients can easily be computed once per frame. There is currently limited support for near-field illumination and area lights, since it is non-trivial to compute the SH coefficients for an area light, and the incident lighting (SH coefficients) varies over the object geometry. We present an efficient closed-form solution for projection of uniform polygonal area lights to spherical harmonic coefficients of arbitrary order, enabling easy adoption of accurate area lighting in PRT systems, with no modifications required to the core PRT framework. Our method only requires computing zonal harmonic (ZH) coefficients, for which we introduce a novel recurrence relation. In practice, ZH coefficients are built up iteratively, with computation linear in the desired SH order. General SH coefficients can then be obtained by the recently developed sparse zonal harmonic rotation method.


    1. J Arvo. 1995. Applications of irradiance tensors to the simulation of non-Lambertian phenomena. In SIGGRAPH 95. 335–342. Google ScholarDigital Library
    2. D. Baum, H. Rushmeier, and J. Winget. 1989. Improving Radiosity Solutions through the use of analytically determined form-factors. In SIGGRAPH 89. 325–334. Google ScholarDigital Library
    3. L. Belcour, G. Xie, C. Hery, M. Meyer, W. Jarosz, and D. Nowrouzezahrai. 2018. Integrating Clipped Spherical Harmonics Expansions. ACM Transactions on Graphics 37, 2 (2018), 19:1–19:12. Google ScholarDigital Library
    4. M Chen and J Arvo. 2001. Simulating Non-Lambertian Phenomena Involving Linearly-Varying Luminaires. In Eurographics Workshop on Rendering. 25–38. Google ScholarDigital Library
    5. J. Dupuy, E. Heitz, and L. Belcour. 2017. A Spherical Cap Preserving Parameterization for Spherical Distributions. ACM Transactions on Graphics (Proc. SIGGRAPH 17) 36, 4 (2017), 139:1–139:12. Google ScholarDigital Library
    6. E. Heitz, J. Dupuy, S. Hill, and D. Neubelt. 2016. Real-Time Polygonal-Light Shading with Linearly Transformed Cosines. ACM Transactions on Graphics (Proc. SIGGRAPH 16) 35, 4 (2016), 41:1–41:8. Google ScholarDigital Library
    7. T MacRobert. 1948. Spherical harmonics: an elementary treatise on harmonic functions with applications. Dover Publications.Google Scholar
    8. R Ng, R Ramamoorthi, and P Hanrahan. 2003. All-Frequency Shadows using Non-Linear Wavelet Lighting Approximation. ACM Transactions on Graphics (Proc. SIGGRAPH 03) 22, 3 (2003), 376–381. Google ScholarDigital Library
    9. D. Nowrouzezahrai, P. Simari, and E. Fiume. 2012. Sparse zonal harmonic factorization for efficient SH rotation. ACM Transactions on Graphics 31, 3 (2012), 23:1–23:9. Google ScholarDigital Library
    10. J Pantaleoni, L Fascione, M Hill, and T Aila. 2010. PantaRay: fast ray-traced occlusion caching of massive scenes. ACM Transactions on Graphics (Proc. SIGGRAPH 10) 29, 4 (2010). Google ScholarDigital Library
    11. R Ramamoorthi. 2009. Precomputation-Based Rendering. Foundations and Trends in Computer Graphics and Vision 3, 4 (2009), 281–369. Google ScholarDigital Library
    12. Z Ren, R Wang, J Snyder, K Zhou, X Liu, B Sun, P Sloan, H Bao, Q Peng, and B Guo. 2006. Real-time Soft Shadows in Dynamic Scenes using Spherical Harmonic Exponentiation. ACM Transactions on Graphics (Proc. SIGGRAPH 06) 25, 3 (2006), 977–986. Google ScholarDigital Library
    13. P Sloan, J Hall, J Hart, and J Snyder. 2003. Clustered Principal Components for Precomputed Radiance Transfer. ACM Transactions on Graphics (Proc. SIGGRAPH 03) 22, 3 (2003), 382–391. Google ScholarDigital Library
    14. P Sloan, J Kautz, and J Snyder. 2002. Precomputed Radiance Transfer for Real-Time Rendering in Dynamic, Low-Frequency Lighting Environments. ACM Transactions on Graphics (Proc. SIGGRAPH 02) 21, 3 (2002), 527–536. Google ScholarDigital Library
    15. P Sloan, B Luna, and J Snyder. 2005. Local, deformable precomputed radiance transfer. ACM Transactions on Graphics (Proc. SIGGRAPH 05) 24, 3 (2005), 1216–1224. Google ScholarDigital Library
    16. J. Snyder. 1996. Area Light Sources for Real-Time Graphics. Technical Report MSR-TR-96-11. Microsoft Research.Google Scholar
    17. D. Stern. 1965. Classification of Magnetic Shells. Journal of Geophysics Research 70, 15 (1965), 3629–3634.Google ScholarCross Ref
    18. B Sun and R Ramamoorthi. 2009. Affine double and triple product wavelet integrals for rendering. ACM Transaction on Graphics 28, 2 (2009). Google ScholarDigital Library
    19. Y Tsai and Z Shih. 2006. All-Frequency Precomputed Radiance Transfer using Spherical Radial Basis Functions and Clustered Tensor Approximation. ACM Transactions on Graphics(Proc. SIGGRAPH 06) 25, 3 (2006), 967–976. Google ScholarDigital Library
    20. K Zhou, Y Hu, S Lin, B Guo, and H Shum. 2005. Precomputed shadow fields for dynamic scenes. ACM Transactions on Graphics (Proc. SIGGRAPH 05) 24, 3 (2005), 1196–1201. Google ScholarDigital Library

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