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Integrating Clipped Spherical Harmonics Expansions

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 expansions clipped to polygonal domains. Our solution, based on zonal decompositions of spherical integrands and discrete contour integration, introduces an important numerical operationg for spherical harmonic 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, non-polynomial 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 spherical harmonic evaluation and discrete contour integration routines, and we release a full source CPU-only and shader-based implementations (< 750 lines of commented code).
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https://hal.inria.fr/hal-01695284
Contributor : Laurent Belcour <>
Submitted on : Monday, January 29, 2018 - 10:59:32 AM
Last modification on : Wednesday, January 6, 2021 - 4:10:05 PM
Long-term archiving on: : Friday, May 25, 2018 - 3:08:15 PM

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  • HAL Id : hal-01695284, version 1

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Laurent Belcour, Guofu Xie, Christophe Hery, Mark Meyer, Wojciech Jarosz, et al.. Integrating Clipped Spherical Harmonics Expansions. ACM Transactions on Graphics, Association for Computing Machinery, 2018, 37 (2). ⟨hal-01695284⟩

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