Integrating Clipped Spherical Harmonics Expansions

1University of Montreal 2Pixar Animation Studios 3Dartmouth College 4McGill University

In ACM Transactions on Graphics (Presented at SIGGRAPH), 2018

Axial moments (far-left) integrate cosine-power lobes and diffuse emitters [Arvo 1995]. We generalize this to cosine-power lobes and cosine-power emitters (center-left), zonal harmonic expansions (center-right), and arbitrary bandlimited (spherical harmonic) emitters and BRDFs (far-right).


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).



Laurent Belcour, Guofu Xie, Christophe Hery, Mark Meyer, Wojciech Jarosz, Derek Nowrouzezahrai. Integrating Clipped Spherical Harmonics Expansions. ACM Transactions on Graphics (Presented at SIGGRAPH), 37(2), March 2018.
    author = "Belcour, Laurent and Xie, Guofu and Hery, Christophe and Meyer, Mark and Jarosz, Wojciech and Nowrouzezahrai, Derek",
    title = "Integrating Clipped Spherical Harmonics Expansions",
    journal = "ACM Transactions on Graphics (Presented at SIGGRAPH)",
    volume = "37",
    number = "2",
    month = "mar",
    year = "2018",
    doi = "10.1145/3015459"
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