Analytic spherical harmonic coefficients for polygonal area lights

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.