Analytic Spherical Harmonic Gradients for Real-Time Rendering with Many Polygonal Area Lights

摘要

coefficients from uniform polygonal lights, enabling near-field area lights tobe included in Precomputed Radiance Transfer (PRT) systems, and in offlinerendering. However, the method is inefficient since coefficients need to berecomputed at each vertex or shading point, for each light, even thoughthe SH coefficients vary smoothly in space. The complexity scales linearlywith the number of lights, making many-light rendering difficult. In thispaper, we develop a novel analytic formula for the spatial gradients of thespherical harmonic coefficients for uniform polygonal area lights. The result isa significant generalization, involving the Reynolds transport theorem toreduce the problem to a boundary integral for whichwe derive a newanalyticformula, showing how to reduce a key term to an earlier recurrence for SHcoefficients. The implementation requires only minor additions to existingcode for SH coefficients. The results also hold implications for recent effortson differentiable rendering. We show that SH gradients enable very sparsespatial sampling, followed by accurate Hermite interpolation. This enables scaling PRT to hundreds of area lights with minimal overhead and real-timeframe rates. Moreover, the SH gradient formula is a new mathematical resultthat potentially enables many other graphics applications.