Fast and accurate spherical harmonics products

Spherical Harmonics (SH) have been proven as a powerful tool for rendering, especially in real-time applications such as Precomputed Radiance Transfer (PRT). Spherical harmonics are orthonormal basis functions and are efficient in computing dot products. However, computations of triple product and multiple product operations are often the bottlenecks that prevent moderately high-frequency use of spherical harmonics. Specifically state-of-the-art methods for accurate SH triple products of order n have a time complexity of O(n5), which is a heavy burden for most real-time applications. Even worse, a brute-force way to compute k-multiple products would take O(n2k) time. In this paper, we propose a fast and accurate method for spherical harmonics triple products with the time complexity of only O(n3), and further extend it for computing k-multiple products with the time complexity of O(kn3 + k2n2 log(kn)). Our key insight is to conduct the triple and multiple products in the Fourier space, in which the multiplications can be performed much more efficiently. To our knowledge, our method is theoretically the fastest for accurate spherical harmonics triple and multiple products. And in practice, we demonstrate the efficiency of our method in rendering applications including mid-frequency relighting and shadow fields.