Integrating Clipped Spherical Harmonics Expansions
Integrating Clipped Spherical Harmonics Expansions
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积分限幅球谐展开式
DOI：
10.1145/3015459
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
D. Nowrouzezahrai
中科院分区：
文献类型：
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作者：
Laurent Belcour;Guofu Xie;C. Hery;Mark Meyer;Wojciech Jarosz;D. Nowrouzezahrai
渲染中的许多应用程序都依赖于球面多边形上的积分函数。我们提出了一种新的数值解决方案，用于计算剪裁到多边形域的球谐函数 (SH) 展开式的积分。我们的解决方案基于球面被积函数的区域分解和离散轮廓积分，为渲染应用中的 SH 展开引入了重要的数值运算。我们的方法简单、高效，并且在带限被积函数的谐波展开中线性缩放。我们将我们的技术应用于渲染问题，包括表面和体积着色、分层产品重要性采样以及交互式渲染的快速基础投影。此外，我们还展示了如何使用控制变量在蒙特卡罗设置中处理一般的非多项式被积函数。我们的技术计算带限球函数的积分，其性能与更通用的数值积分方法相比（或更快），适用于离线和交互式渲染环境中的广泛问题。我们的实现很简单，仅依赖于独立的 SH 评估和离散轮廓积分例程，并且我们发布了仅 CPU 和基于着色器的完整源代码实现（<750 行注释代码）。
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).