The Phong and Modified Phong specular BRDFs, although of limited physical basis, are nevertheless some of the simplest BRDFs exhibiting glossy and specular qualities to understand and to implement, making them useful for validation and teaching. Unfortunately, although it is well-known how to make these BRDFs conserve energy (that is, never gain energy), making them energy-normalized (that is, never lose nor gain energy) is far more difficult. Lesser-known algorithms exist, but require the specular exponent n to be integer-valued, and have O(n) runtime cost. We express these algorithms as mathematical formulae and generalize to the real-valued specular exponent case. We then simplify and optimize to finally attain an algorithm that is O(1). Energy normalization makes the Phong BRDFs more physically plausible and therefore both more practically and theoretically useful—and our improvements allow for this energy normalization to be done efficiently and without arbitrary limitations.

中文翻译：

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Phong 镜面反射 BRDF 的恒定时间能量归一化

Phong 和 Modified Phong 镜面反射 BRDF 虽然物理基础有限，但仍然是一些最简单的 BRDF，展示了光泽和镜面反射特性，需要理解和实施，使它们可用于验证和教学。不幸的是，虽然众所周知如何使这些 BRDF 节能（即永远不会获得能量），但使它们能量标准化（即永远不会失去或获得能量）要困难得多。存在鲜为人知的算法，但要求镜面反射指数 n 为整数值，并且运行时间成本为 O(n)。我们将这些算法表示为数学公式，并推广到实值镜面指数情况。然后我们简化和优化以最终获得 O(1) 的算法。