Constant-time energy-normalization for the Phong specular BRDFs
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ORIGINAL ARTICLE
Constant-time energy-normalization for the Phong specular BRDFs Ian Mallett1
· Cem Yuksel1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Keywords Phong · BRDF · Phong BRDF · Modified Phong BRDF · Energy conservation · Energy normalization · Physically based rendering
1 Introduction The importance of a reflectance model, i.e. the bidirectional reflectance distribution function (BRDF), to be energyconserving is well-understood in computer graphics. An energy-conserving BRDF ensures that the material does not reflect more light than it receives—that is, the material is permitted to absorb some of the light it receives, but it cannot add light. In the absence of any absorption, real materials still exhibit some darkening in their appearance based on the view angle. When light hits a surface, the Fresnel equations define the portion that is reflected or transmitted. Collectively, the Fresnel coefficient, multiplied by a distribution function that describes the distribution of reflectance, gives the BRDF. For a material to be physically realistic, the distribution function must integrate to one—that is, any darkening in the BRDF should be due to the Fresnel term, not due to artifacts of
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Ian Mallett [email protected] https://geometrian.com/ Cem Yuksel [email protected] http://www.cemyuksel.com/
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the distribution; when there is no absorption, the distribution function must scatter all light it receives. We refer to this criterion as energy normalization. Note that energy normalization is a stronger criterion than energy conservation, and is arguably part of the definition of the BRDF itself [12]. Energy normalization has received less attention in computer graphics than energy conservation, as its solution is usually more difficult. In production (such as AAA video games, special effects, and feature animation), microfacet BRDFs [2,15] are common. Energy normalization for a microfacet BRDF can be physically interpreted, mostly, as accounting for multiple scattering off of secondary microface
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