A Microfacet Based Coupled Specular-Matte BRDF Model with Importance Sampling

Abstract
This paper presents a BRDF model based on the analysis of the photon collisions with the microfacets of the surface. The new model is not only physically plausible, i.e. symmetric and energy conserving, but provides other important features of real materials, including the off-specular peak and the mirroring limit case. Using theoretical considerations the reflected light is broken down to a specular component representing single reflections and a matte component accounting for multiple reflections and re-emissions of previously absorbed photons. Unlike most of the previous models, the proportion of the matte 无光泽的占比 and specular components is not constant but varies with the viewing angle 根据视角而变化. In order to keep the resulting formulae 公式 simple, several approximations are made, which are quite accurate but allow for tabulation 列表显示, fast calculation and even for accurate importance sampling. 快速计算、精确的重要性采样。

1. introduction
   in order to render realistic images, we have to use realistic material models. material models are usually defined by Bidirectional Reflectance Distribution Functions (BRDFs) that describe the chance of reflection for different pairs of incoming and outgoing light directions. when introducing BRDF models, the following notations are used: N上箭头 is the unit normal vector of the surface, L上箭头 is a unit vector pointing towards the light source. V上箭头 is a unit vector pointing towards the camera, H上箭头 is a unit vector that is halfway between L and V, θ’ is the angle between L and N, θ is the angle between V and N, α is the angle between H and N and β is the angle between V and H and similarly between L and H. angles θ’ and θ are also called the incoming and outgoing angles, respectively.

A BRDF can be derived from the probability density function of the reflection. suppose that a photon comes from direction L. The refletion density w(L,V) describes the probability that the photon leaves the surface at a differential solid angle around V given it comes from L: 注意这里的动词leave，是离开表面。

Pr{photon is reflected to dwv around v } coming from L}.

the BRDF is the reflection density divided by the cosine of the outgoing angle:

A realistic BRDF model is expected not to violate physics laws including the Helmholtz-symmetry, and the law of energy conservation, and to mimic the features of real materials. The Helmholtz principle states that light and view directions can be exchanged in the BRDF, i.e.
According to energy conservation, a surface——provided that it is not a light source——can not reflect more photons (more power) to the complete hemisphere Ω than what have been received, thus the albedo defined by

can not be greater than 1. for isotropic models 对于各项同性的模型, reflections of illuminations from incident directions with the same θ’ have rotational symmetry 旋转也是对称的, thus the albedo of a given point depends only on incident angle θ’. to emphasize this, the albedo will be denoted by α(θ’). Due to the symmetry of the BRDF, the albedo can also be presented in the following form:

note that this is the integral of the rendering equation if the incoming radiance is 1, thus the albedo can also be interpreted as the response to homogenous unit illumination 均匀单元照明.

Concerning other important features of real materials, the following properties are worth mentioning. if the surface is perfectly smooth, then the reflection is described by the Fresnel function that is based on the refraction index and the extinction coefficient. At grazing angles glossy materials become quite good mirrors, that is, specularity increases while matte 无光泽的；不光滑的 behavior fades (figure 1).

interestingly, the maximum of the reflectance is not at the idea reflection direction at grazing angles, but is lower than that. this phenomenon is called the off-specular peak.

there are two uses of BRDFs in rendering. on the one hand, when the illumination of a point is known, then the reflected radiance can be obtained by multiplying the indicent irradiance by the BRDF to obtain the reflected radiance. on the other hand, random-walk algorithms generate random light-paths to find the illumination coming from multiple reflections. when doing so, a random direction should be found in a way that the variance 方差 of the random estimation is small. the variance can be reduced by importance sampling which requires the probability density of random directions to mimic fr.cosθ for shooting (i.e. when the light path is followed from the light-source towards the eye) and fr.cosθ’ for gathering (i.e. when the light-path is followed from the eye towards the light-source).

BRDF models can be elaborated in two different ways. they can be simple mathematical constructions, when the validity is determined by checking the symmetry and the energy conservation. these models are called physically plausible empirical models. on the other hand, BRDFs can also be constructed by the careful analysis of the light-surface interaction, which results in the so called physically based models. physically based models are more natural, but are usually quite complex. empirical models are simpler and some of them are also good for importance sampling, but may have artificial features.

in this paper a BRDF model is presented that meets all the requirements of physical plausibility and all the other mentioned features. to derive the model, physically based analysis is carried out, which is similar to that presented by Cook and Torrance and by Shirley et. al. however, the results are generalized and simplified with the aim of allowing fast computations and importance sampling.

2. the model of microfacet reflection
   reflection can be rigorously 严格地 analyzed by modeling the surface irregularities by probability distributions, as has been proposed by Torrance, Sparrow, Cook and Blinn. In their model, the surface is assumed to consist of randomly oriented perfect mirrors, so-called microfacets. this model was also generalized to more accurately mimic the scattering of electromagnetic waves by He et.al. in.

A photon may be absorbed or reflected when colliding with a microfacet. reflected photons may leave the surface or may meet other microfacets, while absorbed photons can be re-emitted into a direction that is independent of the incident direction. the single reflection of photons is responsible for the specular effects and is relatively easy to analye. in this paper a physically based formula is proposed for this specular component. the other component is called the matte component, which includes the multiple scattering and re-emission, and is very difficult to describe analytically. this component is usually handled as constant diffuse factor, but this is against practical observations that show that the ratio of the powers of the matte and specular reflection is not constant but specular reflections become dominant at grazing angles. this type of interdependence 相互依存 was first discussed in where a coupling was established with the idea mirror component. in this paper a simple and physically plausible construction is proposed for the matte component that is couple with the glossy specular reflection. in the next sections the derivation of the specular component is presented first, then the matte component is defined to represent the multiple reflections and re-emissions which are missing from the specular part. thus the specular and matte 无光泽的；不光滑的 part are not independent but are coupled dependng on the viewing angle.

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