1. BRDF 的 作用：
A BRDF, bi-directional reflectance distribution function, is a tool for describing the distribution of reflected light at a surface. Given an incoming light ray at a point on a surface, the BRDF is used to calculate how much of that light ray will be reflected in a particular outgoing direction. Specifically, the BRDF is an approximation of the BSSRDF, bi-directional sub-surface scattering reflectance distribution function. The BRDF ignores sub-surface scattering and assumes that the light striking the surface at some point will be reflected from that same point.
The bidirectional reflectance distribution function (BRDF) gives a formalism for describing
reflection froma surface. Consider the setting in Figure 5.18: we’d like to know how much
radiance is leaving the surface in the direction ωo toward the viewer, Lo(p, ωo), as a result
of incident radiance along the direction ωi , Li(p, ωi).
The BRDF. The bidirectional reflectance distribution function is a four-dimensional
function over pairs of directions ωi and ωo that describes how much incident light along ωi is scattered
from the surface in the direction ωo.
2. BRDF 的定义
The BRDF was first defined by Fred Nicodemus around 1965.[1] The definition is:
The BRDF, is defined as the outgoing radiance divided by the irradiance. The units of the BRDF are therefore inverse steradians. A BRDF describes the relation between the incoming and outgoing radiances at a given point P on the surface. This is because the BRDF is a function of irradiance and the irradiance takes into account all incoming radiances. Because the BRDF is a ratio, the values are independent of the strength and geometry of the light source. This is very important, because it allows BRDF values calculated or measured on a material in one set of lighting conditions to be applied to that material under any lighting conditions.
3. BRDF 的 定义是怎么来的
参考 ： 《Directional reflectance and emissivity of an opaque surface》
Consider a radiation field, where the radiance Ni is a function of both position and direction, incident on the surface of an opaque body where some of the radiation is absorbed and the rest is reflected (as used here, "reflected" includes diffuse reflectance or scattering) to form a second radiation field, where the radiance Nr of the reflected radiation is also a function of position and direction.
Nr is directly proportional to Ni in the sense that, if the value of Ni is multiplied by a constant that is independent of position and direction,the resulting values of Nr will all be multiplied by the same constant factor.
(Ni 其实就是 Li， Nr 其实就是Lo)
consider only the radiant power incident on a particular element dA of a reflecting surface through an elementary beam of
solid angle dwi .(考虑 光 通过 dwi 方向入射 到 dA上的 radiant power （Radiant Flux）)
This incident radiant power（Radiant Flux） is given by :
dPi = Ni * dwi * cos(pi) * dA
单位是 ： [w]
Correspondingly, the irradiance at dA is
dHi = Ni * dwi * cos(pi)
单位是 ： [w * cm^(-2)]
Then the radiant intensity of the surface element dA, due to reflection (scattering) of radiatlon from this incident elementary beam, in the direction dwr is
（这里是求出在 dA 上的 radiant intensity）
dJr = dPi / dwr , 单位是 [ w * sr ^(-1) ]
进一步来说，我们不知道 dwr 是什么，所以 假设 F= 1 / dwr
dJr = F * dPi
两边同时除以 dA 得到
dJr / dA = F * dPi / dA
简化得到
dNr = F * dHi 单位是 【w * cm^(-2) * sr ^(-1)】
=》
F = dNr / dHi 单位是 【sr ^(-1)】
(其实就是 F = dNr（Radiance） / dHi （Irradiance）)
所以：
F is the partial retfectance or "reflection-distribution function' of the
surface element dA for radiation incident from the direction dwi，and reflected (scattered) in the direction dwr，