1.5: Reflectance Functions

In the most general case of diffuse reflection, the reflectance of a surface will depend on both the direction of the incident radiation and that of the reflected radiation. The bidirectional reflectance distribution function, f_r, links the irradiance E to the reflected radiance, such that

$$L_{r}=f_{r}\left(\mu, \varphi ; \mu_{0}, \varphi_{0}\right) E\left(\mu_{0}, \varphi_{0}\right).$$

For a surface irradiated with flux density F, the irradiance is simply the component of the flux density perpendicular to the surface

$$E=\mu_{0} \mathbf{F},$$

so that, abbreviated, we can write

$$L_{r}=f_{r} \mu_{0} \mathbf{F}$$

One of the simplest examples of a reflectance rule is that of a Lambertian reflecting surface for which the radiance is isotropic, so that

$$L_{r}=\frac{\lambda_{0}}{\pi} \mu_{0} \mathbf{F},$$

where λ₀ is sometimes referred to as the Lambertian albedo . Although it is not strictly physically correct, it is convenient (Chandrasekhar, p147) to identify λ₀ with the single scattering albedo ϖ₀, so for Lambert’s law the BRDF is

$$f_{r}=\frac{\varpi_{0}}{\pi}.$$

For the most part, we shall refer all reflectance rules used to a BRDF; alternative reflectance functions will be discussed in Section 1.7.