Components of the BRDF
		Specular component
		Uniform diffuse component

Directional diffuse component

The directional diffuse component spreads reflected light out to the entire hemisphere above a reflecting surface. The component has a very strongly-directional, nonuniform character. It is responsible for highlights of light sources and blurry reflection images on a rough surface. It arises from diffraction and scattering by the surface roughness. The reflected radiance can be expressed as a product:

where G is a geometric factor, and D is a distribution function. The |F|² and S terms are the Fresnel and shadowing terms previously discussed.

Note that the incident solid angle dω_i appears in Equation (7). In the interactive figures that have a dω_i the coefficient of dω_i is approximated using a midpoint rule, and dω_i is evaluated assuming a circular cross-section. The values of solid angle shown for “dOmega” on the control panels have been multiplied by a scale factor of ten. The actual solid angles are one-tenth of the values shown. The maximum slider value of unity corresponds to a maximum incident cone angle of 20.4°. This represents an incident solid angle of 0.1 steradians, or 1/20π = 1.6% of the incident hemisphere.

The illustration below demonstrates how the geometric factor G varies with incidence angle θ_i. The incidence direction is shown with the red line. The semicircle is the unit hemisphere. The function G arises from geometric projections of the incident and reflected directions onto the surface and is given in [HE91].

 
Figure 11: Interactive display of geometric factor G

The distribution function D is given by the following infinite summation:

where h is a function depending on the surface parameter τ, wavelength λ, and incidence and reflection angles. It is given by:

The illustration below shows how the distribution function D varies with θ_i, σ, τ, and λ. The semicircle is the unit hemisphere. The incident direction is shown with the red line on the left; the dashed line to the right indicates the specular direction with respect to the mean plane of the surface.

 
Figure 12: Interactive display of the distribution function D

Finally, let's see how the entire directional diffuse component dI_r,dd varies with the parameters θ_i, σ, τ, λ, and material. This is shown in Figure 13, where the incident light cone is on the left and the specular-ray direction for the mean surface is indicated on the right (dashed line). The dashed semicircle is added for reference, and corresponds to Lambertian, ideal-diffuse, 100% reflection of the incident beam. For such a reference reflector, the reflected radiance is where dEr is the reflected energy flux. The π-factor arises when converting energy flux to radiance for a Lambertian surface. For a 100% reflector, dE_r = dE_i, with dE_i given by Equation (2). Thus,

and we see that the radius of the reference hemisphere varies with θ_i and dω_i.

 
Figure 13: Interactive display of entire directional diffuse term

		Components of the BRDF
		Specular component
		Uniform diffuse component

Last modified: Mon Sep 5 23:19:34 EDT 2011

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