PHYSICALLY BASED SURFACE MODELS

The most widely used model of surfaces that are not perfectly smooth and uniform is the Cook-Torrance [COOK 1982] model. What they did was to assume that

• The geometry of the roughness of the surface is larger than that of the wavelength of the light.

• The geometry is considered to be made up of v-shaped facets.

• The facets are randomly oriented.

• The facets are mirrorlike.

Using such a model, there are three different ways that a light ray can interact with the entire internal reflecting surface of the v-shaped geometry (termed "microfacets") depending on the angle of the v.

• The ray can reflect with no interference.

• The ray could be partially shadowed by other geometry.

• The ray could get blocked by part of the geometry.

These three cases are shown in Figure 3.34.

Figure 3.34: The three types of surface reflections that can occur in the Cook-Torrance surface model.

Roughness Distribution Function

When using this model, we need a way to specify the distribution of the slopes of the facets. This is termed the slope distribution function, D. Blinn [BLINN 1977] used a Gaussian distribution function to model the slope distribution.

where c is some arbitrary constant and cos(α) = n • h. The parameter m is the RMS slope parameter, for which smaller values (0.2) indicate a smooth surface, whereas larger values (0.8) indicate a rougher surface.

Cook and Torrance [COOK 1982] used a Beckmann distribution function, which they state can successfully model both rough and smooth dielectrics and conductors.

This model has the advantage of not requiring a constant but just relies on one parameter m to specify the surface roughness.

There are other models such as the Trowbridge-Reitz model [TROWBRIDGE 1975], which models the microfacets as ellipsoids. You can even consider the Phong specular term to be a distribution function with the roughness specified as the exponential power value.

Geometric Attenuation Function

In addition to describing how the geometry of a rough surface is laid out, the Cook-Torrance model can be used to calculate the amount of light actually hitting on the microfacets. Blinn [BLINN 1977] has a very nice derivation of the geometry involved. Basically, there are three different cases to consider.

1. There is no interference in any of the light.

2. Some of the incoming light is blocked (shadowed).

3. Some of the reflected light is blocked (masked).

Blinn calculated the amount of light that would get blocked in each case. These are basically functions of the light direction and the facet normal. You then calculate these three values and select the minimum as the geometric attenuation function, G.