Cook-Torrance Components

General form of the Cook-Torrance shading model:

$$r = k_a + \sum_{i=0}^n l_c * (\vec n \cdot \vec l) * ( d * r_d + s * r_s )$$ $$s + d = 1$$

$s$ in this equation is the specular material property, which we can use directly from our .pov files.

Diffuse reflection is often modelled using Lambertian:

$$r_d = k_d$$

There are other options, however, including Oren-Neyer.

Cook-Torrance specular reflectance (also sometimes simply called microfacet reflectance or microfacet BRDF) has the form:

$$r_s = \frac{D * G * F}{4 * (\vec n \cdot \vec l) * (\vec n \cdot \vec v)}$$

$D$ is the Normal Distribution Function, which describes the concentration of microfacets which are oriented such that they could reflect light. (1)

$G$ is the Geometric Shadowing-Masking Function, which describes the percentage of microfacets which are neither shadowed or masked by neighboring microfacets. (1)

$F$ is the Fresnel Function, which describes the amount of light reflected by the material (as opposed to being absorbed or transmitted by the material).

Symbols

$\vec n$ is the macrosurface normal.

$\vec l$ is the light vector.

$\vec v$ is the view vector.

$\vec m$ is the microsurface normal - in these computations (i.e. for implementation purposes) the half-vector is used.

$\theta_m$ is the angle between $\vec m$ and $\vec n$

$\alpha$ is the roughness constant. $\alpha$ is defined as such using the roughness material property from our .pov files: (4)

$$\alpha = roughness^2$$

Normal Distribution Functions

The $D$ function is always positive. (2) It is also normalized to satisfy certain geometric properties. (3)

This normalization is the source of the $\frac{1}{\pi \alpha^2}$ factors you see in the Blinn-Phong and Beckmann variations.

$D(m)$ is the concentration of microfacets oriented with subsurface normal $\vec m$. For our microfacet model we plug in $\vec h$ since we want $D(h)$, the concentration of microfacets which can reflect light from $\hat l$ to $\hat v$. (1)

Blinn-Phong

$$\Large D_{blinn} = \frac{1}{\pi \alpha^2} (\vec n \cdot \vec m)^{power}$$

Here, $power$ is the “shininess” or “specular power”. There doesn’t seem to be a standard way to compute this from $\alpha$ or roughness. Past iterations of CSC 473 have used the reciprical of POV-Ray roughness:

$$power = \frac{1}{roughness}$$

I recommend (and use) the equation provided by Karis’: (4)

$$power = \left( \frac{2}{\alpha^2} - 2 \right)$$

Beckmann

This is the NDF which Cook and Torrance originally advocated for in (5).

$$\Large D_{beckmann} = \frac{1}{\pi \alpha^2} \frac {e^{\frac{-tan^2(\theta_m)}{\alpha ^2}}} {(\vec n \cdot \vec m)^4}$$

Note that for normalized vectors, the following are equivalent:

$$cos^4(\theta_m) = (\vec n \cdot \vec m)^4$$

And by trigonometric identity, the following are equivalent:

$$-tan^2(\theta_m) = \left(\frac{(\vec n \cdot \vec m)^2 - 1}{(\vec n \cdot \vec m)^2}\right)$$

With those substitutions, you find the common form of the Beckmann NDF.

GGX

$$\Large D_{GGX} = \frac {\alpha^2 \chi^+ (\vec n \cdot \vec m)} {\pi \left((\vec n \cdot \vec m)^2 *(\alpha^2 + tan^2(\theta_m))\right)^2}$$

Geometric Functions

Cook-Torrance

$$\Large G_{cook-torrance} = min \left( 1, \frac{2 (\vec n \cdot \vec m) (\vec n \cdot \vec v)}{(\vec v \cdot \vec h)}, \frac{2 (\vec n \cdot \vec m) (\vec n \cdot \vec l)}{(\vec v \cdot \vec h)} \right)$$

GGX

$$\Large G_{GGX} = G_1(\vec v) * G_1(\vec l)$$ $$\Large G_1(x) = \chi^+(\frac{\vec x \cdot \vec m}{\vec x \cdot \vec n}) \frac{2}{1+\sqrt {1 + \alpha^2 tan^2(\theta_x))}}$$

$\chi^+$ is the positive characteristic function:

$$\large \chi^+(a) = \begin{cases} 1 & \text{if $a > 0$} \\ 0 & \text{if $a \leq 0$} \end{cases}$$

Fresnel Functions

Schlick’s Approximation

$$F_0 = \frac{(n - 1)^2}{(n + 1)^2}$$ $$F = F_0 + (1 - F_0)*(1 - (\vec v \cdot \vec h))^5$$

References

1: Naty Hoffman, Background: Physics and Math of Shading, 2012. From the SIGGRAPH 2012 Practical Physically Based Shading in Film and Game Production

2: Walter et al., Microfacet Models for Refraction through Rough Surfaces, 2007. Introduced the GGX form of D and G functions.

3: Nathan Reed, How Is The NDF Really Defined?, 2013. Describes where the normalization factor of D function comes from.

4: Brian Karis, Specular BRDF Reference, 2013. Reference of D, F, and G functions. Written by a UE4 graphics programmer.

5: Robert Cook & Kenneth Torrance, A Reflectance Model for Computer Graphics, 1982. Original Cook-Torrance paper. Many of the equations in this paper have slightly different forms in modern usage, so beware.