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Chapter 69: Cook-Torrance GGX

Power exponent

In traditional Blinn-Phong we had a simple power parameter (usually called shininess) that we used to control the size and intensity of the specular highlight.

In Cook-Torrance we want to have a more general parameter that can be used in multiple places. Cook-Torrance defines a constant $\alpha$ that indicates the roughness of the material, with 0 indicating ideal smooth surfaces and 1 indicating maximum roughness. In practice you never want to use absolute 0 or 1 since these edge cases tend to produce divide-by-zero and other issues.

We can then relate our new $\alpha$ parameter to our old power variable as such:

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

roughness is one of the material properties defined by our .pov files. We’ll use UE4’s convention for determining $\alpha$ from roughness:

$$\alpha = \text{roughness}^2$$

Note that for traditional Blinn-Phong (specifically, not just $D_{blinn}$ but if we aren’t doing Cook-Torrance at all), you should still use the above power equation for shininess, but you don’t need to square the roughness constant:

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

This is an arbitrary convention but I have found it works reasonably well!

GGX Equations

Normal Distribution Function

$$\large D_{GGX} = \chi^+ (\vec n \cdot \vec h) \frac { \alpha^2 } { \pi \left((\vec n \cdot \vec h)^2 *(\alpha^2 - 1) + 1\right)^2}$$

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

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

Geometric Shadowing Function

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

$\theta_x$ is the angle between $\hat x$ and $\hat n$.

$tan^2(\theta) = sec^2(\theta) - 1$ $= \frac{1}{cos^2(\theta)} - 1$ $= \frac{1 - cos^2(\theta)}{cos^2(\theta)}$

$$tan^2(\theta_x) = \frac{1 - (\vec x \cdot \vec n)^2}{(\vec x \cdot \vec n)^2}$$

Implementation Details

Every dot product in these equations should be “saturated”, i.e. clamped between 0 and 1.

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