2.1 Roughness

When considering the surface normal, it is common to model the roughness of the object, representing sub-pixel-scale geometry. In a very smooth object, all parts of the pixel have exactly the same surface normal and $\vec n$ is a simple constant. But most objects are at least a little rough, meaning some parts of the pixel have a noticeably different normal than other parts, and $\vec n$ is better treated as a distribution of normals over the area of the pixel.

The roughness of a material has more impact than just changing $\vec n$: a tiny bump on one part of the material might cast a shadow over another part (called self shadowing), and some light scattered off of a bump might hit another bump to add a kind of local light source called multiscattering).

Roughness is traditionally represented in equations with the symbol $\alpha^2$ because early papers used a definition of roughness that was the square root of what was needed for the math. In 2012, Brent Burley at Walt Disney Animation Studios published a whitepaper which included the recommendation that artists preferred a roughness parameter that was the square root of $\alpha$, and that has become mainstream since. This means that $\alpha^2 = \text{roughness}^4$, where $\text{roughness}$ is the value provided by artists, materials, and roughness textures.

3.1 Fresnel

In 1994, Christophe Schlick published¹¹ DOI 10.1111/1467-8659.1330233 several computationally-efficient energy-preserving visually-close-enough approximations of PBR functions. Some of these have become the standard implementation for PBR.

Schlick’s equation for Fresnel reflectivity is

$$\begin{split} F_0 &= \left(1 - \text{ior} \over 1 + \text{ior}\right)^2 \\ F_r &= F_0 + (1 - F_0)\big(1 - |\vec v \cdot \vec h|\big)^5 \end{split}$$

where $\text{ior}$ is the index of refraction of the material²² Often assumed to be 1.5, a common value for organic material like wood and plastic, giving $F_0 = 0.04$ and $\vec h$ is the halfway vector, $\vec \ell + \vec v$ normalized.

$F_r$ is a ratio of light that is specularly reflected: when $F_r$ is 0, no light is reflected and all makes it into the material, and when $F_r$ is 1, all light is reflected and none makes it into the material.

Note that roughness is not mentioned in this approximation. While roughness does have some impact on Fresnel, it is generally not considered to be visually important enough to be worth the extra computation.

To fully preserve energy, light that is specularly reflected due to Fresnel should not be used to illuminate the material. However, that lighting reduction is often omitted for efficiency, meaning that objects may be a bit brighter than they should be. In 2017 Christopher Kulla and Alejandro Conty presented³³ In a SIGGRAPH course (DOI 10.1145/3084873.3084893, which does not publish course content; one author has the slides on his personal page. their finding that the lighting loss did not seem to have a simple equation and when they wanted to add it they used an piecewise trilinear equation with 4096 terms.

3.2 Reflection

There are many models of reflection, in part because there are many micro-geometries that might make a material rough.

Since the 2010s, the most popular model has been the GGX model, so named (with no explanation of why they picked that name) by Bruce Walter, Stephen Marschner, Honsong Li, and Kenneth Torrance in 2007⁴⁴ DOI 10.5555/2383847.2383874. The same model was also derived and published much earlier, but without getting much buy-in, by T. S. Towbridge and Karl Reitz in 1975⁵⁵ DOI 10.1364/JOSA.65.000531.

The Towbridge-Reitz/GGX model is the product of two functions. One is a microfacet distribution function: $$D = \frac{\alpha^2}{\big(1 + (\vec n \cdot \vec h)^2 (\alpha^2 - 1)\big)^2 \pi}$$ The other is a self-shadowing/occluding function or visibility function⁶⁶ This page has the visibility function. The self-shadowing/occluding function has an extra factor $4 |\vec n \cdot \vec l| \, |\vec n \cdot \vec v|$ in the numerator and is traditionally given the letter $G$ instead of $V$., typically based on an derivation published by B. Smith in 1967⁷⁷ DOI 10.1109/TAP.1967.1138991: $$V = \frac{1}{ \left(|\vec n \cdot \vec \ell| + \sqrt{\alpha^2 + (1-\alpha^2)(\vec n \cdot \vec \ell)^2}\right) \left(|\vec n \cdot \vec v| + \sqrt{\alpha^2 + (1-\alpha^2)(\vec n \cdot v)^2}\right) }$$

If any of $\vec n \cdot \vec h$, $\vec h \cdot \vec l$, or $\vec h \cdot \vec v$ is negative, no reflection occurs.

The product $V D$ gives the ratio of light that is reflected directly from the provided light source ($\vec \ell$) to the provided viewer ($\vec v$). Any remaining light ($1 - V D$) is still reflected, but towards other viewers.

The common form of Towbridge-Reitz/GGX as given above is not fully energy-conserving because it lacks a term for multiscattering, meaning very rough surfaces come out darker than they should. In 2017 Christopher Kulla and Alejandro Conty presented⁸⁸ In a SIGGRAPH course (DOI 10.1145/3084873.3084893, which does not publish course content; one author has the slides on his personal page. an additional multiscattering term, based not on a function of any kind but rather on precomputing the multiscattering energy loss of the above formula on a grid of viewing angles and roughnesses and looking up the energy to add back in via a texture lookup in that grid.

3.3 Diffuse

Lambert’s law is based on simple trigonometry: a unit area of light impacting a surface at angle $\theta$ is spread out over an area of $1 / \cos(\theta)$ of the surface and thus reflects with intensity proportional to $\cos(\theta)$. Integrating over all viewing angles shows $\cos(\theta)$ alone would diffuse $\pi$ units of light for each unit of inbound light, which violates conservation of energy, so we can derive the correct diffusion intensity as $\cos(\theta) / \pi$. $\cos(\theta)$ is easy to compute: it’s just $\vec n \cdot \vec \ell$, giving us $$\vec n \cdot \vec \ell \over \pi$$

Note that roughness is not mentioned in this approximation. While roughness does have some impact on diffuse lighting, the normal distribution, self-shadowing, and multiscattering terms come close to cancelling one another out, leaving a minor enough visual effect that it is generally not considered to be worth the extra computation.

4.1 Image-based lighting

Environment maps are generally provided as high dynamic range (HDR) images with full 360°, $4\pi$ steradian coverage. There are multiple HDR image formats and multiple ways to project an entire surrounding sphere onto a rectangular image, but these are not generally used as-is: rather, they are first preprocessed into a few cubemaps.

A cubemap is an set of 6 square images that together cover the surface of a cube. They are indexed by 3-vectors: the face to use is chosen by finding which component of the vector is largest and the texel on that face is chosen by dividing the other two components by that largest value.

The pre-processing converts the environment map into more than one cube map: one for each major type of lighting.

For diffuse lighting, the cube map is indexed by surface normal. To find the light intensity and color of a given diffuse cube map texel we iterate over the environment map and apply the diffuse lighting equation for the surface normal mapping to that cube map texel, adding the effect of all such lightings. An important consideration here is that in most environment maps, not all pixels represent the same solid angle and smaller pixels contribute less light to the diffuse cube map.

For reflective lighting, the cube map is indexed by the reflection of the view vector over the surface normal, $\vec r = (2 \vec n \cdot \vec v) \vec n - \vec v$. For zero-roughness materials, each texel of the cube map is simply the environment map pixel. For higher roughness, the cube map needs to be blurred as specified in the Towbridge-Reitz/GGX distribution. Such bluring is computationally expensive and is done as part of the cube map creation, with higher roughnesses stored as smaller mip maps. Computing the blur needs to not only follow Towbridge-Reitz/GGX but also the non-uniform solid angles of pixels, and in practice is often done by using a Monte Carlo quasirandom sampling of the Towbridge-Reitz/GGX distribution at a given roughness.

Once the cube maps are created, image-based lighting becomes

1. Compute Fresnel the same as with point lights.
2. For diffuse lighting, check the diffuse cube map instead of point light computation.
3. For reflection, check the reflection cube map at a mip map level based on roughness instead of the point light computation.
4. Combine those with absorbtion based on the fresnel, metalicity, and absorbtion color of the material.
5. If there are also point or area lights, compute and add their contribution too.