Physically-based BRDF

Introduction

Physically based rendering have been known for many years, but the "ad-hoc" rendering models (such as Phong) are still widely used in game. These "ad-hoc" models require laborious tweaking to produce high-quality images. However, physically based, energy-conserving rendering models easily create materials that hold up under a variety of lighting environments.

Surprisingly, physically based models are not more difficult to implement or evaluate than the traditional "ad-hoc" ones.

Reflectance equation

The most common used rendering model in game describes only reflectance, not including terms such as SSS. The reflectance equation is:

$LaTeX: L_0(\mathbf{v})=\int_{\Omega} \rho(\mathbf{l},\mathbf{v}) \otimes L_i(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i$

Here $LaTeX: \rho(\mathbf{l},\mathbf{v})$ is BRDF, $LaTeX: L_i(\mathbf{l})$ is the contribution from light source, $LaTeX: (\mathbf{n} \cdot \mathbf{l})$ is the angle between light and surface normal. This integration results the sum of all light sources contribute to a surface point.

Diffuse term

The simplest BRDF is the Lambert. The well-known Lambertian BRDF in game is present as $LaTeX: (\mathbf{n} \cdot \mathbf{l})$. However, it is part of reflectance equation, and lambertian term is actually a constant value:

$LaTeX: \rho_{lambert}(\mathbf{l},\mathbf{v})=\frac{\mathbf{c}_{diff}}{\pi}$

Punctual Light Sources

These are classic point, directional, and spot lights in compute games. These local light source can be abstracted to a concept named "punctual light sources". They are infinitely small light with centain direction. Since what we want is the lighting reach a surface point, we don't need to consider the attenuation. A punctual light source are parameterized by the light color $LaTeX: \mathbf{c}_{light}$ and the light direction vector $LaTeX: \mathbf{l_c}$. The light color $LaTeX: \mathbf{c}_{light}$ is specified as the color a white Lambertian surface would have when illuminated by the light from a direction parallel to the surface normal ($LaTeX: \mathbf{l_c}=\mathbf{n}$).

How to calculate the contribution from this light source to a point? Here we will start by defining a tiny area light source centered on $LaTeX: \mathbf{l_c}$, with a small angular extent $LaTeX: \varepsilon$. This tiny area light illuminates a shaded surface point with the incoming radiance function $LaTeX: L_{tiny}(\mathbf{l})$. The incoming radiance function has the following two properties:

$LaTeX: \forall\mathbf{l}|\angle(\mathbf{l}, \mathbf{l_c})>\varepsilon, L_{tiny}(\mathbf{l})=0$ $LaTeX: if \mathbf{l_c}=\mathbf{n}, \mathbf{c}_{light}=\frac{1}{\pi}\int_{\Omega} L_{tiny}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i$

The first property says that no light is incoming for any light directions which form an angle greater than $LaTeX: \varepsilon$ with $LaTeX: \mathbf{l_c}$. The second property follows from the definition of $LaTeX: \mathbf{c}_{light}$. Since the surface is white, $LaTeX: \mathbf{c}_{diff}=1$. Applying reflectance equation and Lambert from last article, there's the second property. Because $LaTeX: \mathbf{c}_{light}$ requires $LaTeX: \mathbf{l_c}=\mathbf{n}$, it is the limit as $LaTeX: \varepsilon$ goes to 0:

$LaTeX: if \mathbf{l_c}=\mathbf{n}, \mathbf{c}_{light}=\lim_{\varepsilon \to 0}(\frac{1}{\pi}\int_{\Omega} L_{tiny}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i)$

Since $LaTeX: \mathbf{l_c}=\mathbf{n}$ and $LaTeX: \varepsilon \to 0$, we can assume $LaTeX: \mathbf{n} \cdot \mathbf{l}=1$, which gives us:

$LaTeX: \mathbf{c}_{light}=\lim_{\varepsilon \to 0}(\frac{1}{\pi}\int_{\Omega} L_{tiny}(\mathbf{l}) d\omega_i)$

That is:

$LaTeX: \lim_{\varepsilon \to 0}(\int_{\Omega} L_{tiny}(\mathbf{l}) d\omega_i) = \pi \mathbf{c}_{light}$

Now we shall apply our tiny area light to a general BRDF, and look at its behavior in the limit as goes to 0:

$LaTeX: L_0(\mathbf{v})=\lim_{\varepsilon \to 0}(\int_{\Omega} \rho(\mathbf{l},\mathbf{v}) \otimes L_{tiny}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i)=\rho(\mathbf{l_c}, \mathbf{v}) \otimes \lim_{\varepsilon \to 0}(\int_{\Omega} L_{tiny}(\mathbf{l}) d\omega_i)(\mathbf{n} \cdot \mathbf{l_c})$

So

$LaTeX: L_0(\mathbf{v})=\pi \rho(\mathbf{l_c}, \mathbf{v}) \otimes \mathbf{c}_{light} (\mathbf{n} \cdot \mathbf{l_c})$

As you see, the tiny area light source term disappears. The remaining of our equation is what we familar with.