Rendering Equation

The rendering equation is a common equation that describes ray-optics-based light propogation which is commonly used for almost all rendering tasks (aside from accurate optics and quantum simualtions). The actual equation is as follows:

$$L_o(\overrightarrow{x},{\overrightarrow{\omega }}_o)=L_e(\overrightarrow{x},{\overrightarrow{\omega }}_o)+{\int }_{\mathrm{\Omega }}{f}_r(\overrightarrow{x},{\overrightarrow{\omega }}_i,{\overrightarrow{\omega }}_o)L_i(\overrightarrow{x},{\overrightarrow{\omega }}_i)({\overrightarrow{\omega }}_i\cdot \overrightarrow{n})d{\overrightarrow{\omega }}_i$$

Simply it can be put as: $L_o=L_e+L_r$.

The magic really lies within the BRDF as it expresses on how light behaves when interacts with the surface. There are some extensions to this function like the bidrectional scattering distribution function to further describe materials like skin. Extensions also exists to describe non-vacuum volumes.

Some downsides include:

• Assumes that wavelength is independent; which allows rendering to only focus on RGB instead of the full spectrum
  • Unable model materials which changes the wavelength
• Assumes that time is independent; also means each frame can be rendered independently and temporal methods are natural
  • Unable model materials which store and re-emit light (e.g. glow-in-the-dark)
  • Unable model relativistic velocities and nanosecond timescales
• Assumes ray-optics: faster to compute
  • Unable model wave interference and diffraction patterns

Many of these can done by externally calculating for a new BRDF or emittance function, however this would need to be done for all cases.

Wave Optics Propagation

The equation to model wave optics is the Rayleigh-Sommerfeld diffraction integral which commonly gets approximated to Fresnel diffraction integral.

$$U({\overrightarrow{x}}^{\prime })=\frac{1}{i\lambda }\int U(\overrightarrow{x})\frac{e^{ikr}}{r}d\overrightarrow{x};r=\Vert {\overrightarrow{x}}^{\prime }-\overrightarrow{x}\Vert ;k=\frac{2\pi }{\lambda }$$