The Rendering Equation
Ever wondered how it’s possible to render such impressive CG images and videos like Final Fantasy: The Spirits Within or Final Fantasy: Advent Children?
The (non-secrect) formula behind such achievements is this:
It’s the so called “Rendering Equation”, which was independently introduced by David Immel and James Kajiya in 1986.
It’s a physically based (classical physics – no relativity or quantum physics) equation that quite nicely represents light distribution within a N-dimensional space (usually 3 or 4-space).
It’s only indirectly applicable to CG, because the (infinite) integrals have to be implemented with discrete and finite sampling methods.
Common mathematical tricks to achieve acceptable render times (at expense of photo-realism) are:
- Markov Chains
- Importance Sampling
- Photon Mapping
- Irradiance Caching
- Monte Carlo / Quasi Monte Carlo techniques
- Adaptive rendering (use computing power only on perceivable effects)
- Spherical harmonics
- many more…
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