We first introduce a non-linear integral scattering equation that describes scattering from complex objects directly in terms of the composition of their lower-level scattering properties. This equation was first derived to solve scattering problems in astrophysics and has gone on to revolutionize approaches to transport problems in a number of fields. We derive this equation in a sufficiently general setting to be able to apply it to a variety of problems in graphics, which typically has problems with higher-dimensionality, more complexity, and less regularity than those in other fields. Methods to solve this equation have a divide-and-conquer flavor to them, in contrast to previous iterative methods based on the equation of transfer (i.e. the rendering equation). We apply Monte Carlo techniques to solve this scattering equation efficiently; to our knowledge, this is the first application of Monte Carlo to solving it in any field.
We next introduce Preisendorfer's Interaction Principle, which subsumes both scattering and light transport based approaches to transfer problems. It leads to a derivation of a set of adding equations that describe scattering from multiple objects in terms of how they scatter light individually. We show how Monte Carlo techniques can be applied to solve these adding equations and apply them to the problem of rendering subsurface scattering.
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