Non-symmetric Scattering in Light Transport Algorithms
Non-symmetric scattering is far more common in computer graphics than is generally recognized, and can occur even when the underlying scattering model is physically correct. For example, we show that non-symmetry occurs whenever light is refracted, and also whenever shading normals are used (e.g. due to interpolation of normals in a triangle mesh, or bump mapping ).
We examine the implications of non-symmetric scattering for light transport theory. We extend the work of Arvo et al.  into a complete framework for light, importance, and particle transport with non-symmetric kernels. We show that physically valid scattering models are not always symmetric, and derive the condition for arbitrary model to obey Helmholtz reciprocity. By rewriting the transport operators in terms of optical invariants, we obtain a new framework where symmetry and reciprocity are the same.
We also consider the practical consequences for global illumination algorithms. The problem is that many implementations indirectly assume symmetry, by using the same scattering rules for light and importance, or particles and viewing rays. This can lead to incorrect results for physically valid models. It can also cause different rendering algorithms to converge to different solutions (whether the model is physically valid or not), and it can cause shading artifacts. If the non-symmetry is recognized and handled correctly, these problems can easily be avoided.
KeywordsComputer Graphic Transport Operator Particle Weight Global Illumination Light Transport
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