Non-symmetric Scattering in Light Transport Algorithms

Non-symmetric Scattering in Light Transport Algorithms

Reference: Eric Veach, Eurographics Rendering Workshop 1996 Proceedings (Porto, Portugal, June 1996), pp. 82-91. Also appears in Rendering Techniques '96, Springer-Verlag, New York, 1996.

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Abstract

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 models to obey Helmholtz reciprocity. By rewriting the transport operators in terms of optical invariants, we obtain a new framework where symmetry and reciprocity mean the same thing.

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.

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Postscript of main text, as it appears in the proceedings. The formatting is cramped due to page limits. (236K)
Postscript of main text, with spacious formatting and several typographical corrections. (250K)
Postscript of color plate, with low-resolution grayscale images. (1100K)
Plate 1(a). A bump-mapped teapot, and a polygonalized sphere with smooth interpolation of shading normals. This is a reference image, showing direct lighting as it would be computed by particle tracing. (JPEG, 100K)
Plate 1(b). The same model (including normals), showing errors caused by assuming that shading normals do not affect the symmetry of BRDF's. (JPEG, 104K)
Plate 2(a). A pool of water with small waves, illuminated by two area light sources (reference image). Simulated using a particle tracing algorithm. (JPEG, 112K)
Plate 2(b). This image has incorrect caustics (too bright), caused by assuming that refraction between air and water is described by a symmetric BTDF. (JPEG, 118K)

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Last modified: June 11, 1996

Eric Veach