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.

### Additional information

If you would like more information about the **rendering algorithm**
used to compute the image above, it is described in
a different paper.
Make sure to get both the **main text** and the **color plate**,
if you want the full paper.

- 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)

If the images are too dark, try the
gamma corrected pages.
All JPEG images were compressed using a quality setting of 90.

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Last modified: June 11, 1996

Eric Veach