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 [5]).
We examine the implications of non-symmetric scattering for light transport theory. We extend the work of Arvo et al. [4] 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.
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References
Al-Gwaiz, M. A. Theory of Distributions. Marcel Dekker, Inc., New York, 1992.
Ambartzumian, R. V. Factorization Calculus and Geometric Probability. Cambridge University Press, 1990.
Arvo, J. Analytic Methods for Simulated Light Transport. PhD thesis, Yale University, Dec. 1995.
Arvo, J., Torrance, K., and Smits, B. A framework for the analysis of error in global illumination algorithms. In Proceedings of SIGGRAPH ’94 (July 1994), ACM Press, pp. 75–84.
Blinn, J. F. Simulation of wrinkled surfaces. In Computer Graphics (SIGGRAPH 78 Proceedings) (Aug. 1978), vol. 12, pp. 286–292.
Chandrasekar, S. Radiative Transfer. Dover Publications, New York, 1960.
Christensen, P. H., Salesin, D. H., and DeRose, T. D. A continuous adjoint formulation for radiance transport. In Fourth Eurographics Workshop on Rendering (Paris, June 1993), pp. 95–104.
Christensen, P. H., Stollnitz, E. J., Salesin, D. H., and DeRose, T. D. Global illumination of glossy environments using wavelets and importance. ACM Trans. Graphics 15 (1996), 37–71.
Cohen, M. F., and Wallace, J. R. Radiosity and Realistic Image Synthesis. Academic Press Professional, San Diego, CA, 1993.
Delves, L. M., and Mohamed, J. L. Computational Methods for Integral Equations. Cambridge University Press, New York, 1985.
Drude, P. The Theory of Optics. Hemisphere Publishing Corp., New York, 1981. Original date of publication 1900. Translated from the German by C. R. Mann and R. A. Millikan.
Fournier, A. From local to global illumination and back. In Rendering Techniques ’95 (New York, 1995), P. M. Hanrahan and W. Purgathofer, Eds., Springer-Verlag, pp. 127-136.
Glassner, A. Principles of Digital Image Synthesis. Morgan Kaufmann, New York, 1995.
Hall, R. Illumination and Color in Computer Generated Imagery. Springer-Verlag, New York, 1989.
Heckbert, P. S. Adaptive radiosity textures for bidirectional ray tracing. Computer Graphics 24, 4 (July 1990), 145–154.
Heckbert, P. S. Simulating Global Illumination Using Adaptive Meshing. PhD thesis, University of California, Berkeley, June 1991.
Keitz, H. A. E. Light Calculations and Measurements. The Macmillan Company, New York, 1971.
Lafortune, E. P., and Willems, Y. D. Bi-directional path tracing. In Proceedings of CompuGraphics (Alvor, Portugal, Dec. 1993), pp. 145–153.
Lafortune, E. P., and Willems, Y. D. A theoretical framework for physically based rendering. Computer Graphics Forum 13, 2 (June 1994), 97–107.
Lewins, J. Importance, The Adjoint Function: The Physical Basis of Variational and Perturbation Theory in Transport and Diffusion Problems. Pergamon Press, New York, 1965.
Nicodemus, F. E. Radiance. Am. J. Phys. 31 (1963), 368–377.
Nicodemus, F. E. Directional reflectance and emissivity of an opaque surface. Applied Optics 4 (1965), 767–773.
Nicodemus, F. E. Reflectance nomenclature and directional reflectance and emissivity. Applied Optics 9 (1970), 1474–1475.
Nicodemus, F. E. Comment on “current definitions of reflectance”. J. Opt. Soc. Am. 66 (1976), 283–285.
Nicodemus, F. E., Ed. Self-Study Manual on Optical Radiation Measurements: Part I—Concepts, Chapters 1 to 3. No. 910-1 in Technical Note. National Bureau of Standards (US), Mar. 1976.
Nicodemus, F. E., Ed. Self-Study Manual on Optical Radiation Measurements: Part I—Concepts, Chapters 4 and 5. No. 910-2 in Technical Note. National Bureau of Standards (US), Feb. 1978.
Nicodemus, F. E., et al. Geometric considerations and nomenclature for reflectance. Monograph 161, National Bureau of Standards (US), Oct. 1977.
Pattanaik, S. N., and Mudur, S. P. Adjoint equations and random walks for illumination computation. ACM Trans. Graphics 14 (Jan. 1995), 77–102.
Phong, B. T. Illumination for computer generated pictures. Communications of the ACM 18, 6 (June 1975), 311–317.
Schröder, P., and Hanrahan, P. Wavelet methods for radiance computations. In Fifth Eurographics Workshop on Rendering (Darmstadt, Germany, June 1994), pp. 303–311.
Shirley, P., Wade, B., Hubbard, P. M., Zareski, D., Walter, B., and Greenberg, D. P. Global illumination via density-estimation. In Rendering Techniques ’95 (New York, 1995), P. M. Hanrahan and W. Purgathofer, Eds., Springer-Verlag, pp. 219–230.
Siegel, R., and Howell, J. R. Thermal Radiation Heat Transfer, second ed. Hemisphere Publishing Corp., New York, 1981.
Smits, B. E., Arvo, J. R., and Salesin, D. H. An importance-driven radiosity algorithm. In Computer Graphics (SIGGRAPH ’92 Proceedings) (July 1992), vol. 26, pp. 273–282.
Snyder, J. M., and Barr, A. H. Ray tracing complex models containing surface tessellations. In Computer Graphics (SIGGRAPH ’87 Proceedings) (July 1987), vol. 21, pp. 119–128.
Steel, W. H. Luminosity, throughput, or etendue? Applied Optics 13 (1974), 704–705.
Taylor, A. E., and Lay, D. C. Introduction to Functional Analysis, second ed. John Wiley & Sons, New York, 1980.
Veach, E., and Guibas, L. Bidirectional estimators for light transport. In Fifth Eurographics Workshop on Rendering (Darmstadt, Germany, June 1994), pp. 147–162.
Veach, E., and Guibas, L. J. Optimally combining sampling techniques for monte carlo rendering. In SIGGRAPH ’95 Conference Proceedings (Aug. 1995), Addison Wesley, pp. 419–428.
von Fragstein, C. 1st eine lichtbewegung stets umkehrbar? Opt. Acta 2 (1955), 16–22.
von Helmholtz, H. Helmholtz S Treatise on Physiological Optics, vol. 1. Dover Publications, New York, 1962. James P. C. Southall, Ed. Original date of publication 1856. Translated from the third German edition.
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Veach, E. (1996). Non-symmetric Scattering in Light Transport Algorithms. In: Pueyo, X., Schröder, P. (eds) Rendering Techniques ’96. EGSR 1996. Eurographics. Springer, Vienna. https://doi.org/10.1007/978-3-7091-7484-5_9
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DOI: https://doi.org/10.1007/978-3-7091-7484-5_9
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