“Energy preserving non-linear filters” by Rushmeier and Ward

Monte Carlo techniques for image synthesis are simple and powerful, but they are prone to noise from inadequate sampling. This paper describes a class of non-linear filters that remove sampling noise in synthetic images without removing salient features. This is achieved by spreading real input sample values into the output image via variable-width filter kernels, rather than gathering samples into each output pixel via a constant-width kernel. The technique is nonlinear because kernel widths are based on sample magnitudes, and this local redistribution of values cannot generally be mapped to a linear function. Nevertheless, the technique preserves energy because the kernels are normalized, and all input samples have the same average influence on the output. To demonstrate its effectiveness, the new filtering method is applied to two rendering techniques. The first is a Monte Carlo path tracing technique with the conflicting goals of keeping pixel variance below a specified limit and finishing in a finite amount of time; this application shows how the filter may be used to “clean up” areas where it is not practical to sample adequately. The second is a hybrid deterministic and Monte Carlo ray-tracing program; this application shows how the filter can be effective even when the pixel variance is not known.

1. J. Arvo and D. Kirk. Particle Transport and Image Synthesis. Proc. of SIGGRAPH ’90 (Dallas,TX, Aug. 6- 10, 1990. Computer Graphics, 24(4):63{66, Aug. 1990.
2. J. Arvo and D. Kirk. Unbiased Sampling Techniques for Image Synthesis. Proc. of SIGGRAPH ’91 (Las Vegas,NV, Jul. 28- Aug. 2). Computer Graphics, 25(4):153{156, Jul. 1991.
3. S. Chen, H. Rushmeier, G. Miller, and D. Turner. A Progressive Multi-Pass Method for Global Illumination. Proc. of SIGGRAPH ’91 (Las Vegas,NV, Jug. 28- Aug. 2). Computer Graphics, 25(4):165{174, Jul. 1991.
4. K. Chiu, M. Herf, P. Shirley, S.Swamy, C.Wang, and K. Zimmerman. Spatially Non-Uniform Scaling Functions for High Contrast Images. In Proc. of Graphics Interface 1993 (Toronto, May 19-21), pages 245{253.
5. C.-H. Chu and E. Delp. Impulsive Noise Suppression and Background Normalization of Electrocardiogram Signals Using Morphological Operators. IEEE Trans. on Biomedical Engineering, pages 262{267, Feb. 1989.
6. M. Dipp~e and E. Wold. Antialiasing Through Stochastic Sampling. Proc. of SIGGRAPH ’85 (San Francisco,CA, Jul. 22- 26, 1991). Computer Graphics, 19(3):69{78, Jul. 1985.
7. J. Kajiya. The Rendering Equation. Proc. of SIG- GRAPH ’86 (Dallas,TX, Aug. 18-22). Computer Graphics, 20(4):143{150, Aug. 1986.
8. M. E. Lee and R. A. Redner. A Note on the Use of Nonlinear Filtering in Computer Graphics. IEEE Computer Graphics and Applications, pages 23{29, May 1990.
9. D. Mitchell. Generating Antialiased Images at Low Sampling Densities. Proc. of SIGGRAPH ’87 (Anaheim,CA, Jul. 27-31). Computer Graphics, 21(4):65{72, Jul. 1987.
10. D. Mitchell. Spectrally Optimal Sampling for Distributed Ray Tracing. Proc. of SIGGRAPH ’91 (Las Vegas,NV, Jul. 28- Aug. 2). Computer Graphics, 25(4):157{164, Jul. 1991.
11. W. Purgathofer. A Statistical Method for Adaptive Sampling. Computers & Graphics, pages 157{162, 1987.
12. J. Tumblin and H. Rushmeier. Tone Reproduction for Realistic Images. IEEE Computer Graphics and Applications, pages 42{48, Nov. 1993.
13. G. Ward. A Contrast-Based Scalefactor for Luminance Display. In P. Heckbert, editor, Graphics Gems IV. Academic Press, 1994.
14. G. Ward. The RADIANCE Lighting Simulation and Rendering System. Proc. of SIGGRAPH ’94 (Orlando,FL, Jul. 24-29). Computer Graphics, Annual Conference Series, 1994.