“Blue-Noise Sampling With Controlled Aliasing” by Heck, Schlömer and Deussen

  • ©Daniel Heck, Thomas Schlömer, and Oliver Deussen




    Blue-Noise Sampling With Controlled Aliasing

Session/Category Title: Sampling




    In this article we revisit the problem of blue noise sampling with a strong focus on the spectral properties of the sampling patterns. Starting from the observation that oscillations in the power spectrum of a sampling pattern can cause aliasing artifacts in the resulting images, we synthesize two new types of blue noise patterns: step blue noise with a power spectrum in the form of a step function and single-peak blue noise with a wide zero-region and no oscillations except for a single peak. We study the mathematical relationship of the radial power spectrum to a spatial statistic known as the radial distribution function to determine which power spectra can actually be realized and to construct the corresponding point sets. Finally, we show that both proposed sampling patterns effectively prevent structured aliasing at low sampling rates and perform well at high sampling rates.


    1. Balzer, M., Schlömer, T., and Deussen, O. 2009. Capacity-constrained point distributions: A variant of Lloyd’s method. ACM Trans. Graph. 28, 3, 86:1–8.
    2. Bracewell, R. N. 1999. The Fourier Transform and its Applications 3rd Ed. McGraw-Hill, New York.
    3. Chen, Z., Yuan, Z., Choi, Y.-K., Liu, L., and Wang, W. 2012. Variational blue noise sampling. IEEE Trans. Vis. Comput. Graph. 18, 1784–1796.
    4. Cook, R. L. 1986. Stochastic sampling in computer graphics. ACM SIGGRAPH Comput. Graph. 5, 1, 51–72.
    5. Crawford, J., Torquato, S., and Stillinger, F. H. 2003. Aspects of correlation function realizability. J. Chem. Phys. 119, 14, 7065–7074.
    6. Dammertz, H., Keller, A., and Dammertz, S. 2008. Simulation on rank-1 lattices. In Monte Carlo and Quasi-Monte Carlo Methods, Springer, 205–216.
    7. Dippé, M. A. Z. and Wold, E. H. 1985. Antialiasing through stochastic sampling. ACM SIGGRAPH Comput. Graph. 19, 3, 69–78.
    8. Fattal, R. 2011. Blue-noise point sampling using kernel density model. ACM Trans. Graph. 30, 4, 48:1–48:12.
    9. Giraud, B. G. and Peschanski, R. 2006. On positive functions with positive Fourier transforms. Acta Physica Polonica B 37, 331.
    10. Glassner, A. S. 1995. Principles of Digital Image Synthesis 1st. Ed. Vol. 1. Morgan Kaufmann, San Francisco, CA.
    11. Illian, J., Penttinen, A., Stoyan, H., and Stoyan, D. 2008. Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley & Sons.
    12. Kansal, A. R., Truskett, T. M., and Torquato, S. 2000. Nonequilibrium hard-disk packings with controlled orientational order. J. Chem. Phys. 113, 12, 4844.
    13. Lagae, A. and Dutré, P. 2008. A comparison of methods for generating Poisson disk distributions. Comput. Graph. Forum 27, 1, 114–129.
    14. Lau, D. L., Ulichney, R., and Arce, G. R. 2003. Blue- and green-noise halftoning models. IEEE Signal Process. Mag. 20, 4, 28–38.
    15. Lloyd, S. P. 1982. Least square quantization in PCM. IEEE Trans. Inf. Theory 28, 2, 129–137.
    16. Mitchell, D. P. 1990. The anti-aliasing problem in ray tracing. Advanced topics in ray tracing. In ACM SIGGRAPH Course Notes.
    17. Mitchell, D. P. 1991. Spectrally optimal sampling for distribution ray tracing. ACM SIGGRAPH Comput. Graph. 91, 25, 4, 157–164.
    18. Mitsa, T. and Parker, K. J. 1992. Digital halftoning technique using a blue-noise mask. J. Opt. Soc. Amer. A 9, 11, 1920–1929.
    19. Öztireli, A. C. and Gross, M. 2012. Analysis and synthesis of point distributions based on pair correlation. ACM Trans. Graph. 31, 6, 170:1–170:10.
    20. Pharr, M. and Humphreys, G. 2010. Physically Based Rendering: From Theory to Implementation 2nd Ed. Morgan Kaufmann Publishers.
    21. Rintoul, M. D. and Torquato, S. 1997. Reconstruction of the structure of dispersions. J. Colloid Interface Sci. 186, 467–476.
    22. Schlömer, T., Heck, D., and Deussen, O. 2011. Farthest-point optimized point sets with maximized minimum distance. In Proceedings of the ACM SIGGRAPH Symposium on High Performance Graphics. 135–154.
    23. Schmaltz, C., Gwosdek, P., Bruhn, A., and Weickert, J. 2010. Electrostatic halftoning. Comput. Graph. Forum 29, 8, 2313–2327.
    24. Torquato, S. and Stillinger, F. H. 2002. Controlling the short-range order and packing densities of many-particle systems. J. Phys. Chem. B 106, 33, 8354–8359.
    25. Uche, O. U., Stillinger, F. H., and Torquato, S. 2006. On the realizability of pair correlation functions. Physica A 360, 21, 21–36.
    26. Ulichney, R. A. 1988. Dithering with blue noise. Proc. IEEE 76, 1, 56–79.
    27. Ulichney, R. A. 1993. Digital Halftoning. MIT Press.
    28. Wei, L.-Y. and Wang, R. 2011. Differential domain analysis for nonuniform sampling. ACM Trans. Graph. 30, 4, 50:1–50:10.
    29. Yellot, Jr., J. I. 1983. Spectral consequences of photoreceptor sampling in the rhesus retina. Science 221, 382–385.
    30. Zhou, Y., Huang, H., Wei, L.-Y., and Wang, R. 2012. Point sampling with general noise spectrum. ACM Trans. Graph. 31, 4, 76:1–76:11.

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