“Recursive Wang tiles for real-time blue noise” by Kopf, Cohen-Or, Deussen and Lischinski

Well distributed point sets play an important role in a variety of computer graphics contexts, such as anti-aliasing, global illumination, halftoning, non-photorealistic rendering, point-based modeling and rendering, and geometry processing. In this paper, we introduce a novel technique for rapidly generating large point sets possessing a blue noise Fourier spectrum and high visual quality. Our technique generates non-periodic point sets, distributed over arbitrarily large areas. The local density of a point set may be prescribed by an arbitrary target density function, without any preset bound on the maximum density. Our technique is deterministic and tile-based; thus, any local portion of a potentially infinite point set may be consistently regenerated as needed. The memory footprint of the technique is constant, and the cost to generate any local portion of the point set is proportional to the integral over the target density in that area. These properties make our technique highly suitable for a variety of real-time interactive applications, some of which are demonstrated in the paper.Our technique utilizes a set of carefully constructed progressive and recursive blue noise Wang tiles. The use of Wang tiles enables the generation of infinite non-periodic tilings. The progressive point sets inside each tile are able to produce spatially varying point densities. Recursion allows our technique to adaptively subdivide tiles only where high density is required, and makes it possible to zoom into point sets by an arbitrary amount, while maintaining a constant apparent density.

1. Cohen, M. F., Shade, J., Hiller, S., and Deussen, O. 2003. Wang tiles for image and texture generation. ACM Transactions on Graphics 22, 3 (Proc. SIGGRAPH 2003), 287–294. Google ScholarDigital Library
2. Cook, R. L. 1986. Stochastic sampling in computer graphics. ACM Transactions on Graphics 5, 1, 51–72. Google ScholarDigital Library
3. Cormen, T. H., Leiserson, C. E., Rivest, R. L., and Stein, C. 2001. Introduction to Algorithms, second ed. The MIT Press, Cambridge, MA, USA. Google ScholarDigital Library
4. Deussen, O., Hiller, S., Van Overveld, C. W. A. M., and Strothotte, T. 2000. Floating points: A method for computing stipple drawings. Computer Graphics Forum 19, 3 (Proc. Eurographics 2000), 40–51.Google ScholarCross Ref
5. Dippé, M. A. Z., and Wold, E. H. 1985. Antialiasing through stochastic sampling. Computer Graphics 19, 3 (Proc. SIGGRAPH 85), 69–78. Google ScholarDigital Library
6. Dunbar, D., and Humphreys, G. 2006. A spatial data structure for fast Poisson-disk sample generation. ACM Transactions on Graphics 25, 3 (Proc. SIGGRAPH 2006). Google ScholarDigital Library
7. Durand, F., Ostromoukhov, V., Miller, M., Duranleau, F., and Dorsey, J. 2001. Decoupling strokes and high-level attributes for interactive traditional drawing. In Rendering Techniques 2001, 71–82. Google ScholarDigital Library
8. Efros, A. A., and Freeman, W. T. 2001. Image quilting for texture synthesis and transfer. In Proc. SIGGRAPH 2001, 341–346. Google ScholarDigital Library
9. Glassner, A. S. 1994. Principles of Digital Image Synthesis. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA. Google ScholarDigital Library
10. Hertzmann, A. 1998. Painterly rendering with curved brush strokes of multiple sizes. In Proc. SIGGRAPH 98, 453–460. Google ScholarDigital Library
11. Hiller, S., Deussen, O., and Keller, A. 2001. Tiled blue noise samples. In Proc. Vison Modeling and Visualization 2001, 265–272. Google ScholarDigital Library
12. Hiller, S., Hellwig, H., and Deussen, O. 2003. Beyond stippling —methods for distributing objects on the plane. Computer Graphics Forum 22, 3 (Proc. Eurographics 2003), 515–522.Google ScholarCross Ref
13. Kalnins, R. D., Markosian, L., Meier, B. J., Kowalski, M. A., Lee, J. C., Davidson, P. L., Webb, M., Hughes, J. F., and Finkelstein, A. 2002. WYSIWYG NPR: drawing strokes directly on 3D models. ACM Transactions on Graphics 21, 3 (Proc. SIGGRAPH 2002), 755–762. Google ScholarDigital Library
14. Keller, A. 2004. Myths of computer graphics. In Monte Carlo and Quasi-Monte Carlo Methods 2004, Springer-Verlag, Berlin, Germany. Google ScholarDigital Library
15. Kollig, T., and Keller, A. 2002. Efficient multidimensional sampling. Computer Graphics Forum 21, 3 (Proc. Eurographics 2002), 557–563.Google ScholarCross Ref
16. Kollig, T., and Keller, A. 2003. Efficient illumination by high dynamic range images. In Rendering Techniques 2003, 45–50. Google ScholarDigital Library
17. Lagae, A., and Dutré, P. 2005. A procedural object distribution function. ACM Transactions on Graphics 24, 4, 1442–1461. Google ScholarDigital Library
18. Lloyd, S. 1983. An optimization approach to relaxation labeling algorithms. Image and Vision Computing 1, 2, 85–91.Google ScholarCross Ref
19. McCool, M., and Fiume, E. 1992. Hierarchical poisson disk sampling distributions. In Proc. Graphics interface ’92, 94–105. Google ScholarDigital Library
20. Mitchell, D. P. 1987. Generating antialiased images at low sampling densities. Computer Graphics 21, 4 (Proc. SIGGRAPH 87), 65–72. Google ScholarDigital Library
21. Mitchell, D. P. 1991. Spectrally optimal sampling for distribution ray tracing. Computer Graphics 25, 4 (Proc. SIGGRAPH 91), 157–164. Google ScholarDigital Library
22. Ostromoukhov, V., Donohue, C., and Jodoin, P.-M. 2004. Fast hierarchical importance sampling with blue noise properties. ACM Transactions on Graphics 23, 3 (Proc. SIGGRAPH 2004), 488–495. Google ScholarDigital Library
23. Praun, E., Hoppe, H., Webb, M., and Finkelstein, A. 2001. Real-time hatching. In Proc. SIGGRAPH 2001, 579–584. Google ScholarDigital Library
24. Salisbury, M. P., Anderson, S. E., Barzel, R., and Salesin, D. H. 1994. Interactive pen-and-ink illustration. In Proc. SIGGRAPH 94, 101–108. Google ScholarDigital Library
25. Secord, A., Heidrich, W., and Streit, L. 2002. Fast primitive distribution for illustration. In Rendering Techniques 2002, 215–226. Google ScholarDigital Library
26. Secord, A. 2002. Weighted voronoi stippling. In Proc. NPAR 2002, 37–43. Google ScholarDigital Library
27. Shirley, P. 1991. Discrepancy as a quality measure for sample distributions. In Proc. Eurographics ’91, 183–193.Google Scholar
28. Sutherland, I. E. 1964. Sketch pad a man-machine graphical communication system. In Proc. SHARE design automation workshop ’64, 6.329–6.346. Google ScholarDigital Library
29. Ulichney, R. A. 1988. Dithering with blue noise. Proc. IEEE 76, 1, 56–79.Google ScholarCross Ref
30. Wang, H. 1961. Proving theorems by pattern recognition II. Bell Systems Technical Journal 40, 1–42.Google ScholarCross Ref
31. Wang, H. 1965. Games, logic and computers. Scientific American 213, 5, 98–106.Google ScholarCross Ref
32. Winkenbach, G., and Salesin, D. H. 1994. Computer-generated pen-and-ink illustration. In Proc. SIGGRAPH 94, 91–100. Google ScholarDigital Library