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Issue No.10 - Oct. (2012 vol.18)

pp: 1784-1796

Zhonggui Chen , Xiamen University, Xiamen

Zhan Yuan , The University of Hong Kong, Hong Kong

Yi-King Choi , The University of Hong Kong, Hong Kong

Ligang Liu , University of Science and Technology of China, Hefei

Wenping Wang , The University of Hong Kong, Hong Kong

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TVCG.2012.94

ABSTRACT

Blue noise point sampling is one of the core algorithms in computer graphics. In this paper, we present a new and versatile variational framework for generating point distributions with high-quality blue noise characteristics while precisely adapting to given density functions. Different from previous approaches based on discrete settings of capacity-constrained Voronoi tessellation, we cast the blue noise sampling generation as a variational problem with continuous settings. Based on an accurate evaluation of the gradient of an energy function, an efficient optimization is developed which delivers significantly faster performance than the previous optimization-based methods. Our framework can easily be extended to generating blue noise point samples on manifold surfaces and for multi-class sampling. The optimization formulation also allows us to naturally deal with dynamic domains, such as deformable surfaces, and to yield blue noise samplings with temporal coherence. We present experimental results to validate the efficacy of our variational framework. Finally, we show a variety of applications of the proposed methods, including nonphotorealistic image stippling, color stippling, and blue noise sampling on deformable surfaces.

INDEX TERMS

Noise measurement, Density functional theory, Optimization, Computer graphics, quasi-Newton method., Point sampling, blue noise, centroidal Voronoi tessellation, capacity-constrained

CITATION

Zhonggui Chen, Zhan Yuan, Yi-King Choi, Ligang Liu, Wenping Wang, "Variational Blue Noise Sampling",

*IEEE Transactions on Visualization & Computer Graphics*, vol.18, no. 10, pp. 1784-1796, Oct. 2012, doi:10.1109/TVCG.2012.94REFERENCES

- [1] S.P. Lloyd, "Least Squares Quantization in PCM,"
IEEE Trans. Information Theory, vol. TIT-28, no. 2, pp. 129-137, Mar. 1982.- [2] M. McCool and F. Eugene, "Hierarchical Poisson Disk Sampling Distributions,"
Proc. Conf. Graphics Interface, pp. 94-105, 1992.- [3] A. Gersho, "Asymptotically Optimal Block Quantization,"
IEEE Trans. Information Theory, vol. TIT-25, no. 4, pp. 373-380, July 1979.- [4] Y. Liu, W. Wang, B. Lévy, F. Sun, D.-M. Yan, L. Lu, and C. Yang, "On Centroidal Voronoi Tessellation—Energy Smoothness and Fast Computation,"
ACM Trans. Graphics, vol. 28, no. 4, pp. 1-17, 2009.- [5] M. Balzer, T. Schlömer, and O. Deussen, "Capacity-Constrained Point Distributions: A Variant of Lloyd's Method,"
Proc. ACM SIGGRAPH Papers, pp. 1-8, 2009.- [6] D.C. Liu and J. Nocedal, "On the Limited Memory BFGS Method for Large Scale Optimization,"
Math. Programming: Series A and B, vol. 45, no. 3, pp. 503-528, 1989.- [7] A. Lagae and P. Dutré, "A Comparison of Methods for Generating Poisson Disk Distributions,"
Computer Graphics Forum, vol. 27, no. 1, pp. 114-129, 2008.- [8] R.L. Cook, "Stochastic Sampling in Computer Graphics,"
ACM Trans. Graphics, vol. 5, no. 1, pp. 51-72, 1986.- [9] T.R. Jones, "Efficient Generation of Poisson-Disk Sampling Patterns,"
J. Graphics Tools, vol. 11, no. 2, pp. 27-36, 2006.- [10] D. Dunbar and G. Humphreys, "A Spatial Data Structure for Fast Poisson-Disk Sample Generation,"
ACM Trans. Graphics, vol. 25, no. 3, pp. 503-508, 2006.- [11] K.B. White, D. Cline, and P.K. Egbert, "Poisson Disk Point Sets by Hierarchical Dart throwing,"
Proc. IEEE Symp. Interactive Ray Tracing, pp. 129-132, 2007.- [12] L.-Y. Wei, "Parallel Poisson Disk Sampling,"
ACM Trans. Graphics, vol. 27, no. 3, pp. 20:1-20:9, 2008.- [13] M.N. Gamito and S.C. Maddock, "Accurate Multidimensional Poisson-Disk Sampling,"
ACM Trans. Graphics, vol. 29, pp. 8:1-8:19, Dec. 2009.- [14] M.F. Cohen, J. Shade, S. Hiller, and O. Deussen, "Wang Tiles for Image and Texture Generation,"
Proc. ACM SIGGRAPH Papers, pp. 287-294, 2003.- [15] A. Lagae and P. Dutré, "An Alternative for Wang Tiles: Colored Edges versus Colored Corners,"
ACM Trans. Graphics, vol. 25, no. 4, pp. 1442-1459, 2006.- [16] J. Kopf, D. Cohen-Or, O. Deussen, and D. Lischinski, "Recursive Wang Tiles for Real-Time Blue Noise,"
ACM Trans. Graphics, vol. 25, pp. 509-518, 2006.- [17] V. Ostromoukhov, C. Donohue, and P.-M. Jodoin, "Fast Hierarchical Importance Sampling with Blue Noise Properties,"
Proc. ACM SIGGRAPH Papers, pp. 488-495, 2004.- [18] V. Ostromoukhov, "Sampling with Polyominoes,"
ACM Trans. Graphics, vol. 26, pp. 78:1-78:6, July 2007.- [19] H. Li, D. Nehaby, L.-Y. Wei, P.V. Sander, and C.-W. Fu, "Fast Capacity Constrained Voronoi Tessellation,"
Proc. Symp. Interactive 3D Graphics and Games, pp. 13:1-13:7, 2010.- [20] C. Schmaltz, P. Gwosdek, A. Bruhn, and J. Weickert, "Electrostatic Halftoning,"
Computer Graphics Forum, vol. 29, no. 8, pp. 2313-2327, 2010.- [21] R. Fattal, "Blue-Noise Point Sampling Using Kernel Density Model,"
ACM Trans. Graphics, vol. 30, pp. 48:1-48:12, Aug. 2011.- [22] T. Schlömer, D. Heck, and O. Deussen, "Farthest-Point Optimized Point Sets with Maximized Minimum Distance,"
Proc. ACM SIGGRAPH Symp. High Performance Graphics, pp. 135-142, 2011.- [23] V. Surazhsky, P. Alliez, and C. Gotsman, "Isotropic Remeshing of Surfaces: A Local Parameterization Approach,"
Proc. 12th Int'l Meshing Roundtable, 2003.- [24] G. Rong, Y. Liu, W. Wang, X. Yin, X.D. Gu, and X. Guo, "GPU-Assisted Computation of Centroidal Voronoi Tessellation,"
IEEE Trans. Visualization and Computer Graphics, vol. 17, no. 3, pp. 345-356, Mar. 2011.- [25] H. Li, K.-Y. Lo, M.-K. Leung, and C.-W. Fu, "Dual Poisson-Disk Tiling: An Efficient Method for Distributing Features on Arbitrary Surfaces,"
IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 5, pp. 982-998, Sept./Oct. 2008.- [26] Y. Fu and B. Zhou, "Direct Sampling on Surfaces for High Quality Remeshing,"
Computer Aided Geometric Design, vol. 26, no. 6, pp. 711-723, 2009.- [27] D. Cline, S. Jeschke, K. White, A. Razdan, and P. Wonka, "Dart Throwing on Surfaces,"
Computer Graphics Forum, vol. 24, no. 4, pp. 1217-1226, 2009.- [28] J. Bowers, R. Wang, L.-Y. Wei, and D. Maletz, "Parallel Poisson Disk Sampling with Spectrum Analysis on Surfaces,"
ACM Trans. Graphics, vol. 29, no. 6, pp. 166:1-166:12, 2010.- [29] A. Öztireli, M. Alexa, and M. Gross, "Spectral Sampling of Manifolds,"
ACM Trans. Graphics, vol. 29, no. 6, pp. 168:1-168:8, 2010.- [30] L.-Y. Wei, "Multi-Class Blue Noise Sampling,"
ACM Trans. Graphics, vol. 29, no. 4, pp. 79:1-79:8, 2010.- [31] C. Schmaltz, P. Gwosdek, and J. Weickert, "Multi-Class Electrostatic Halftoning," technical report, Dept. of Math., Saarland Univ., Oct. 2011.
- [32] Q. Du, V. Faber, and M. Gunzburger, "Centroidal Voronoi Tessellations: Applications and Algorithms,"
SIAM Rev., vol. 41, pp. 637-676, 1999.- [33] M. Balzer, "Capacity-Constrained Voronoi Diagrams in Continuous Spaces,"
Proc. Sixth Int'l Symp. Voronoi Diagrams in Science and Eng., pp. 79-88, 2009.- [34] J.A. Nelder and R. Mead, "A Simplex Method for Function Minimization,"
The Computer J., vol. 7, no. 4, pp. 308-313, Jan. 1965.- [35] J. Nocedal and S.J. Wright,
Numerical Optimization, second ed. Springer, 2006.- [36] T. Schlömer and O. Deussen, "Towards a Standardized Spectral Analysis of Point Sets with Applications in Graphics," technical report, Univ. of Konstanz, 2010.
- [37] "Cgal, Computational Geometry Algorithms Library," http:/www.cgal.org, 2012.
- [38] G. Peyré and L.D. Cohen, "Geodesic Remeshing Using Front Propagation,"
Int'l J. Computer Vision, vol. 69, pp. 145-156, Aug. 2006.- [39] H. Edelsbrunner and N.R. Shah, "Triangulating Topological Spaces,"
Int'l J. Computational Geometry and Applications, vol. 7, no. 4, pp. 365-378, 1997.- [40] D.-M. Yan, B. Lévy, Y. Liu, F. Sun, and W. Wang, "Isotropic Remeshing with Fast and Exact Computation of Restricted Voronoi Diagram,"
Proc. Symp. Geometry Processing, pp. 1445-1454. 2009,- [41] Q. Du, M.D. Gunzburger, and L. Ju, "Constrained Centroidal Voronoi Tessellations for Surfaces,"
SIAM J. Scientific Computing, vol. 24, pp. 1488-1506, May 2002.- [42] G. Turk, "Re-Tiling Polygonal Surfaces,"
Proc. 19th Ann. Conf. Computer Graphics and Interactive Techniques, pp. 55-64, 1992.- [43] H. Flanders, "Differentiation under the Integral Sign,"
The Am. Math. Monthly, vol. 80, no. 6, pp. 615-627, 1973.- [44] H. Li, L.-Y. Wei, P.V. Sander, and C.-W. Fu, "Anisotropic Blue Noise Sampling,"
ACM Trans. Graphics, vol. 29, no. 6, pp. 167:1-167:10, 2010. |