Fast hierarchical importance sampling with blue noise properties

This paper presents a novel method for efficiently generating a good sampling pattern given an importance density over a 2D domain. A Penrose tiling is hierarchically subdivided creating a sufficiently large number of sample points. These points are numbered using the Fibonacci number system, and these numbers are used to threshold the samples against the local value of the importance density. Pre-computed correction vectors, obtained using relaxation, are used to improve the spectral characteristics of the sampling pattern. The technique is deterministic and very fast; the sampling time grows linearly with the required number of samples. We illustrate our technique with importance-based environment mapping, but the technique is versatile enough to be used in a large variety of computer graphics applications, such as light transport calculations, digital halftoning, geometry processing, and various rendering techniques.

[1]  Thomas Ertl,et al.  Computer Graphics - Principles and Practice, 3rd Edition , 2014 .

[2]  Alexander Keller,et al.  Tiled Blue Noise Samples , 2001, VMV.

[3]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[4]  Alexander Keller,et al.  Efficient Multidimensional Sampling , 2002, Comput. Graph. Forum.

[5]  Peter Shirley,et al.  Discrepancy as a Quality Measure for Sample Distributions , 1991, Eurographics.

[6]  Alan L. Mackay,et al.  Crystallography and the penrose pattern , 1982 .

[7]  Wolfgang Heidrich,et al.  Fast Primitive Distribution for Illustration , 2002, Rendering Techniques.

[8]  Pierre Alliez,et al.  Isotropic Remeshing of Surfaces: A Local Parameterization Approach , 2003, IMR.

[9]  P J Steinhardt,et al.  The physics of quasicrystals , 1987 .

[10]  Paul E. Debevec,et al.  Rendering synthetic objects into real scenes: bridging traditional and image-based graphics with global illumination and high dynamic range photography , 1998, SIGGRAPH '08.

[11]  Don P. Mitchell,et al.  Spectrally optimal sampling for distribution ray tracing , 1991, SIGGRAPH.

[12]  R. Penrose.  Pentaplexity A Class of Non-Periodic Tilings of the Plane , 1979 .

[13]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science , 1991 .

[14]  Alexander Keller,et al.  Efficient Bidirectional Path Tracing by Randomized Quasi-Monte Carlo Integration , 2002 .

[15]  Leonidas J. Guibas,et al.  Robust Monte Carlo methods for light transport simulation , 1997 .

[16]  Eugene Fiume,et al.  Hierarchical Poisson disk sampling distributions , 1992 .

[17]  G. C. Shephard,et al.  Tilings and Patterns , 1990 .

[18]  Victor Ostromoukhov,et al.  A simple and efficient error-diffusion algorithm , 2001, SIGGRAPH.

[19]  Mark Meyer,et al.  Interactive geometry remeshing , 2002, SIGGRAPH.

[20]  Andrew S. Glassner,et al.  Principles of Digital Image Synthesis , 1995 .

[21]  J. Socolar,et al.  Simple octagonal and dodecagonal quasicrystals. , 1989, Physical review. B, Condensed matter.

[22]  Alexander Keller,et al.  Efficient Illumination by High Dynamic Range Images , 2003, Rendering Techniques.

[23]  Roger D. Hersch,et al.  Rotated dispersed dither: a new technique for digital halftoning , 1994, SIGGRAPH.

[24]  Dinesh Manocha,et al.  Fast computation of generalized Voronoi diagrams using graphics hardware , 1999, SIGGRAPH.

[25]  Steven K. Feiner,et al.  Computer graphics: principles and practice (2nd ed.) , 1990 .

[26]  Robert Ulichney,et al.  Dithering with blue noise , 1988, Proc. IEEE.

[27]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[28]  S. A. Lloyd.  An optimization approach to relaxation labelling algorithms , 1983, Image Vis. Comput..

[29]  Ronald L. Graham,et al.  Concrete Mathematics, a Foundation for Computer Science , 1991, The Mathematical Gazette.

[30]  Bingfeng Zhou,et al.  Improving mid-tone quality of variable-coefficient error diffusion using threshold modulation , 2003, ACM Trans. Graph..

[31]  Robert Ulichney,et al.  Digital Halftoning , 1987 .

[32]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[33]  Oliver Deussen,et al.  Wang Tiles for image and texture generation , 2003, ACM Trans. Graph..

[34]  Serge J. Belongie,et al.  Structured importance sampling of environment maps , 2003, ACM Trans. Graph..

[35]  Andrew S. Glassner.  Andrew Glassner's notebook , 2004, IEEE Computer Graphics and Applications.

[36]  Robert L. Cook,et al.  Stochastic sampling in computer graphics , 1988, TOGS.