Zoltan Szego †*, Yoshihiro Kanamori ‡, Tomoyuki Nishita † † The University of Tokyo, *Google Japan Inc., ‡ University of Tsukuba

 Background  Related Work  Our Method  Results  Conclusions and Future Work

 Background  Related Work  Our Method  Results  Conclusions and Future Work

 Sampling is essential in CG  rendering, image processing, object placement etc. HalftoningLight sampling on HDR environment maps

 Desired sampling patterns  Equally distant samples … e.g. Poisson disk  Low energy in low frequency of the Fourier spectrum … Blue noise cf. Totally randomEqually distant → Blue noise → White noise

 Blue noise property  Observed in natural objects  Considered optimal for human eyes Layout of human eye photoreceptors [Yellott, 1983]

 Quality measures for blue noise spectra  Radial average power spectrum ▪ The larger the central ring, the better  Anisotropy ▪ The lower and flatter, the better Spectrum Radial average power spectrum Anisotropy ring

 Efficient, high-quality blue noise sampling  Adaptive sampling should be supported UniformAdaptive

 Support for sampling in various domains  2D  3D (volumetric sampling)  On curved surfaces (spheres, polygonal meshes) 2D3DOn curved surfaces

 Background  Related Work  Our Method  Results  Conclusions and Future Work

 Two major approaches  Dart throwing ▪ Random sampling of equidistant samples  Tiling ▪ Tiling of precomputed samples

 Dart throwing [Cook, 1986]  Used for distributed ray tracing  High computational cost  Quality improvement: Lloyd’s relaxation … more costly  Parallel Poisson disk [Wei, 2008]  GPU-based acceleration  # of samples cannot be determined  Only supports 2D and 3D Our method # of samples can be specified Supports 2D, 3D, and curved surfaces

 Wang tiles [Kopf et al., 2006]  Requires precomputation  Low quality  Polyominoes [Ostromoukhov, 2007]  Requires complicated precomputation Our method High quality No precomputation

 Background  Related Work  Our Method  Results  Conclusions and Future Work

 Input: seed points  Given by the user  Output: blue noise samples  Features:  Deterministic (reproducible with the same seeds)  No precomputation  Supports various sampling domains

 Sequentially sample at the most sparse region  The largest empty circle problem [Okabe et al., 2000]  Can be solved using Delaunay triangulation ▪ Correspond to finding the largest circumcircle in Delaunay triangles 2D example

 Loop: 1. Find the largest empty circle 2. Add a sample at the center 2D example

 Loop: 1. Find the largest empty circle 2. Add a sample at the center 3. Update Delaunay triangles 2D example

 Acceleration for search: Use of heap  To find the largest circumcircle in O(1)  Costs for insert / delete: O(log N)  Support for adaptive sampling  Scale the radii stored in the heap using density functions  The greater the density, the higher the priority Heap of circumcircles’ radii Density function

 Regular patterns peaks in the spectrum

 Reason of the artifacts  Iterative subdivisions of equilateral triangles  Our solution: 1. Detect an equilateral triangle 2. Displace the new sample from the center of its circumcircle (see our paper for details)

 Artifact-free 100,000 samples

 Sparse samples at boundaries  Reason  Very thin triangles around boundaries  Our solution:  Use of periodic boundaries Tiled samples (tiled just for illustration)

 Periodic boundaries  Toroidal (torus-like) domain

 Pros:  Sparse regions disappear  Edge lengths of triangles become balanced ▪ Overall centers of circumcircles lie within their triangles ▪ Allows us to specify the position of the new sample in O(1)  Cons:  A little additional cost for modifying coordinates

 Exploit multi-core CPUs  Uniform subdivision of 2D domain  Further subdivision  Costs: O(N log N) 4 M log M < N log N (if M = N/4)  4x4 subdivision is the fastest for a 4-core CPU ▪ 1.69 times faster for 100K samples

 3D domain: [0, 1) 3  2D → 3D  Triangles → Tetrahedra (Delaunay Tetrahedralization)  Circumcircles → Circumspheres  Similar to 2D algorithm Delaunay tetrahedralization

 Sampling domain:  Spherical surfaces  Polygonal mesh surfaces  Initial seeds:  Vertices of simplified mesh  Similar to 2D  New samples are projected onto the surface Samples on a sphere Simplified Given mesh Initial seeds

 Background  Related Work  Our Method  Results  Conclusions and Future Work

 Uniform sampling # of samples ： 20K Time ： 92 ms Experimental environment: Intel Core 2 Quad Q GHz, 2GB RAM

Our method: 378 msecWang tiles [2006]: 1.35 msec Radial average Anisotropy

Radial average Anisotropy Our method: 378 msecDart throwing [2007]: 420 msec ours

 20K samples in 3D

 Spectra for 10K samples in 3D Low energy spheres in the center → blue noise property

 Sampling on a sphere  Initial mesh: an equilateral octahedron Density function DenseSparse

 Sampling on HDR environment maps  Blighter region → denser samples

 Sampling on HDR environment maps  Blighter region → denser samples

 Background  Related Work  Our Method  Results  Conclusions and Future Work

 High-quality blue noise sampling using Delaunay triangulation  Find centers of largest circumcircles of Delaunay triangles  Adaptive sampling by scaling circumcircles’ radii  Support for sampling on various domains: 2D, 3D, and curved surfaces

 GPU acceleration using CUDA  Fast Lloyd’s relaxation using the connectivity of Delaunay triangles

Thank you

 Adaptive sampling for halftoning Density function = grayscale image Halftone image with 100K samples

A polygonal mesh (two tori)

A simplified mesh

Delaunay triangles

Generated samples (vertices)