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

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Background Related Work Our Method Results Conclusions and Future Work

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Background Related Work Our Method Results Conclusions and Future Work

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

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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

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

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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

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

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

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Background Related Work Our Method Results Conclusions and Future Work

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

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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

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

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Background Related Work Our Method Results Conclusions and Future Work

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

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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

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

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

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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

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Regular patterns peaks in the spectrum

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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)

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Artifact-free 100,000 samples

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

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Periodic boundaries Toroidal (torus-like) domain

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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

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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 12 34 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

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

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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

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Background Related Work Our Method Results Conclusions and Future Work

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Uniform sampling # of samples ： 20K Time ： 92 ms Experimental environment: Intel Core 2 Quad Q6700 2.66GHz, 2GB RAM

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

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

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20K samples in 3D

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

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

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Sampling on HDR environment maps Blighter region → denser samples

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Sampling on HDR environment maps Blighter region → denser samples

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Background Related Work Our Method Results Conclusions and Future Work

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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

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

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Thank you

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

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A polygonal mesh (two tori)

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A simplified mesh

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Delaunay triangles

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Generated samples (vertices)

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