blue noise sampling via delaunay triangulation
DESCRIPTION
Zoltan Szego †* , Yoshihiro Kanamori ‡ , Tomoyuki Nishita † † The University of Tokyo, *Google Japan Inc . , ‡ University of Tsukuba. Blue Noise Sampling via Delaunay Triangulation. Contents. Background Related Work Our Method Results Conclusions and Future Work. Contents. - PowerPoint PPT PresentationTRANSCRIPT
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Blue Noise Sampling via Delaunay Triangulation
Zoltan Szego†*, Yoshihiro Kanamori‡, Tomoyuki Nishita††The University of Tokyo, *Google Japan Inc., ‡University of Tsukuba
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Contents
Background Related Work Our Method Results Conclusions and Future Work
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Contents
Background Related Work Our Method Results Conclusions and Future Work
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Background
Sampling is essential in CG rendering, image processing, object
placement etc.
HalftoningLight sampling on HDR environment maps
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Background
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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Background
Blue noise property Observed in natural objects Considered optimal for human eyes
Layout of human eye photoreceptors [Yellott, 1983]
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Background
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 averagepower spectrum
Anisotropy
ring
ring
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Our Goal
Efficient, high-quality blue noise sampling Adaptive sampling should be supported
Uniform
Adaptive
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Our Goal
Support for sampling in various domains 2D 3D (volumetric sampling) On curved surfaces (spheres, polygonal
meshes)
2D 3D On curved surfaces
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Contents
BackgroundRelated Work Our Method Results Conclusions and Future Work
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Related Work
Two major approaches Dart throwing▪ Random sampling of equidistant samples
Tiling▪ Tiling of precomputed samples
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Related Work
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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Related Work
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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Contents
Background Related WorkOur Method Results Conclusions and Future Work
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Overview
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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Overview
Sequentially sample atthe 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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Basic Algorithm
Loop:1. Find the largest
empty circle2. Add a sample
at the center
2D example
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Basic Algorithm
Loop:1. Find the largest
empty circle2. Add a sample
at the center3. Update
Delaunay triangles
2D example
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Basic Algorithm
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 heapusing density functions
The greater the density, the higher the priority
Heap of circumcircles’ radii
Density function
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Artifact #1
Regular patterns peaks in the spectrum
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Modification #1
Reason of the artifacts Iterative subdivisions of equilateral
triangles
Our solution:1. Detect an equilateral triangle2. Displace the new sample
from the center of its circumcircle(see our paper for details)
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Artifact #2
Sparse samplesat boundaries
Reason Very thin triangles
around boundaries
Our solution: Use of periodic boundaries
Tiled samples(tiled just for illustration)
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Modification #2
Periodic boundaries Toroidal (torus-like) domain
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Modification #2
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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Parallelization
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
1 2
3 4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
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Sampling in 3D
3D domain: [0, 1)3
2D → 3D Triangles → Tetrahedra
(Delaunay Tetrahedralization)
Circumcircles→ Circumspheres
Similar to 2D algorithm
Delaunay tetrahedralization
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Sampling on Curved Surfaces 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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Contents
Background Related Work Our MethodResults Conclusions and Future Work
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Results
Uniform sampling
# of samples : 20KTime : 92 ms
Experimental environment:Intel Core 2 Quad Q6700 2.66GHz, 2GB RAM
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Comparison – 50,000 samples –
Our method: 378 msec Wang tiles [2006]: 1.35 msec
Radial average Radial averageAnisotropy Anisotropy
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Comparison – 50,000 samples –
Radial average Radial averageAnisotropy Anisotropy
Our method: 378 msec Dart throwing [2007]: 420 msec
ours
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Results
20K samples in 3D
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Results
Spectra for 10K samples in 3D
Low energy spheres in the center → blue noise property
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Results
Sampling on a sphere Initial mesh: an equilateral
octahedron
Density functionDense Sparse
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Results
Sampling on HDR environment maps Blighter region → denser samples
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Contents
Background Related Work Our Method ResultsConclusions and Future Work
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Conclusions
High-quality blue noise samplingusing 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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Future Work
GPU acceleration using CUDA
Fast Lloyd’s relaxation using the connectivity of Delaunay triangles
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Thank you
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