Zoltan Szego †*, Yoshihiro Kanamori ‡, Tomoyuki Nishita † † The University of Tokyo, *Google...
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Transcript of Zoltan Szego †*, Yoshihiro Kanamori ‡, Tomoyuki Nishita † † The University of Tokyo, *Google...
Blue Noise Sampling via Delaunay Triangulation
Zoltan Szego†*, Yoshihiro Kanamori‡, Tomoyuki Nishita†
†The University of Tokyo, *Google Japan Inc., ‡University of Tsukuba
Contents
Background Related Work Our Method Results Conclusions and Future Work
Contents
Background Related Work Our Method Results Conclusions and Future Work
Background
Sampling is essential in CG rendering, image processing, object
placement etc.
HalftoningLight sampling on HDR environment maps
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
Background
Blue noise property Observed in natural objects Considered optimal for human eyes
Layout of human eye photoreceptors [Yellott, 1983]
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
Our Goal
Efficient, high-quality blue noise sampling Adaptive sampling should be supported
Uniform
Adaptive
Our Goal
Support for sampling in various domains 2D 3D (volumetric sampling) On curved surfaces (spheres, polygonal
meshes)
2D 3D On curved surfaces
Contents
BackgroundRelated Work Our Method Results Conclusions and Future Work
Related Work
Two major approaches Dart throwing▪ Random sampling of equidistant samples
Tiling▪ Tiling of precomputed samples
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
Related Work
Wang tiles [Kopf et al., 2006] Requires precomputation Low quality
Polyominoes [Ostromoukhov, 2007] Requires complicated precomputation
Our method• High quality• No precomputation
Contents
Background Related WorkOur Method Results Conclusions and Future Work
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
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
Basic Algorithm
Loop:1. Find the largest
empty circle2. Add a sample
at the center
2D example
Basic Algorithm
Loop:1. Find the largest
empty circle2. Add a sample
at the center3. Update
Delaunay triangles
2D example
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
Artifact #1
Regular patterns peaks in the spectrum
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)
Artifact #2
Sparse samplesat boundaries
Reason Very thin triangles
around boundaries
Our solution: Use of periodic boundaries
Tiled samples(tiled just for illustration)
Modification #2
Periodic boundaries Toroidal (torus-like) domain
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
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
3
4
Sampling in 3D
3D domain: [0, 1)3
2D → 3D Triangles → Tetrahedra
(Delaunay Tetrahedralization)
Circumcircles→ Circumspheres
Similar to 2D algorithm
Delaunay tetrahedralization
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
Contents
Background Related Work Our MethodResults Conclusions and Future Work
Results
Uniform sampling
# of samples : 20KTime : 92 ms
Experimental environment:Intel Core 2 Quad Q6700 2.66GHz, 2GB RAM
Comparison – 50,000 samples –
Our method: 378 msec Wang tiles [2006]: 1.35 msec
Radial average Radial averageAnisotropy Anisotropy
Comparison – 50,000 samples –
Radial average Radial averageAnisotropy Anisotropy
Our method: 378 msec Dart throwing [2007]: 420 msec
ours
Results
20K samples in 3D
Results
Spectra for 10K samples in 3D
Low energy spheres in the center → blue noise property
Results
Sampling on a sphere Initial mesh: an equilateral
octahedron
Density functionDense Sparse
Results
Sampling on HDR environment maps Blighter region → denser samples
Contents
Background Related Work Our Method ResultsConclusions and Future Work
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
Future Work
GPU acceleration using CUDA
Fast Lloyd’s relaxation using the connectivity of Delaunay triangles
Thank you