Constrained Farthest Point Optimization Renjie Chen Craig Gotsman Technion Israel Institute of Technology SGP12 Tallinn Bluenoise distribution AKA Poisson disk distribution ID: 686068
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Slide1
Parallel Blue-noise Sampling by Constrained Farthest Point Optimization
Renjie Chen Craig GotsmanTechnion – Israel Institute of Technology
SGP’12 @ TallinnSlide2
Blue-noise distributionAKA Poisson disk distribution
UniformUniform point densityLarge minimal mutual distanceIrregularity
No correlations between points
2
Random
Blue-noise
Hexagonal grid
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Parallel Blue-noise Sampling by Constrained FPOSlide3
OutlineIntroduction & related workFPD & FPOLocal characterization of FPD
Constrained FPOExperimental results & conclusion32012/8/16
Parallel Blue-noise Sampling by Constrained FPOSlide4
Blue-noise distributionPower spectrum analysis
Ulichney, 1987: study of the frequency domain characteristics Periodograms
of Fourier transform on point distributions
4Power spectrum
Radially averaged power spectru
m
Anisotropy
Blue-noise
spectra
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Parallel Blue-noise Sampling by Constrained FPOSlide5
Blue-noise Spectra
5
stably flat
structural residual peaks
lacking low-frequency
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Parallel Blue-noise Sampling by Constrained FPOSlide6
Applications6
Sampling & meshing
Rendering
NPR Stippling
HDR Imaging
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Parallel Blue-noise Sampling by Constrained FPOSlide7
Related Work
7Dart-throwing
based
Relaxation basedPrecomputed tiles
Poisson disk distribution
(Crow,
1977
)
Lloyd’s relaxation
(McCool et al, 1992)
Poisson disk tiles
(Shade
et al
, 2000)
Dart-throwing
(Cook, 1986)
ODT
(Chen
et al
, 2004)
Edge-based tiles
(
Lagae
and
Dutre
, 2005a)
Boundary Sampling
(Dunbar and Humphreys 2006)
CCVT
(
Balzer
et al
, 2009)
Template tiles
(
Lagae
and
Dutre
, 2005b)
Parallel Poisson disk sampling
(Wei
et al
, 2008)
FPO
(
Schlömer
et al
, 2011)
Corner-based tiles
(
Lagae
and
Dutre
, 2006)
Maximal Poisson-disk sampling
(
Ebeida
et al
, 2011, 2012)
CCDT
(
Xu
et al
,
2011
)Recursive Wang Tiles(Kopf et al, 2006)………………
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Parallel Blue-noise Sampling by Constrained FPOSlide8
Dart-throwingIncrementally generate
samples randomlyHard to control sampling
sizeComputationally expensive
8
All points are separated from each other by a minimum distance
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Parallel Blue-noise Sampling by Constrained FPOSlide9
Relaxation Approach
Starting from an initial distributionMove points following some criteria until convergeHard to possess blue noise characteristic
9
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Parallel Blue-noise Sampling by Constrained FPOSlide10
Precomputed tiles
Construct a few tiles of Blue-noise patternSeamlessly tile the tiles Inferior Blue-noise quality
Lacking variety
10
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Parallel Blue-noise Sampling by Constrained FPOSlide11
FPD - Farthest Point Distribution
Farthest point (of a point set X)The point with
maximal distance(to X)
FPD
Each point is a
farthest
point
11
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Parallel Blue-noise Sampling by Constrained FPOSlide12
FPD - Farthest Point Distribution
12
Blue-noise Spectra
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Parallel Blue-noise Sampling by Constrained FPOSlide13
FPO - Farthest Point Optimization
Main algorithmWhile X is not convergedforeach xX
move x to fp(
X\{x}) 13
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Parallel Blue-noise Sampling by Constrained FPOSlide14
FPO - Farthest Point OptimizationFarthest Point
14
X
– the point setLargest empty circle
DT
(
X
)
Priority queue
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Parallel Blue-noise Sampling by Constrained FPOSlide15
Local FPO
15
Pro
Linear complexity?
Con
Requires maintaining a global DT
Slow convergence
Difficult to parallelize
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Parallel Blue-noise Sampling by Constrained FPO
?Slide16
Constrained Farthest PointLocal FP and global FP can be different
16
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Parallel Blue-noise Sampling by Constrained FPOSlide17
Local characterization of FPD
Local farthest point property
Local covering property
17
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Parallel Blue-noise Sampling by Constrained FPO
Slide18
Local characterization of FPDLocal farthest point property
Local covering property 18
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Parallel Blue-noise Sampling by Constrained FPOSlide19
Local characterization of FPD
Local
farthest
point property
Local
covering property
19
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Parallel Blue-noise Sampling by Constrained FPO
Slide20
CFPO – constrained farthest point optimizationLocal farthest
point20
x'
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Parallel Blue-noise Sampling by Constrained FPO
x
Slide21
CFPO – constrained farthest point optimizationLocal
covering21Uniform
random initialization
1 iteration
2 iterations
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Parallel Blue-noise Sampling by Constrained FPOSlide22
Parallel CFPO
22
1 iteration
20 iterations
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Parallel Blue-noise Sampling by Constrained FPOSlide23
Local CFPO
23
Largest empty circle
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Parallel Blue-noise Sampling by Constrained FPOSlide24
Convergence
24
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Parallel Blue-noise Sampling by Constrained FPOSlide25
Results – Blue-noise spectra
25
FPO
CFPO
simplified
CFPO
δ(
X
)
=
0.925
,
=
0.934
δ(
X
)
=
0.925
,
=
0.931
δ(
X
)
=
0.924
,
=
0.930
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Parallel Blue-noise Sampling by Constrained FPOSlide26
Results - Performance
26
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Parallel Blue-noise Sampling by Constrained FPO80x
8xSlide27
ConclusionCFPOLocal
optimizationGlobal propertyEquivalency with FPOEasy parallelizationFuture WorkNon-uniform/anisotropic distributionHigher dimension
Non-Euclidean manifold
272012/8/16Parallel Blue-noise Sampling by Constrained FPOSlide28
Thanks for your attention!
28
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Parallel Blue-noise Sampling by Constrained FPO29
δ(
X
)
=
0.655
,
=
0.844
CCDT
CCVT
BS
δ(
X
)
=
0.710
,
=
0.854
δ(
X
)
=
0.799
,
=
0.863
δ(
X
)
=
0.666
,
=
0.818
DT
CFPO
δ(
X
)
=
0.925
,
=
0.931