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Parallel  Blue-noise Sampling by Parallel  Blue-noise Sampling by

Parallel Blue-noise Sampling by - PowerPoint Presentation

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Parallel Blue-noise Sampling by - PPT Presentation

Constrained Farthest Point Optimization Renjie Chen Craig Gotsman Technion Israel Institute of Technology SGP12 Tallinn Bluenoise distribution AKA Poisson disk distribution ID: 686068

noise blue constrained sampling blue noise sampling constrained fpo parallel 2012 point farthest local tiles cfpo property fpd disk

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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

2012/8/16

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

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide5

Blue-noise Spectra

5

stably flat

structural residual peaks

lacking low-frequency

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide6

Applications6

Sampling & meshing

Rendering

NPR Stippling

HDR Imaging

2012/8/16

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)………………

2012/8/16

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

2012/8/16

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

2012/8/16

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

2012/8/16

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

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide12

FPD - Farthest Point Distribution

12

Blue-noise Spectra

2012/8/16

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

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide14

FPO - Farthest Point OptimizationFarthest Point

14

X

– the point setLargest empty circle

DT

(

X

)

Priority queue

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide15

Local FPO

15

Pro

Linear complexity?

Con

Requires maintaining a global DT

Slow convergence

Difficult to parallelize

2012/8/16

Parallel Blue-noise Sampling by Constrained FPO

?Slide16

Constrained Farthest PointLocal FP and global FP can be different

16

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide17

Local characterization of FPD

Local farthest point property

Local covering property

17 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2012/8/16

Parallel Blue-noise Sampling by Constrained FPO

 Slide18

Local characterization of FPDLocal farthest point property

Local covering property 18

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide19

 

Local characterization of FPD

Local

farthest

point property

Local

covering property

19

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2012/8/16

Parallel Blue-noise Sampling by Constrained FPO

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 Slide20

CFPO – constrained farthest point optimizationLocal farthest

point20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x'

 

 

2012/8/16

Parallel Blue-noise Sampling by Constrained FPO

x

 

 

 

 

 

 

 

 

 Slide21

CFPO – constrained farthest point optimizationLocal

covering21Uniform

random initialization

1 iteration

2 iterations

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide22

Parallel CFPO

22

1 iteration

20 iterations

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide23

Local CFPO

23

Largest empty circle

 

 

 

 

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide24

Convergence

24

2012/8/16

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

 

2012/8/16

Parallel Blue-noise Sampling by Constrained FPOSlide26

Results - Performance

26

2012/8/16

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

2012/8/16Parallel Blue-noise Sampling by Constrained FPOSlide29

2012/8/16

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