Parallel Poisson Disk Sampling
Li-Yi Wei
Microsoft
Parallelism
Processors are becoming parallel
Intel Larrabee, NVIDIA, AMD/ATI, IBM/Sony Cell, etc.
So are programming interfaces
BSGP, CUDA, CAL, Ct, DX, OpenGL, etc.
As well as applications
To take advantage of parallel environment
Parallelization
Traditional parallelization methods
Sequential consistency [Lampert]
- sorting, FFT, matrix, etc.
Not all algorithms need to be seq-consistent
Graphics, computer vision, image/video, statistics
Approximate solutions might suffice
Opportunities for new parallelization methods
First pick: Poisson disk sampling
A set of samples that are
as random as possible
remain a minimum distance
r away from each other
Why pick this problem?
important algorithm
seemly non-parallelizable
Importance of Poisson disk sampling
Best quality for N samples [Cook 1986]
Natural object distribution (retina cells, ecology)
Blue noise spectrum
void in low freq
noise in high freq
Applications in
Rendering, imaging, geometry processing, etc.
Optimal spectrum (given # samples)
All with 1600 samples
spectrum
samples
Blue noise: aliasing → noise
regular grid
jittered grid
Poisson disk
Spatial sampling
sin( x 2  y 2 )
(zone plate)
aliasing
noisy
regular grid
jittered grid
Poisson disk
Methods
Dart throwing [Cook 1986]
Loop:
Random sample from the
entire domain
Accept sample if not in
conflict with existing ones
O High quality
Ground truth
X Slow speed
Inherently sequential
Speed improvement
Computation on the fly (sequential)
Scalloped regions [Dunbar & Humphreys 2006]
Onion layers [Bridson 2007]
Hierarchical dart throwing [White et al. 2007]
Pre-computed data set (parallel access)
Penrose tiling [Ostromoukhov et al 2004]
Wang tiles [Cohen et al. 2003; Lagae & Dutre 2005; Kopf et al. 2006]
Polyominoes [Ostromoukhov 2007]
X Potential large data set + quality issue
Features of our approach
Parallel computation
Entirely on the fly
(no pre-computed data)
Good spectrum quality
Like dart throwing
+ Adaptive sampling
+ Any dimension
Parallel GPU run time
(in slow motion)
Multi-resolution synthesis
Our basic idea
Samples from a grid
1 sample per grid cell
Sample grid cells far
apart in parallel
Watch out for bias!
Tricks to avoid bias
Algorithm in gradual steps
Uniform sampling, sequential
Uniform sampling, parallel
Adaptive sampling
Sequential sampling
Basic data structure
Choose grid cell size dd so
that each cell has at most
one sample
r
d
r = minimum spacing
n = dimension
Inspired by
[Bridson 2007]
Texture synthesis
2
r
Sequential sampling
scan-line order + single resolution
Bias!
Scanline order
Grid sampling
Sequential sampling
random order + single resolution
Removes scanline bias
But still grid-cell biased
scanline
random
Sequential sampling
random order + multi-resolution
Removes both biases
random scanline
scanline, grid
1 level
3 level
5 level
Sequential sampling
Summary for bias removal
Two sources of bias
Grid sampling
fixed by multi-resolution
random scanline
Traversal order
fixed by random order
1 level
3 level
5 level
Parallel sampling
Key insight
Sample cells sufficiently
far away in parallel
2D example:
r
Cells 2d apart cannot
conflict with each other
split cells → phase groups
d
r
2
Phase group partition
grid partition
random order
random partition
grid partition
scanline order
6
7
8
6
7
8
3
2
8
4
2
7
1
3
2
1
3
2
3
4
5
3
4
5
4
6
1
3
6
1
8
4
6
8
4
6
0
1
2
0
1
2
9
5
0
9
5
0
5
0
7
5
0
7
6
7
8
6
7
8
3
2
8
4
2
7
1
3
2
1
3
2
3
4
5
3
4
5
4
6
1
3
6
1
8
4
6
8
4
6
0
1
2
0
1
2
5
0
7
5
0
8
5
0
7
5
0
7
O easy to compute
X bias! (scanline)
O good quality
X hard to compute
(sequential)
O easy to compute
O good quality
Parallel sampling
Summary
for each level low to high
for each phase group p
parallel: for each cell in p
if cell contains no sample
draw one sample randomly
from the cell domain
add the sample if not
conflicting existing ones
Adaptive sampling
Slightly more involved
than uniform sampling
Parallelizable as well
Results
our method
dart throwing
Spectrum comparison - 2D
power spectrum (10 run)
radial mean
radial variance
Sampling in higher dimensions
Algorithm applicable to 2+ dimension
power spectrum
3D samples
radial mean
radial variance
Performance
O: on the fly
P: pre-computed dataset
# samples per second
O Our method
(NVIDIA 8800 GTX)
O Boundary sampling
[Dunbar & Humphreys 2006]
O Hierarchical dart throwing
[White et al. 2007]
P Wang tiling
[Kopf et al. 2006]
P Polyominoes
[Ostromoukhov 2007]
2D
3D
4D
5D
6D
4.06 M
555 K
42.9 K
2.43 K
179
0.20 M
X
X
X
X
0.21 M
X
X
X
X
1~3M
X
X
X
X
>1 M
X
X
X
X
Wang tiling
Corner tiling P-pentominoes Our method
[Kopf et al. 2006]
[Lagae & Dutre 2006]
[Ostromoukhov 2007]
Limitations
Only empirical, but no theoretical proof yet
Slow in high dimensions, adaptive sampling
Hard to control exact # of samples
No fine-grain sample ranking
e.g. progressive zoom-in [Kopf et al. 2006]
Euclidean space only (no manifold surface)
Future work for parallel algorithm
Sequential consistency [Lampert]
too strict for some applications
A looser sense of consistency?
parallel texture synthesis [Lefebvre & Hoppe 2005]
random number generation [Tzeng & Wei 2008]
Acknowledgements
Ares Lagae
Stanley Tzeng
Johannes Kopf
Eric Stollnitz
Victor Ostromoukhov
Brandon Lloyd
Eric Andres
Dwight Daniels
Zhouchen Lin
Jianwei Han
Ting Zhang
Baining Guo
Kun Zhou
Harry Shum
Xin Tong
Reviewers
Jian Sun