2. Parallel Poisson Disk Sampling
Li-Yi Wei
Microsoft

3. 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

4. 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

5. 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

6. 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.

7. Optimal spectrum (given # samples) Blue noise: aliasing → noise
regular grid
jittered grid
Poisson disk
All with 1600 samples
samples
spectrum

8. Spatial sampling aliasing noisy regular grid jittered grid
(zone plate)
aliasing
noisy
regular grid
jittered grid
Poisson disk

9. Methods

10. Dart throwing [Cook 1986] Loop: O High quality X Slow speed
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

11. 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

12. 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

13. 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

14. Algorithm in gradual steps
Uniform sampling, sequential
Uniform sampling, parallel
Adaptive sampling

15. Sequential sampling Basic data structure
Choose grid cell size d so that each cell has at most one sample
r = minimum spacing
n = dimension
Inspired by
[Bridson 2007]
Texture synthesis

16. Sequential sampling scan-line order + single resolution
Bias!
Scanline order
Grid sampling

17. Sequential sampling random order + single resolution
Removes scanline bias
But still grid-cell biased
scanline
random

18. Sequential sampling random order + multi-resolution
Removes both biases
scanline, grid
1 level
3 level
5 level
scanline
random

19. Sequential sampling Summary for bias removal
Two sources of bias
Grid sampling
fixed by multi-resolution
Traversal order
fixed by random order
scanline
random
1 level
3 level
5 level

20. Sequential sampling Summary
for each level low to high
visit cells in a random order
if cell contains no sample
draw one sample randomly from the cell domain
add the sample if not conflicting existing ones

21. Parallel sampling Key insight
for each level low to high
visit cells in a random order
if cell contains no sample
draw one sample randomly from the cell domain
add the sample if not conflicting existing ones
visit cells in parallel!

22. Parallel sampling Key insight
Sample cells sufficiently far away in parallel
2D example:
Cells apart cannot conflict with each other
split cells → phase groups

23. Phase group partition grid partition scanline order random partition
random order 6 7 8 9 3 4 8 1 7 2 6 5 1 3 2 3 4 5 8 4 6 1 2 5 7
O easy to compute
X bias! (scanline)
O good quality
X hard to compute
(sequential)
O easy to compute
O good quality

24. 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

25. Adaptive sampling Slightly more involved than uniform sampling
Parallelizable as well

26. Adaptive sampling Multi-resolution as in uniform sampling
But uses adaptive tree instead of uniform grid
Subdivide only if possible to add more samples

27. Results

28. Spectrum comparison - 2D
dart throwing
our method
power spectrum (10 run)
radial mean
radial variance

29. Sampling in higher dimensions
Algorithm applicable to 2+ dimension
power spectrum
radial mean
radial variance
3D samples

30. Performance O: on the fly P: pre-computed dataset # samples per second
O Our method
(NVIDIA 8800 GTX)
4.06 M
555 K
42.9 K
2.43 K
179
O Boundary sampling
[Dunbar & Humphreys 2006]
0.20 M X
O Hierarchical dart throwing
[White et al. 2007]
0.21 M
P Wang tiling
[Kopf et al. 2006]
1 ~ 3 M
P Polyominoes
[Ostromoukhov 2007]
>1 M

31. Wang tiling Corner tiling P-pentominoes Our method [Kopf et al. 2006]
[Lagae & Dutre 2006]
P-pentominoes
[Ostromoukhov 2007]
Our method

32. 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)

33. 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]

34. Acknowledgements
Ares Lagae Johannes Kopf Victor Ostromoukhov Eric Andres Zhouchen Lin Ting Zhang Kun Zhou Xin Tong Jian Sun
Stanley Tzeng Eric Stollnitz Brandon Lloyd Dwight Daniels Jianwei Han Baining Guo Harry Shum Reviewers

36. Jittered grid One sample per grid cell Uniform sample O Very fast
X Quality not so good

37. Jittered grid vs. Poisson disk (Recap)
samples
spectrum
regular grid
jittered grid
Poisson disk

38. Combining strengths
Quality of dart throwing
Speed of jittered grid

39. Relaxation vs Dart throwing
G-hexominoes
[Ostromoukhov 2007]
☺ spatial uniform
Our method
☺ spectrum
radial mean
radial variance
sample layout

40. Future work Parallelizing important algorithms
Sequential consistency [Lampert]
Dwarfs [Landscape of Parallel Computing 2006]
- linear algebra, spectral, N-body, grids, Monte-Carlo
New dwarfs that do not need to be seq-con?

42. How to start a bar flight
I saw people fighting in a bar for the dart board.
I asked them why.
They told me when they play together, their darts landed too close to each other.

43. How to stop the bar flight
Parallel GPU run time
(slow motion)
4M Poisson disk samples / sec
in parallel!
I told them if they play by my rule, I can ensure their darts to land at least a certain distance away from each other.
The run also runs on a GPU, producing more than 4 million darts per second in parallel.

45. Poisson Disk Sampling Popular pattern Prior methods sequential
[Cook 1986]
Prior methods sequential
e.g. dart throwing
A Poisson disk set contains samples that are randomly distributed, but remain a minimum distance away from each other.
It is a popular sampling pattern, used in many graphics applications.
However, it is also hard to generate. Existing methods are sequential, such as dart throwing, where samples are drawn randomly, but rejected if too close to existing samples, and accepted otherwise.

46. Dart Throwing in Parallel
Fast – 4 million samples/sec on GPU
Entirely on the fly, no pre-compute
Good quality – like (sequential) dart throwing
Texture synthesis …
To address this, I have figure out a method to throw darts in parallel.
It is faster than existing methods, and produce more than 4 million samples per second.
The quality is also good, just like traditional dart throwing.
The algorithm is actually derived from texture synthesis (old dog with new tricks).

47. 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

48. 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

50. Limitations