1. Renjie Chen Craig Gotsman Technion – Israel Institute of Technology
Tallinn
Parallel Blue-noise Sampling by Constrained Farthest Point Optimization
Good after everyone. Thanks for staying until this almost the last talk.
I’m Renjie Chen from Technion.
In the following, I’ll talk about my work with Craig Gotsman about parallel blue-noise sampling using constrained farthest point optimization
First let’s see what’s blue-noise distribution.
Renjie Chen Craig Gotsman
Technion – Israel Institute of Technology

2. Blue-noise distribution
AKA Poisson disk distribution
Uniform
Uniform point density
Large minimal mutual distance
Irregularity
No correlations between points
2018/9/18
Parallel Blue-noise Sampling by Constrained FPO
Blue-noise distribution is also know as Poisson disk distribution.
It must satisfy 2 properties.
First, the distribution should be uniform. That is the point density should be more or less the same everywhere. And the minimum distance between any two points should be sufficient large.
Second, the distribution should be irregular. In other words, there should be no correlations between points like the regular hexagonal grid shown here.
Random
Blue-noise
Hexagonal grid

3. Outline Introduction & related work FPD & FPO
Local characterization of FPD
Constrained FPO
Experimental results & conclusion
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Parallel Blue-noise Sampling by Constrained FPO
Here’s the outline of this talk.
First, I will give a brief introduction for the blue-noise distribution and some related work.
Next, I will talk about the most important prior work, based on which we carry on this work, that is the FPD, farthest point distribution and FPO, farthest point optimization.
After that, I will give a local characterization for FPD. Based on which, we developed the main algorithm.
Then I will describe the main algorithm in detail, that is the constrained farthest point optimization.
Finally I will show some experimental results and conclude the talk.

4. Blue-noise distribution
Power spectrum analysis
Ulichney, 1987: study of the frequency domain characteristics
Periodograms of Fourier transform on point distributions
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Radially averaged power spectrum
Parallel Blue-noise Sampling by Constrained FPO
The term Blue-noise was first introduced by Ulichney in 1987 in his book, study of the frequency domain characteristics.
It’s based on the periodograms of the Fourier transform on point distributions.
Ulichney proposed 2 statistics for the power spectrum of point distributions, that is the radially averaged power spectrum and anisotropy, which can be roughly thought as the change of the average and the normalized variance of the spectrum over different frequency.
Anisotropy
Power spectrum
Blue-noise spectra

5. Blue-noise Spectra structural residual peaks stably flat
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Parallel Blue-noise Sampling by Constrained FPO
structural residual peaks
Here we show the two Blue-noise measurements for 3 typical point distributions, that is regular Hexagonal grid, random and blue-noise distribution.
For a blue-noise distribution, the average power spectrum lacks low frequency,
and it has structural residual peaks.
While the anisotropy should be stably flat.
stably flat
lacking low-frequency

6. Applications Sampling & meshing NPR Stippling Rendering HDR Imaging
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Sampling & meshing
NPR Stippling
Parallel Blue-noise Sampling by Constrained FPO
There are many applications for the blue-noise distributions.
Such as sampling and meshing,
Non-photorealistic stippling,
rendering,
anti-aliasing,
and high dynamic range imaging.
Rendering
HDR Imaging

7. Related Work Dart-throwing based Relaxation based Precomputed 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 FPO
There has been many prior work in the field of blue-noise distribution.
Most of them can be categorized into 3 different types, that is dart-throwing based approaches,
relaxation based methods
and precomputed tiles.
And our work is mostly based on the Farthest point optimization proposed by Scholmer and others last year.

8. All points are separated from each other by a minimum distance
Dart-throwing
Incrementally generate samples randomly
Hard to control sampling size
Computationally expensive
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All points are separated from each other by a minimum distance
Parallel Blue-noise Sampling by Constrained FPO
First let’s see the most well-known algorithm, that is dart throwing.
Basically, this algorithm incrementally generate samples randomly. In the result, all points will be separated from each other by a minimum distance.
This approach has 2 shortcomings.
First, it’s difficult to control the sampling size, that is the number of points in the distribution.
Second, it’s computationally expensive, as one can imagine, the chance that a new randomly generated point get accepted into the distribution will decrease to 0 as the size of the distribution increase.

9. Relaxation Approach Starting from an initial distribution
Move points following some criteria until converge
Hard to possess blue noise characteristic
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Parallel Blue-noise Sampling by Constrained FPO
Now let’s see the other popular algorithm for generating blue-noise distributions.
In the algorithm, first, an initial distribution will be produced, in most cases, simply by random generation.
Then the algorithm move the points iteratively following some criteria until the point set converges and stop moving.
The main problem with the algorithm is that it’s hard for the point set to possess blue noise property.

10. Precomputed tiles Construct a few tiles of Blue-noise pattern
Seamlessly tile the tiles
Inferior Blue-noise quality
Lacking variety
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Parallel Blue-noise Sampling by Constrained FPO
The third type of algorithm for generating blue noise distribution is based on precomputed tiles.
In the algorithm, fist a few tiles of blue noise pattern are constructed.
Then the blue noise tiles are tiled seamlessly to form larger point set.
The problem with this algorithm is the resulting point distribution is of inferior blue noise quality,
and it lacks variety of different patterns,
due to the finite number of patterns that the tiles can offer.

11. FPD - Farthest Point Distribution
Farthest point (of a point set X)
The point with maximal distance(to X)
𝑓𝑝 𝑋 = argmax 𝑥 𝑑 𝑥,𝑋
𝑑 𝑥,𝑋 = min 𝑦∈𝑋 𝑑 𝑥,𝑦
FPD
Each point is a farthest point
𝑥≡ 𝑓𝑝 𝑋\{𝑥} = argmax 𝑦 𝑑 𝑦,𝑋\{𝑥}
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Parallel Blue-noise Sampling by Constrained FPO
Now let’s see farthest point distribution, which is the best blue noise distribution so far.
A farthest point, such as the red point shown in the figure
Of a point set X, the black point set,
Is the point with the maximal distance
to the point set X.
Denote the farthest point of X as fp(X), then the function fp maximize the distance from the point in question to the point set X.
Here the distance from a point to a point set is defined in the usual sense, that is the minimum distance from the point to all the points inside the point set.
A farthest point distribution is the point set where each point is a farthest point, from the remaining point set, that is the point set excluding itself.
In other words, each point in a FPD maximize the distance from itself to the remaining point set.

12. FPD - Farthest Point Distribution
Blue-noise Spectra
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Parallel Blue-noise Sampling by Constrained FPO
FPD possess excellent blue-noise spectra, as shown here. It has more obvious structural residual peaks than most other algorithms, such as dart throwing and CCVT.

13. FPO - Farthest Point Optimization
Main algorithm
While X is not converged
foreach x∈X
move x to fp(X\{x})
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Parallel Blue-noise Sampling by Constrained FPO
The intuitive way to produce FPD is through farthest point optimization.
In this algorithm, each point is iteratively moved to the farthest point, of the remaining point set, until the point set stops moving and converges.
Here we show the evolution of the point set while the algorithm is running.

14. FPO - Farthest Point Optimization
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Parallel Blue-noise Sampling by Constrained FPO
The key problem that FPO needs to solve is how to compute the farthest point.
Given a point set, in Toroidal space,
the farthest point resides in the center of the largest empty circle,
Which can be found using the Delaunay triangulation of the point set.
The largest empty circle is actually the circumcircle of some Delaunay triangle.
To find the Delaunay triangle with the largest circumcircle, we can use a priority queue. Then the largest empty circle can be found efficiently, in log time.
X – the point set
Largest empty circle
DT(X)
Priority queue

15. ? Local FPO Pro Con Linear complexity?
Requires maintaining a global DT
Slow convergence
Difficult to parallelize
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Parallel Blue-noise Sampling by Constrained FPO
A naive way to simplify the farthest point optimization is to look for the farthest point locally, instead of checking all the triangles in Delaunay triangulation, we can check a subset of it,
such as the triangles incident to the vertices in the two-rings of the current point.
Although this simple variant seems to be of linear complexity. However since the operations of Delaunay removing and insertion of a point are not always of constant complexity. Instead in some extreme cases, they can be of linear complexity. Therefore this simple variant is not guaranteed to give linear complexity.
Also this simple variant have several shortcomings,
First, it requires maintaining a global Delaunay triangulation, which means this is not really a local algorithm.
Second, because it move points to sub-optimal positions, the algorithm has slower convergence.
Last, because of the global Delaunay triangulation, it’s difficult to parallelize the algorithm.

16. Constrained Farthest Point
Local FP and global FP can be different
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Parallel Blue-noise Sampling by Constrained FPO
In the following I will discuss the main algorithm in this work in detail.
This main question in this work is about whether we can compute the FPD locally.
For any given local point set, we can compute its farthest point.
Obviously it can be different from the farthest point of the whole point set.
As shown in this figure, the red point is the farthest point of the whole point set, including the black points and the blue points.
However the farthest point of the local point set, the blue point set, falls on the convex hull of the local point set.
Here we should note that the farthest point of a point set should be inside the convex hull of the point set, since otherwise, the farthest point could be infinity.
Luckily if we change the constrain area of the farthest point to the black circle, centered at the current point, the red point will again become the farthest point of the local blue point set.

17. Local characterization of FPD
Local farthest point property
Local covering property
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min 𝑥,𝑦∈𝑋 𝑥−𝑦
Parallel Blue-noise Sampling by Constrained FPO
𝛿 𝑋
Now we give the local characterization for FPD, based on which we developed the main algorithm in this work.
Consider the local neighborhood of the red point, shaded gray in the figure.
Shrink the gray circle by a distance of delta X, we have the black circle.
delta X is the minimal mutual distance between any two points of X
Then the red point is the farthest point of the blue point set constrained inside the black circle.
This is the local farthest point property.

18. Local characterization of FPD
Local farthest point property
Local covering property
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𝛿 𝑋
Parallel Blue-noise Sampling by Constrained FPO
And for the local covering property,
It requires that the black circle is fully covered by the disks centered at the local point set,
Including both the blue points and the current point, shown in red.
The local characterization says that for each point in a FPD, we can find this local neighborhood, such that the two properties are satisfied.
And on the other hand, given a point distribution, if we can find a local neighborhood for each point such that the two local properties are satisfied, then it is a FPD

19. Local characterization of FPD
Local farthest point property
Local covering property
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𝛿 𝑋
Parallel Blue-noise Sampling by Constrained FPO
The local neighborhood can be other shapes than simple circles,
here we show some local neighborhood with arbitrary shape,
On which the local characterization still holds.
Note that the erosioned version of the local neighborhood, with black curved boundary,
is derived by shrinking the local neighborhood by a distance of delta X.

20. CFPO – constrained farthest point optimization
Local farthest point
2018/9/18 x
Parallel Blue-noise Sampling by Constrained FPO x'
Now we show how to enforce the two local properties for a point set, in order to produce a blue-noise distribution.
We use the uniform grid data structure to define the local neighborhood.
Then for the local farthest point property, we simply iteratively move each point to the farthest point of a local point set, constrained in a smaller area.
To make the algorithm simpler, we replace delta X with the theoretical maximum of the minimal mutual distance. Then the grids can stay fixed all the time.
𝛿 𝑋
∆ 𝑁 +𝜖 𝐿

21. CFPO – constrained farthest point optimization
Local covering
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Parallel Blue-noise Sampling by Constrained FPO
For the local covering property, we achieve it by initialize the point set properly.
Instead of generating the initial point set purely randomly, we use the uniform grid structure to generate the point set uniformly.
As shown in the figure, we randomly place one or two points inside each grid randomly.
Since our algorithm is guaranteed to increase the minimal mutual distance, the circles centered at the point set will increase, therefore cover more and more areas.
As an example shown here, after only 2 iterations, the whole domain is fully covered by the circles, implying the local covering property satisfied.
Uniform random initialization
1 iteration
2 iterations

22. Parallel CFPO 1 iteration 20 iterations 2018/9/18
Parallel Blue-noise Sampling by Constrained FPO
1 iteration
Now we show the pipeline of our parallel CFPO algorihtm.
First we generate the initial point set randomly and uniformly in the uniform grids.
Then, we choose a set of grids which are sufficient apart, and move the points inside to corresponding constrained local farthest points at the same time.
After that, we choose a different set of grids, and optimize the points inside them again.
As we can see, the point distribution is greatly improved after a whole iteration of parallel CFPO.
And after 20 iterations, the point set is very close to a FPD.
20 iterations

23. Local CFPO Largest empty circle 2018/9/18
Largest empty circle
Parallel Blue-noise Sampling by Constrained FPO
Here we show the core of our parallel CFPO algorithm, that is computing the local constrained farthest point.
Here, we consider the red point, its local neighborhood is defined as the 3 by 3 grid centered at it, colored in green. Then we collect all the point set inside the local neighborhood, except the red point. After that we define the local constrain area to be the black square.
This local constrained farthest point is essentially the largest empty circle problem. Which can be solved using the Delaunay triangulation of the local point set.

24. Convergence 2018/9/18 Parallel Blue-noise Sampling by Constrained FPO
Here we show the convergence behavior of our algorithm.
The cyan curves corresponds to the CFPO.
As we can see, it converges almost in the same way as FPO, after 3 or 4 iterations, and it also converges to almost the same delta X, which is the minimal mutual distance.

25. Results – Blue-noise spectra
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δ(X)=0.925, δ (𝑋)=0.934
FPO
Parallel Blue-noise Sampling by Constrained FPO
δ(X)=0.925, δ (𝑋)=0.931
CFPO
Here we show the some results of the parallel CFPO algorithm.
It’s easy to see that it gives almost identical results as FPO.
In the last row, we show the results of a simplified version of our algorithm.
In which, we constrain each point to be inside the grid that it resides. Therefore we can skip the updating step for the grid data structure.
There’s only one flaw in the blue noise spectra of the result, where the anisotropy has a peak in small frequency. Which we contribute to the underlying grid structure that the distribution possesses.
δ(X)=0.924, δ (𝑋)=0.930
simplified CFPO

26. Results - Performance 8x 80x 2018/9/18
Parallel Blue-noise Sampling by Constrained FPO
Here we show the runtime of our algorithm under different configurations.
The serial version of our algorithm on CPU is 8 times faster than FPO, and
Our GPU implementation gives up to 80 times speedup over FPO.
In the right, we show the almost perfect speedup of the CPU version of the algorithm using multi-core CPU.

27. Conclusion CFPO Local optimization Global property
Equivalency with FPO
Easy parallelization
Future Work
Non-uniform/anisotropic distribution
Higher dimension
Non-Euclidean manifold
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Parallel Blue-noise Sampling by Constrained FPO
To conclude
In this talk, we introduced constrained farthest point optimization.
This algorithm does only local optimization,
But it can ensure the global properties for the results.
We proved that it’s equivalent to FPO.
And we show it’s very easy to parallelize.
As for the future work, we want to extend the algorithm to non-uniform and anisotropic point distribution,
and generalize it to higher dimensions and non-Euclidean manifolds.

28. Thanks for your attention!
2018/9/18
Thanks for your attention!
Parallel Blue-noise Sampling by Constrained FPO

29. Parallel Blue-noise Sampling by Constrained FPO
δ(X)=0.655, δ (𝑋)=0.844
CCDT
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δ(X)=0.710, δ (𝑋)=0.854
CCVT
δ(X)=0.666, δ (𝑋)=0.818 DT
Parallel Blue-noise Sampling by Constrained FPO
δ(X)=0.799, δ (𝑋)=0.863 BS
δ(X)=0.925, δ (𝑋)=0.931
CFPO