blue noise sampling using an sph-based method

Varying Kernel Functions

Blue Noise Sampling using an SPH-based Method

Min Jiang, Yahan Zhou, Rui Wang, Richard Southern, Jian Jun ZhangBournemouth University, University of Massachusetts Amherst

Sponsored by

Thank you Steve for your introduction. Now I am going to introduce a new blue noise sampling method using smooth particle hydrodynamic (SPH) which is currently very popular in fluid simulation1

BackgroundBlue NoiseRefers to sample sets have random and yet uniform distribution. Anti-aliasing->Many applications: image synthesis, physically-based simulation, [Chen et al 2013] non-photorealistic rendering, and geometry processingSPHSPH was originally introduced for astrophysical simulationPopular in fluid simulation Computes particle distributions to minimize the internal pressure variance.

[Aircraft ditching, 2013]

Sponsored by SA2015.SIGGRAPH.ORG

Blue Noise Refers to sample sets have random and yet uniform distribution. It has great advantage of anti-aliasing and the sample patterns are also visually pleasing. Therefore, it is very popular in many applications SPH was originally introduced for the simulation of astrophysical problems. In the last two decades SPH has enjoyed popularity as one of the key methods for fluid simulation. it computes particle distributions to minimize the pressure variance. We found that this results in sample points with a high quality blue-noise spectrum. Inspired by this, we tailored the SPH method for blue noise sampling.

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aliasing


Before we move on, I would like to explain the first question what is Aliasing. Lets see the image. Which one is better?Because the first one has jagged appearance of diagonal lines. And the second one, the sharp edge is blended into the background, so reduced the artefacts. The second pic is called anti-aliasing. They used a so called dithering method. Please check it If you would like to know more.

Why is the aliasing and why it is almost unavoidable because most Image signal is not bandlimited, signal reconstructed from samples will def be different from the original continuous signal, this will leads to some form of aliasing. One solution is to use sampling patterns with a blue noise power spectrum, which reduces the visibility of this aliasing by mapping it to incoherent noise --- the random yet uniform blue noise

%high image frequencies cannot always be removed by prefiltering which necessarily leads to some form of aliasing 2 ways of anti-aliasing:Smoothing the jagged appearance of diagonal lines in a bitmapped image. blend the sharp edge into the background, also know as dithering;Smoothing a distorted signal by applying various techniques that filter out unwanted noise.---fielter ----blue noise 3

BackgroundClassic blue noise difficult to steer or vary distribution propertiesSPH sampling enables a continuous trade-off between the Nyquist frequency and oscillationSPH sampling vs CCVT [Balzer and coworker, 2009]a lot faster, even compared with Lloydscontrollable sampling pattern include CCVT profileSPH sampling vs Kernel [Fattal, 2011]avoids hexagonal patterns without randomness parameteruser specified sampling number


Classic blue noise sampling methods have difficulty controlling the distribution properties of blue noise. In this paper, we use a fluid simulation framework enables a continuous tradeoff between the Nyquist frequency and oscillation, where users can generate desired blue noise patterns between a Lloyds pattern and a CCVT profile, and even beyond these.

Our parallel implementation of SPH runs much faster than Lloyds and the number of samples can be precisely controlled by the user. We do not need a randomness parameter to avoid hexagonal patterns compared with the kernel paper by Fattal.

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DefinitionBlue noise:1. the power spectrum of the sample points is noisy and without concentrated spikes, and 2. the spectrum is close to zero for low frequencies.


To make it more clearly that whats going on here, lets look at the definition of blue noise first. I am not going to read it. Ok. I dont know what you feel about it. To me, it doesnt sounds like a definition, because it is not specified 5

Blue noise -ccvt


Blue noise-Lloyds


Noise and AliasingHeck et al. 2013Effective Nyquist FrequencyOscillation

Radial Means


If they are both called blue noise. How can we differentiate them. To analyze it, we will look at two properties of a blue noise spectrum: effective Nyquist frequency and oscillation.In the radial mean of a power spectrum,8


Effective NyquistFrequency


The effective Nyquist frequency corresponds to the empty region in the low frequency part,9


Oscillation


And oscillation refers to the oscillation after the initial jump, which is marked as green here.As is discussed by Heck et al, a blue-noise sampling with high effective Nyquist frequency can reduce the low-frequency error in the integration, while high oscillation would results in structured aliasing. Ideally the perfect blue noise sample set shall have high Nyquist frequency and low oscillation, which corresponds to both low noise and aliasing. But those two goals are usually conflicting, thus users have to make trade-off between them.

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The core idea - algorithm

Iteration 0Iteration 5Iteration 50


Basically, we run SPH in a bounded area. We initialize the samples with white noise. We propose a new boundary correction method for SPH to fit the sampling context using a cohesion force and a correction force. We replace the viscosity force in SPH with a damping parameter to velocity, since smooth dynamic flow of a simulation is irrelevant and it also speeds up the convergence time. The gravity is not needed here.

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Density Calculation


This function is the standard density calculation formula in SPH and this function is the physical law of density calculation Assuming the mass is the same for each point, the volume of points will be the same if the density of points is the same.This is the kernel function with the smooth length h12

The core idea - algorithmPressure

Density difference


The pressure force is one of the fundamental forces in SPH and will be the most important equation for SPH sampling.

Pressure is created whenever there is a pressure difference: while a density lower than rest density, pressure brings the points together, and a higher density will result in repulsion forces. As such, the pressure force aims to equalize the density differences throughout the particle distribution.

The pressure of each point is calculated from the difference between the actual density of point and the rest density. We refer it as density difference which is the key of the various blue noise properties.

% For blue noise sampling, to avoid the attracting pressure forces we adopt the same concept with fluid simulation and clamp the negative pressure to 0 in each iteration.

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The core idea - convergence


We consider the sampling to have converged when the point displacement is less than epsilon. Upon convergence, the points have the following properties: All points have approximately the same density and the pressure force of all points reaches zero.

This implies that upon convergence, for any point, one of the following condition must stand. Either the sum of gradient kernel equals to zero or the particle density equals to the rest density

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The core idea - convergence

Lloyds AlgorithmCCVT


When most samples converge with the kernel condition, it actually asks for all the forces acting on a sample to be symmetrical, the points in this case will favour a regular distribution haxagonal pattern. This idea is consistent with Lloyds algorithm which moves the points towards the centre of the cell. The results are very similar with the fully-converged Lloyds relaxation;

While most samples converge with the density condition, as long as the density equals to the rest density, it will allow random distribution. This idea corresponds with CCVT method which has the constraint capacity while ours have the same density. In this case the resulting point distribution will tend towards CCVT profile.

%According to the density definition at the beginning, the same density means the same volume.

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The core idea - convergenceTrade-offUser Defined Density Difference


In practice we cant precisely control every point to converge with certain condition. There will be a trade-off between those two.

The different trade-offs can be achieved by varying the density difference: lamda. When the density difference is large, the points can only converge with the kernel condition. While if we set the rest density close to the actual density of each point, the points will quickly converge with the density condition.

Therefore, different from the SPH in fluid simulation, we update the rest density every timestep to keep the user defined density difference. Different trade-off will results in different points distribution, we will discuss it later.

%To make the rest density close to the actual density of all the particles. as the average density of all points rather than setting beforehand. %Therefore, if there is a user defined density difference k, we set the rest density as the average density minus the density difference.

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Noise and Aliasing

Density Difference2825500SamplesSpectrum


This is achieved by varying the density difference parameter.

Here we show the samples with density difference, from 2 to 500.

For the samples with low density difference, like the samples on the left, we can observe that the inner empty region in its spectrum is relatively small, which means a small effective Nyquist frequency. Meanwhile, the number of outer rings in the spectrum is also small, which corresponds to low oscillation.

Samples with high density difference, like the right most example, has both high effective Nyquist frequency and oscillation.

%By tuning the density difference, we can easily control the trade-off between effective Nyquist frequency and oscillation, as is shown here.

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Noise and Aliasing

Trade-off Coordinate


For existing blue noise sampling methods, we can plot their trade-off coordinates between effective Nyquist frequency and oscillation.

For example, dart throwing has low effective Nyquist frequency, which is plotted as the blue point on the left. The CVT method, or the Lloyds relaxation, has both high effective Nyquist frequency and oscillation, plotted as the green point on the right. Most of the existing methods can be represented as a dot in the trade-off coordinate.But, there are few discussions about how to have direct control over the trade-off.

Our algorithm can provide such control. As a result, the trade-off in our algorithm, is more than a dot in the trade-off coordinate,

%While most of the existing blue noise sampling methods are implicitly making such trade-offs,

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Noise and Aliasing



But a continuous curve. User can select a desired trade-off for a specific application.19

Sampling process - surface and volumeNo correctionBoundary correction

With Correction Force

With correction and cohesion forces[Akinci et al 2013]


Lets go through the sampling process first. Due to the insufficient sampling along the object boundary, the points on the boundary become highly disordered.

There are a few papers dealing with the boundary correction of SPH. However, they do not deal with the regular distribution of points on the boundary. To achieve this, we introduce a correction force. This is achieved by constraining the forces to along the tangential direction for surface particles.

Using only the correction force traps samples on the boundary - potentially causing discontinuities between surface points and interior points. For this reason, we applied the cohesion force of Akinci to both interior and surface points. We can see the results are better

%Kij is a symmetric correction factor which is larger for samples lacking neighbours. Thus the cohesion force is amplified for boundary samples. 20

Using geodesic distance [Bowers 2010 ]Run SPH with pressure and correction forceMap samples onto surface Sampling process - surface

Poisson Disk Sampling


Our method can achieve surface sampling without the interior points.

We demonstrate this by replacing the distance metric with geodesic distance using Bowers light weight algorithm. We run SPH for all surface samples, drive the samples movement with only pressure and correction force; within every time step of the SPH process we map the samples back onto the surface;

we compared our surface sampling results with Poisson disk sampling of [Bowers et al. 2010] on a sphere. Ours tends more towards a CCVT profile.

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Sampling process - surface

DDF of our surface samplingDDF of CCVT [Balzer et al]

Difference domain function


We further compared the difference domain function(DDF) of the bunny surface sampling with the DDF of 2D CCVT profile. They look qualitatively similar.

%the difference is mainly due to the geodesic distance approximation when evaluating surface DDF.

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Sampling process - adaptive

Iteration 0Iteration 3Iteration 6Iteration 10Iteration 20Iteration 50Iteration 100Iteration 160


It is easy to extend SPH sampling to support adaptive sampling. We apply a distance field scale to the distance calculation. the distance field in 2D can be calculated from the intensity image, or in 3D it can be defined as the inverse of the size function. The distance between two points will become this with scaled function s(x), which is later used in all the kernel functions. As a result, points will become more sparse in high scale regions and denser in low scale regions.

Now let yahan take over and share a bit of my pressure here.

%The update of the sample position will need to correspond to the distance field scale as well%Notice that when computing the convergence criterion, one still needs to use the unscaled distance instead of the new one.23

Varying Kernel FunctionsChoice of kernels directly affects sampling patterns


After discussing how to do sampling with SPH, we may easily raise a question: how can we change the sampling patterns with our method?

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Here we will focus on two methods to change sampling patterns: by either tuning the parameters or changing the kernel functions. 25



By changing the parameters, we can achieve different blue noise patterns with different trade-off between noise and aliasing properties.

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As is described by Heck et al, one can characterize the blue noise sampling property by looking at its effective Nyquist frequency and oscillation. To understand what it means,27


Spectrum


we firstly need to look at the Fourier power spectrum of a sample set,28

Noise and Aliasing


For each blue noise sampling method, one can plot its trade off between effective Nyquist frequency and oscillation in this trade-off coordinate graph. 29

Noise and Aliasing

CVTTrade-off Coordinate


For example, the Lloyds relaxation, which is also known as centroid Voronoi tessellation, has high effective Nyquist Frequency, but it comes at the cost of high oscillation. Thus it corresponds to a dot on the top-right of the trade-off coordinate.30

Noise and Aliasing

CCVTTrade-off Coordinate


The Capacity-Constrained Voronoi Tessellations has both lower effective Nyquist frequency and oscillation. 31

Noise and Aliasing

StepTrade-off Coordinate


The blue noise with step function has almost zero oscillation, but again it comes at the cost of lower effective Nyquist frequency.32

Noise and Aliasing

SamplesSpectrum


The different trade-offs can be achieved by varying the density difference parameter, where the density difference in our algorithm refers the difference between the rest density and the actual density which is calculated as the average density of all points.

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After discussing how the choice of parameters affects noise and aliasing, we will turn to how kernels change sampling patterns.

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Different Kernels


After discussing how the choice of parameters affects noise and aliasing, we will turn to how kernels change sampling patterns.We try to replace the kernel gradient of the pressure force with 4 different kernels, including the box kernel, tent kernel, quadratic kernel and an off-center double-peak kernel.35

Different Kernels

Kernels

Samples

Spectrum


Each of such kernels will give a different sampling pattern.Here we are showing the results.

The top row shows the kernel, the middle shows the samples, and the bottom shows the Fourier power spectrum. The tent kernel and quadratic kernel give results with blue noise type distributions, but the other kernels generate distributions that are more like red or green noise.

further experiments and theoretical analysis would be interesting to understand the exact relationship between the kernel and the resulting distribution.36

Results - Performance


Here we will show some experiment results and applications of our algorithm.

Our method can be easily parallelized on the GPU. All our experiments are implemented on CUDA. We use a simple uniform grid for the neighbor search, and our time complexity per iteration is O(N logN) which is the same as Lloyds relaxation. Our method can generate more than 16 thousand samples within one second for the CCVT profile, which is much faster than the fast CCVT method by Li et al.

%Currently our method is slower than the tile-based sampling method by Wachtel et al, but their method is difficult to extend to surface and volumetric sampling.

%with 1GB onboard RAM37

Results and Applications Multi-class Sampling


For applications, the first one we discuss is the multi-class blue noise sampling problem. This problem is introduced by Wei in 2010, it aims at generating a multi-class sampling in which each individual class as well as their unions exhibit blue noise characteristics. An example is shown here. The left image shows the union of a 2-class sampling, which exhibits blue noise properties. The middle and right column show the individual samples which also give blue noise properties. Our SPH sampling method can handle this problem easily thanks to the natural property of mixing fluid. In details, we use different distance function to handle within-class and between-class sample distances. The between-class sample distances are scaled by a factor of square root 3. Details can be found in our paper.38


Iteration 0

Iteration 5

Iteration 10

Iteration 20

Iteration 30

Iteration 50

Iteration 100

Iteration 150

Iteration 312


For applications, the first one we discuss is the multi-class blue noise sampling problem. This problem is introduced by Wei in 2010, it aims at generating a multi-class sampling in which each individual class as well as their unions exhibit blue noise characteristics.

An example is shown here. The middle and left column show two sample sets which give blue noise properties. The right image shows the union of them, which also exhibits blue noise properties. | Here are the corresponding spectrums. One can see that both union and individual classes provide blue noise properties.

Our SPH sampling method can handle this problem easily thanks to the natural property of mixing fluid. This is done by using different distance functions to handle within-class and between-class sample distances.

%The between-class sample distances are scaled by a factor of square root 3.

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Iteration 0

Iteration 5

Iteration 10

Iteration 20

Iteration 30

Iteration 50

Iteration 100

Iteration 150

Iteration 312



An example is shown here. The middle and left column show two sample sets which give blue noise properties. The right image shows the union of them, which also exhibits blue noise properties. | Here are the corresponding spectrums. One can see that both union and individual classes provide blue noise properties.

Our SPH sampling method can handle this problem easily thanks to the natural property of mixing fluid. This is done by using different distance functions to handle within-class and between-class sample distances.


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Multi-class sampling


Take two class sampling as an example, if we want samples within the same class to be uniformly sampled, the distance between sample in the same class r1 and the distance of sample in different classes r2 in 2D should satisfy r1= root 3 times r 2

This implies that we can achieve multi-class sampling by modifying the concept of adaptive sampling. , the scale for samples from different class s2 has to fulfil 3s1 = s2 in order to achieve r1= root 3 times r 2

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Iteration 0

Iteration 5

Iteration 10

Iteration 20

Iteration 30

Iteration 50

Iteration 100

Iteration 150

Iteration 312



An example is shown here. The right image shows the union of a 2-class sampling, which exhibits blue noise properties. The middle and left column show the individual samples which also give blue noise properties. Our SPH sampling method can handle this problem easily thanks to the natural property of mixing fluid.

We use different distance function to handle within-class and between-class sample distances. Our method is much simpler to implement than Wei et al. Details can be found in our paper.

Here are the corresponding spectrums. One can see that both union and individual classes provide blue noise properties.


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We can also achieve multi-class sampling with more classes. Here is our result of a 5-class sampling43

Results and Applications Remeshing


Our surface sampling algorithm enables us to achieve excellent surface sampling, which can be used for remeshing.

A good remeshing algorithm should keep sharp features of the mesh while maintaining samples uniformly distributed. Our algorithm can easily generate uniform samples due to blue noise property. The points will be pushed in or out at sharp features, leading the samples to be attracted to these areas.

Here we compare our method with the bilateral blue noise sampling proposed by Chen et al for the remeshing task. The surface of a bowl model is resampled with 5000 points, then remeshed using the Ball Pivoting algorithm.

The results show that our algorithm preserves the features of the bowl very well even without the need for any additional techniques such as sub-sampling in regions of high curvature.

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Results and Applications Adaptive Volume Sampling


Another application of our algorithm is adaptive volume sampling.

It is very useful for a variety of applications, such as variational tetrahedralization and volume rendering. Our method can sample the volume with different size functions based on the particular applications.

%Here we use the feature size as is defined by Adams et al.

%Here we show an example where we sample the interior and the surface of the bunny adaptively by determining the feature size using the medial axis of the Manifold. We get small sample points on surface features, and big sample points in the interior of the mesh. Other size functions are also applicable.

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ConclusionSampling inspired by SPHVariety of blue noiseAdaptive, multi-class, surface, volume samplingFast, easy to implement in parallel


In summary, in this paper we present a new sampling method based on SPH. || Our method is able to generate a variety of blue noise samples with distribution properties that range from Lloyds relaxation to CCVT. || The flexibility of SPH allows spatially-varying point density, leading to adaptive sampling. Our method also supports multi-class blue noise sampling, surface sampling and volume sampling. || It is easy to be parallelized and runs fast.

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Future WorksControllable target spectrumImprove trade-off between Nyquist Frequency and oscillationOther fluid dynamic frameworks


There are several directions for future research. An interesting problem is to study how to automatically compute a kernel function that would lead to a user specified target distribution.

|| Another possible direction is to study how to improve the trade-off between Nyquist Frequency and oscillation.

|| Finally, we would like to try other fluid simulation frameworks for the blue noise sampling and to study their trade-off tendency and potential performance improvement.

%The relationship between the kernel and its resulting sample spectrum warrants further investigation. %This would make it possible to use our SPH-based method to generate samples with a noise with an arbitrarily designed spectral by the user. %The trade-off of our algorithm is currently not competitive with those of ideal blue noise or step blue noise and the single-peak blue noise discussed in Heck et al. [2013]. 47

Thank you!


This concludes our talk. Thank you for your attention.48

Sampling process - adaptive


We additionally compared the quality of adaptive sampling with Balzer et al. [2009] and Fattal [2011] using a quadratic ramp as the scale function.

In the coarse region, our method does not produce as good results as in the dense region. It is because the choice of smooth radius limits the influence range. If the smooth radius is too large, it will cause the points to blur the given density function.49

Results and Applications Performance

2D bunny3D Sphere3D bunnyBaby faceMulti-class3D bowlAdaptive Bunny


In terms of performance, our method can be easily parallelized on the GPU. We use a simple uniform grid for the neighbour searching. The time complexity of our method is O(N log N) for each iteration, where N is the number of samples.

In the table below we show the computation times of our method for generating 16384 samples with Lloyd and CCVT profiles respectively, and compare them with previous work.

Other supported features of our method, such as surface and adaptive sampling, incur additional computational costs. In the table below we show the computation times of different examples presented in this paper.50



According to Nyquist-shannon theorem, the sampling rate has be twice the sampling bandwidth. From this we can get the relationship between the total sample numbers and the smoothing length in kernel function. We have to make sure there is enough samplers and large enough smooth length to make SPH working. However, when there is small feature inside the object, h shouldnt be too big, otherwise the sampling on the boundary will fail. 51

Noise and Aliasing


The desired trade-off can be easily achieved by looking up the corresponding density difference value. For convenience, we plot the density difference parameter against effective Nyquist frequency of the 2D domain,52

Noise and Aliasing


We evaluated it experimentally;

fit a curve onto it. The fitting parameters of the curve can be found in the paper. By calculating the density difference parameter from the fitted curve, one can easily achieve a specific effective Nyquist frequency.53

Noise and Aliasing



If we plot the trade-off for all density difference parameters, we can get our trade-off curve. As we mentioned before, with our method, users can have control over such trade-off. 54

Density difference vs. Effective Nyquist Frequency and Oscillation


Effective nyquist frequency and oscillation changes as we increase the density difference parameter. 55

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blue noise sampling using an sph-based method

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