blue-sampler

Generate large stealthy point patterns on the unit torus $[0, 1)^D$. 📐

Stealthy point patterns exhibit vanishing density fluctuations at low frequencies, making them particularly suited for Monte Carlo integration, image stippling, and any application requiring well-distributed, low-discrepancy points.

The main blue-noise samplers (RGBN and NUFFT) offer linear complexity in the number of points.
They can generate e.g. 1 million 2D (resp 3D) points in under 10 minutes (resp 30 minutes) on a standard CPU, and up to 30× faster on GPU (30s resp 1 minute).

Note: The sampling methods implemented here support adaptive sampling from a target distribution. This feature is still experimental beyond 2d distributions

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Positioning and scope

One can find extensive Blue-noise and hyperuniform sampling methods in the literature, all involving different trade-offs between spectral quality, computational cost, dimensionality, hardware requirements, implementation complexity, and support for adaptive sampling. To briefly cite some of them: Perturbed lattices are simple and fast; Poisson-disk is a mature and popular tool; Void-and-Cluster masks offer instant execution but are restricted to regular grids; Relaxation methods (such as Lloyd or CCVT) are classic but slow to converge and prone to structural artifacts; Optimal Transport (BNOT) provides natural adaptive sampling but is computationally heavy; Gaussian blue noise (GBN) achieves ultra-high spatial quality but has quadratic complexity; and Fast reciprocal space Correlator (FReSCo) combines ultra-high quality with near-linear complexity, though it requires installing heavy external dependencies or familiarity with docker containers.

blue-sampler is not intended as a universal replacement for these methods. It is a lightweight Python framework for experiments on blue noise and hyperuniformity, targeting simple installation (native Python code) and device flexibility (CPU/GPU), high spectral quality (achieving the commonly accepted "stealthy" criterions $S(k) \lesssim 10^{-3} - 10^{-4}$), adaptive sampling, and support for 3D and higher dimensions.

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📦 Installation

pip install blue-sampler

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Quick Start

import blue_sampler as blue

# Generate 10,000 2D blue-noise points
x = blue.sample_points(N=10_000, D=2)
blue.plot(x) 

# Structure factor
blue.plot_structure_factor(x) 

# Image stippling
x = blue.im2points("plots/zebra.jpg") #return points
x = blue.im2quads("plots/vangogh.jpg")   #quadrilaterals

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🖼️ Examples

Uniform sampling, Gaussian Blue Noise (3k points)

Image stippling with points (20k points)

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📋 Available Samplers

Main sampling methods

x = blue.sample_points(N, D, method="rgbn") #(N, D)

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Alternative Samplers

Sobol sequence 📏

x = blue.sobol(N, D) #(N, D)

Low-discrepancy quasi-random sequence.

Clusters

# Raw clusters
cl = blue.sample_clusters(N, D) #(N, K, D)
blue.plot_clusters(cl) 

# Convert to point set
x = blue.cluster2points(cl) #(N, m, D)

STIT Tessellations (2D only) 🧩

# Raw STIT tessellation (quadrilaterals)
ts = blue.sample_tessels(N) #(N, 4, D=2)
blue.plot_tessels(ts) 

# Convert to point set
x = blue.tessel2points(ts) #(N, m, D=2)

Pinwheel Tilings (2D only) 𖣘

# Base pinwheel triangle
pw0 = blue.pinwheel_base()  #(3, D=2)

# Triangulation level 4
pw4 = blue.pinwheel_transform(pw0, depth=4) #(4*5**depth, 3, D=2)
blue.plot_polygons(pw4) 

#===============================
# Convert Pinwheel to point set:
#===============================
#sample points from the BASE pinwheel
x0 = blue.tessel2points(pw0) #(m, D=2)
#then replicate the sample on the full triangulation,
#in a fractal way
x4 = blue.pinwheel_transform(x0, depth = 4) #(4*5**depth, m, D=2)

Note: The conversion from geometric objects (polygons or clusters) to point sets is performed using a standard moment matching technique. cluster2points(x, p = 3) and tessel2points(x, p = 3) will sample m points per batch that mimic the statistical {0, 1, ... p-1} moments of the batch.

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Supported Dimensions

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Documentation & Links 🔗

• 📖 Documentation: https://blue-sampler.readthedocs.io
• 📦 PyPI: https://pypi.org/project/blue-sampler
• 🌐 Project website: https://for-a-few-dpps-more.github.io/rgbn/
• 💻 GitHub: https://github.com/For-a-few-DPPs-more/rgbn

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📚 References

The algorithms and mathematical tools implemented in blue-sampler are based or inspired from the following works.

• Gaussian Blue Noise (repulsive interaction kernel)
  A. G. M. Ahmed, J. Ren, and P. Wonka.
  Gaussian Blue Noise.
  ACM Transactions on Graphics (SIGGRAPH Asia), 41(6), 2022.
  DOI: 10.1145/3550454.3555519

• FReSCo (Non uniform FFT)
  A. Shih, M. Casiulis, and S. Martiniani.
  Fast Generation of Spectrally-Shaped Disorder.
  Physical Review E, 110(3):034122, 2024.
  DOI: 10.1103/PhysRevE.110.034122

• STIT tessellations and moment matching
  L. Lotz and M. A. Klatt.
  Persistence of asymptotic variance under transport: from hyperfluctuation to stealthy hyperuniformity.
  arXiv:2605.22803, 2026.

• Aperiodic tiling for hyuperuniformity (here pinwheel)
  A. Gabrielli, B. Jancovici, M. Joyce, J. L. Lebowitz, L. Pietronero, and F. Sylos Labini.
  Generation of Primordial Cosmological Perturbations from Statistical Mechanical Models.
  Physical Review D, 67(4):043506, 2003.

• SquareNet (v1.3.11). 2026.
  Grid data structure for point clouds, codeveloped with RGBN
  for efficient neighbor query and fourier trnasform in Python.

• Sobol sequences
  Wrapped from scipy.stats.qmc.Sobol (SciPy).

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License

MIT