Golden Ratio Sequences for Low-Discrepancy Sampling

Schretter, Colas; Kobbelt, Leif; Dehaye, Paul-Olivier

doi:10.1080/2165347x.2012.679555

Cited by 17 publications

(8 citation statements)

References 6 publications

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“…In prior work, we explored applications of these golden ratio sequences for generating randomized integration quasi-lattices [14] and for non-uniform sampling [15]. Figure 2 compares the first elements of the van der Corput generator and the golden ratio sequence with s = 0.…”

Section: Integro-approximationsmentioning

confidence: 98%

Van der Corput and Golden Ratio Sequences Along the Hilbert Space-Filling Curve

Schretter

Gerber

et al. 2016

Springer Proceedings in Mathematics &Amp; Statistics

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This work investigates the star discrepancies and squared integration errors of two quasi-random points constructions using a generator one-dimensional sequence and the Hilbert space-filling curve. This recursive fractal is proven to maximize locality and passes uniquely through all points of the d-dimensional space. The van der Corput and the golden ratio generator sequences are compared for randomized integro-approximations of both Lipschitz continuous and piecewise constant functions. We found that the star discrepancy of the construction using the van der Corput sequence reaches the theoretical optimal rate when the number of samples is a power of two while using the golden ratio sequence performs optimally for Fibonacci numbers. Since the Fibonacci sequence increases at a slower rate than the exponential in base 2, the golden ratio sequence is preferable when the budget of samples is not known beforehand. Numerical experiments confirm this observation.

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Section: Integro-approximationsmentioning

confidence: 98%

Van der Corput and Golden Ratio Sequences Along the Hilbert Space-Filling Curve

Schretter

Gerber

et al. 2016

Springer Proceedings in Mathematics &Amp; Statistics

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“…Several authors have independently proposed methods for creating quasi-random sequences based on the golden ratio. [19][20][21] Here, we use the two-dimensional R 2 sequence from Ref. [22].…”

Section: Random and Quasi-random Sampling Sequencesmentioning

confidence: 99%

High‐throughput determination of grain size distributions by EBSD with low‐discrepancy sampling

Long,

Holbrook,

Hufnagel

et al. 2023

Journal of Microscopy

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Because microstructure plays an important role in the mechanical properties of structural materials, developing the capability to quantify microstructures rapidly is important to enabling high‐throughput screening of structural materials. Electron backscatter diffraction (EBSD) is a common method for studying microstructures and extracting information such as grain size distributions (GSDs), but is not particularly fast and thus could be a bottleneck in high‐throughput systems. One approach to accelerating EBSD is to reduce the number of points that must be scanned. In this work, we describe an iterative method for reducing the number of scan points needed to measure GSDs using incremental low‐discrepancy sampling, including on‐the‐fly grain size calculations and a convergence test for the resulting GSD based on the Kolmogorov–Smirnov test. We demonstrate this method on five real EBSD maps collected from magnesium AZ31B specimens and compare the effectiveness of sampling according to two different low discrepancy sequences, the Sobol and R2 sequences, and random sampling. We find that R2 sampling is able to produce GSDs that are statistically very similar to the GSDs of the full density grids using, on average, only 52% of the total scan points. For EBSD maps that contained monodisperse GSDs and over 1000 grains, R2 sampling only required an average of 39% of the total EBSD points.

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“…For instance, this family of lattices is interesting in what concerns discrepancy, or the question on the most evenly-distributed sets of points, cf. [Schretter et al, 2011]. Now we can reverse the direction nature → math to math → nature and ask the question: Is there an object in nature (e. g., a sort of fruit) that corresponds to our L 0 v ψ ,h lattice?…”

Section: Cylindrical Modelmentioning

confidence: 99%

Multidimensional generalization of phyllotaxis

Lodkin

2019

CAP

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The regular spiral arrangement of various parts of biological objects (leaves, florets, etc.), known as phyllotaxis, could not find an explanation during several centuries. Some quantitative parameters of the phyllotaxis (the divergence angle being the principal one) show that the organization in question is, in a sense, the same in a large family of living objects, and the values of the divergence angle that are close to the golden number prevail. This was a mystery, and explanations of this phenomenon long remained “lyrical”. Later, similar patterns were discovered in inorganic objects. After a series of computer models, it was only in the XXI century that the rigorous explanation of the appearance of the golden number in a simple mathematical model has been given. The resulting pattern is related to stable fixed points of some operator and depends on a real parameter. The variation of this parameter leads to an interesting bifurcation diagram where the limiting object is the SL(2;Z)-orbit of the golden number on the segment [0,1]. We present a survey of the problem and introduce a multidimensional analog of phyllotaxis patterns. A conjecture about the object that plays the role of the golden number is given.

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Golden Ratio Sequences for Low-Discrepancy Sampling

Cited by 17 publications

References 6 publications

Van der Corput and Golden Ratio Sequences Along the Hilbert Space-Filling Curve

Van der Corput and Golden Ratio Sequences Along the Hilbert Space-Filling Curve

High‐throughput determination of grain size distributions by EBSD with low‐discrepancy sampling

Multidimensional generalization of phyllotaxis

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