Estimation of the pair correlation function of random point fields via frequency domain
DOI:
https://doi.org/10.5566/ias.3900Keywords:
digital image analysis, random point field, pair correlation function, structure factorAbstract
Applying a Wiener-Khimtchine type theorem for random point fields, the estimation of the pair correlation function via frequency domain is presented, which offers certain advantages over classical estimation, especially for large data sets. The discretization of the point data, i.e., its mapping ontu a grid, can be viewed as a digital image, where this mapping includes a regularization of the data. Using a fast Fourier transform and its co-transform, the estimation of the pair correlation function is consistently embedded in the field of digital image analysis. Finally, bulding upon this technique, an estimator of the density of the Bartlett spectrum is derived, whose mormalization is known in scattering theory as the structure factor. The suitability of the estimators is demonstrated with examples.
References
ASTM D7723 (2018). Standard Test Method for Rubber Property - Macro-Dispersion of Fillers in Compounds. ASTM International, West Conshohocken, PA.
Azais J, Wschebor M (2009). Level Sets and Extrema of Random Processes and Fields. Hoboken, NJ: Wiley & Sons.
Baddeley AJ, Rubak E, Turner R (2015). Spatial Point Patterns. Methodology and Applications with R. Boca Raton, FL: Chapman & Hall/CRC.
Baddeley AJ, Davies M, Hazelton ML (2025). An improved estimator of the pair correlation function of a spatial point process. Biometrika 112:asaf021.
Bartlett MS (1963). The spectral analysis of point processes. J R Stat Soc B25:264-96.
Bartlett MS (1964). The spectral analysis of two-dimensional point processes. Biometrika 51:299-311.
Bartlett MS (1975). The statistical analysis of spatial patterns. Monogr. Appl. Prob. Statist. London: Chapman & Hall.
Beliaev D, Cammarota V, Wigman I (2019). Two-point function for critical points of a random plane wave. Int Math Res Notes 2019:2661-89.
Boudjemaa A (2023). A charged Coulomb Bose gas with dipole-dipole interactions. Phys Letters A 465:128712.
Brémaud P, Massoulié L, Ridolfi A (2005). Power spectra of random spike fields and related processes. Adv Appl Prob 37:1116-46.
Brémaud P (2014). Fourier-Analysis & Stochastic Processes. New York: Springer.
Brémaud P, Massoulié L (2002). Power spectra of general shot noises and Hawkes point processes with a random excitation. Adv Appl Prob 34:205-22.
Chiu SN, Stoyan D, Kendall WS, Mecke J (2013). Stochastic Geometry and its Application, 3rd Ed. Chichester: Wiley.
Cloetens P, Barrett R, Baruchel J, Guigay JP, Schlenker M (1996). Phase objects in synchrotron radiation hard X-ray imaging. Phys D Appl Phys 29:133--46.
Daley DJ, Vere-Jones D (1988). An introduction to the theory of point processes. New York: Springer.
Estrade A, Fournier J (2024). Gaussian fields through geometrical properties. Hal archives ouvertes-04481440:1-50.
Fiksel T (1988). Edge-corrected density estimators for point processes. Statistics 19:67-76.
Gabrielli A, Torquato S (2024). Voronoi and void statistics for superhomogeneous point processes. Phys Rev E Stat Nonlin Soft Matter Phys:041105.
Graham I (1980). Spectral analysis and distance methods in the study of archaeological distributions. Archaeol Sci 7:105-29.
Grainger JP, Rajala TA, Murrell DJ, Olhede SC (2025). Spectral estimation for point processes and random fields. Biometrika 103:asaf089.
Hawat D, Gautier G, Bardenet R, Lachièze-Rey R (2023). On estimating the structure factor of a point process, with applications to hyperuniformity. Statist Comp 33.
Jähne B (2005). Digital Image Processing, 6th Ed. Berlin, Heidelberg: Springer.
Koch K (2002). Spektralanalyse zufälliger abgeschlossener Mengen. Universität Siegen: Diploma Thesis.
Koch K, Ohser J, Schladitz K (2003). Spectral theory for random closed sets and estimating the covariance via frequency space. Adv Appl Prob 35:603-13.
Ladgham S, Lachieze-Rey R (2023). Local repulsion of planar Gaussian critical points. Stoch Proc Appl 166:104221.
Lantuéjoul C (2013). Geostatistical Simulation - Models and Algorithms. Berlin: Springer.
Maier JD, Gaus K, Wagner J (2024). Measurable structure factors of dense dispersions containing polydisperse optically inhomogeneous particles. J Appl Crystallogr 57:1503-1513.
Massaro M, del Campo A (2025). Spatial form factor for point patterns: Poisson point process, Coulomb gas, and vortex statistics. Phys Rev Res 7:023107.
Matérn B (2013). Spatial Variation, 2nd Ed. Lecture Notes in Statistics, Vol. 36. New York: Springer Science & Business Media.
Mugglestone M, Renshaw E (1996). A practical guide to the spectral analysis of spatial point processes. Comp Statist Data Analysis 21:43-65.
Ohser J, Lacayo-Pineda J, Putman M, Rack A, Dobrovolskij D (2019).
Estimation of filler macro-dispersion in rubber matrix by radiometric stereo microscopy. J Microsc 274:32-44.
Ohser J, Mücklich F (2000). Statistical Analysis of Microstructures in Materials Science. Chichester, New York: J Wiley & Sons.
Ohser J, Schladitz K (2009). 3D Images of Materials Structures - Processing and Analysis. Weinheim, Berlin: Wiley VCH.
Paganin D, Mayo SC, Gureyev TE, Miller PR, Wilkins SW (2002).
Simultaneous phase and amplitude extraction from a single defocused
image of a homogeneous object. Microscopy 206:33-40.
Pinkney C, Euán C, Gibbert A, Shojaie A (2024). Regularised spectral estimation for high-dimensional point processes. arXiv240312908 :preprint.
Rack A, García-Moreno F, Schmitt C, Betz O, Cecilia A, Ershov A, Rack T, Banhart J, Zabler S (2010). On the possibilities of hard X-ray imaging with high spatio-temporal resolution using polychromatic synchrotron radiation. J X Ray Sci Tech 18:429-41.
Rajala TA, Olhede SC, Graininger JP, Murrell DJ (2023). What is the Fourier transform of a spatial point process? IEEE Trans Inf Theory 69:5219-52.
Shinozuka M, Jan CM (1972). Digital simulation of random processes and its application. J Sound Vibration 25:111-28.
Stoyan D, Kendall W, Mecke J (1985). Stochastic Geometry and Its Applications 1st Ed. Chichester: J. Wiley & Sons.
Vere-Jones D (1974). An elementary approach to the spectral theory of stationary random measures. In: Harding EF, Kendall DG, eds., Stochastic Geometry. Chichester: J. Wiley & Sons, 307-21.
Walters JK, Honeybone PJR, Huxley DW, Newport RJ, Howells DS (1993).
A high-resolution neutron-diffraction study of the structure of amorphous hydorgenated carbon, a-c:h. J Phys Condens Matter 5:L387-L392.
Weitkamp T, Haas D, Wegrzynek D, Rack A (2011). ANKAphase: software for single-distance phase~retrieval from inline X-ray phase-contrast radiographs. J Synchrotron Radiation 18:617-29.
Wertheim M (1963). Exact solution of the Percus-Yevick integral equation for hard spheres. Phys Rev Letters 10:321-3.
Downloads
Published
Data Availability Statement
Estimating the pair correlation function via frequency domain is a novel method that has not been yet published in this form. The method offers significant computational advantages over previous methods. I belive that the publication of this article could make estimating the pair correlation function (and its counterpart in frequency domain: the density of the Bartlett spectrum) even more attractive to those familiar with digital image analysis.
Issue
Section
License
Copyright (c) 2026 Joachim Ohser
This work is licensed under a Creative Commons Attribution 4.0 International License.
How to Cite
