The effect of discretization on the mean geometry of a 2D random field
Annales Henri Lebesgue, Volume 4 (2021), pp. 1295-1345.

Metadata

KeywordsPerimeter, Total curvature, Euler Characteristic, excursion sets, discrete geometry, stationary random field, image analysis, Gaussian random field

Abstract

The study of the geometry of excursion sets of 2D random fields is a question of interest from both the theoretical and the applied viewpoints. In this paper we are interested in the relationship between the perimeter (resp. the total curvature, related to the Euler characteristic by Gauss–Bonnet Theorem) of the excursion sets of a function and the ones of its discretization. Our approach is a weak framework in which we consider the functions that map the level of the excursion set to the perimeter (resp. the total curvature) of the excursion set. We will be also interested in a stochastic framework in which the sets are the excursion sets of 2D random fields. We show in particular that, under some stationarity and isotropy conditions on the random field, in expectation, the perimeter is always biased (with a 4/π factor), whereas the total curvature is not. We illustrate all our results on different examples of random fields.


References

[Adl00] Adler, Robert J. On excursion sets, tube formulas and maxima of random fields, Ann. Appl. Probab., Volume 10 (2000) no. 1, pp. 1-74 | MR | Zbl

[AFP00] Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, Clarendon Press, 2000 | Zbl

[AN01] Ahsanullah, Mohammad; Nevzorov, Valery B. Ordered random variables, Nova Science Publishers, Inc., Huntington, NY, 2001 | MR | Zbl

[AT07] Adler, Robert J.; Taylor, Jonathan E. Random fields and geometry, Springer Monographs in Mathematics, Springer, 2007 | Zbl

[AW09] Azaïs, Jean-Marc; Wschebor, Mario Level sets and extrema of random processes and fields, John Wiley And Sons, 2009 | Zbl

[BB99] Bilodeau, Martin; Brenner, David Theory of multivariate statistics, Springer Texts in Statistics, Springer, 1999 | MR | Zbl

[BD16] Biermé, Hermine; Desolneux, Agnès On the perimeter of excursion sets of shot noise random fields, Ann. Probab., Volume 44 (2016) no. 1, pp. 521-543 | MR | Zbl

[BD20] Biermé, Hermine; Desolneux, Agnès Mean Geometry for 2D random fields: level perimeter and level total curvature integrals, Ann. Appl. Probab., Volume 30 (2020) no. 2, pp. 561-607 | MR | Zbl

[BDBDE19] Biermé, Hermine; Di Bernardino, Elena; Duval, Céline; Estrade, Anne Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields, Electron. J. Stat., Volume 13 (2019) no. 1, pp. 536-581 | DOI | MR | Zbl

[Bie19] Biermé, Hermine Introduction to random fields and scale invariance, Stochastic geometry (Lecture Notes in Mathematics), Volume 2237, Springer, 2019, pp. 129-180 | DOI | MR

[Cao03] Cao, Frédéric Geometric Curve Evolution and Image Processing, Lecture Notes in Mathematics, 1805, Springer, 2003 | MR | Zbl

[DBEL17] Di Bernardino, Elena; Estrade, Anne; León, José R. A test of Gaussianity based on the Euler Characteristic of excursion sets, Electron. J. Stat., Volume 11 (2017) no. 1, pp. 843-890 | DOI | MR | Zbl

[DC76] Do Carmo, Manfredo P. Differential Geometry of Curves and Surfaces, Prentice Hall, 1976 | Zbl

[EG92] Evans, Lawrence C.; Gariepy, Ronald F. Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, 1992 | Zbl

[EL16] Estrade, Anne; León, José R. A central limit theorem for the Euler characteristic of a Gaussian excursion set, Ann. Probab., Volume 44 (2016) no. 6, pp. 3849-3878 | DOI | MR | Zbl

[GJ87] Gundersen, Hans J. G.; Jensen, E. B. The efficiency of systematic sampling in stereology and its prediction, Journal of Microscopy, Volume 147 (1987) no. 3, pp. 229-263 | DOI

[Gra71] Gray, Stephen B. Local Properties of Binary Images in Two Dimensions, IEEE Trans. Comput., Volume C-20 (1971) no. 5, pp. 551-561 | DOI | Zbl

[KSS06] Klenk, Simone; Schmidt, Volker; Spodarev, Evgueni A new algorithmic approach to the computation of Minkowski functionals of polyconvex sets, Comput. Geom., Volume 34 (2006) no. 3, pp. 127-148 | DOI | MR | Zbl

[KV18] Kratz, Marie; Vadlamani, Sreekar Central limit theorem for Lipschitz–Killing curvatures of excursion sets of Gaussian random fields, J. Theor. Probab., Volume 31 (2018) no. 3, pp. 1729-1758 | DOI | MR | Zbl

[LR19] Lachièze-Rey, Raphaël Bicovariograms and Euler characteristic of random fields excursions, Stochastic Processes Appl., Volume 129 (2019) no. 11, pp. 4687-4703 | DOI | MR | Zbl

[Mat75] Matheron, Georges Random sets and integral geometry, Wiley series in probability and mathematical statistics: Probability and mathematical statistics, John Wiley & Sons, 1975 | MR | Zbl

[Pra07] Pratt, William K. Digital Image Processing: PIKS Scientific Inside, John Wiley & Sons, 2007 | Zbl

[PS15] Pausinger, Florian; Svane, Anne M. A Koksma–Hlawka inequality for general discrepancy systems, J. Complexity, Volume 31 (2015) no. 6, pp. 773-797 | DOI | MR | Zbl

[Ros79] Rosenfeld, Azriel Digital topology, Am. Math. Mon., Volume 86 (1979) no. 8, pp. 621-630 | DOI | MR | Zbl

[Ser82] Serra, Jean Image Analysis and Mathematical Morphology, Academic Press Inc., 1982 | Zbl

[SKM87] Stoyan, Dietrich; Kendall, Wilfrid S.; Mecke, Joseph Stochastic geometry and its applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, 1987 | MR | Zbl

[Sva14] Svane, Anne M. Estimation of intrinsic volumes from digital grey-scale images, J. Math. Imaging Vis., Volume 49 (2014) no. 2, pp. 352-376 | DOI | MR | Zbl

[Sva15] Svane, Anne M. Local digital algorithms for estimating the integrated mean curvature of r-regular sets, Discrete Comput. Geom., Volume 54 (2015) no. 2, pp. 316-338 | DOI | MR | Zbl

[Thä08] Thäle, Christoph 50 years sets with positive reach—a survey, Surv. Math. Appl., Volume 3 (2008), pp. 123-165 | MR | Zbl

[Ton90] Tong, Yung Liang The Multivariate Normal Distribution, Springer Series in Statistics, Springer, 1990 | Zbl

[Wor96] Worsley, Keith J. The geometry of random images, Chance, Volume 9 (1996) no. 1, pp. 27-40 | DOI