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


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


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.


[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 1765203 | Zbl 1171.60338

[AFP00] Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, Clarendon Press, 2000 | Zbl 0957.49001

[AN01] Ahsanullah, Mohammad; Nevzorov, Valery B. Ordered random variables, Nova Science Publishers, Inc., Huntington, NY, 2001 | MR 2019107 | Zbl 1114.62330

[AT07] Adler, Robert J.; Taylor, Jonathan E. Random fields and geometry, Springer Monographs in Mathematics, Springer, 2007 | Zbl 1149.60003

[AW09] Azaïs, Jean-Marc; Wschebor, Mario Level sets and extrema of random processes and fields, John Wiley And Sons, 2009 | Zbl 1168.60002

[BB99] Bilodeau, Martin; Brenner, David Theory of multivariate statistics, Springer Texts in Statistics, Springer, 1999 | MR 1705291 | Zbl 0930.62054

[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 3457393 | Zbl 1343.60060

[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 4108116 | Zbl 1464.60049

[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 | Article | MR 3911693 | Zbl 1406.60076

[Bie19] Biermé, Hermine Introduction to random fields and scale invariance, Stochastic geometry (Lecture Notes in Mathematics), Volume 2237, Springer, 2019, pp. 129-180 | Article | MR 3931585

[Cao03] Cao, Frédéric Geometric Curve Evolution and Image Processing, Lecture Notes in Mathematics, 1805, Springer, 2003 | MR 1976551 | Zbl 1290.35001

[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 | Article | MR 3629017 | Zbl 1362.62098

[DC76] Do Carmo, Manfredo P. Differential Geometry of Curves and Surfaces, Prentice Hall, 1976 | Zbl 0326.53001

[EG92] Evans, Lawrence C.; Gariepy, Ronald F. Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, 1992 | Zbl 0804.28001

[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 | Article | MR 3572325 | Zbl 1367.60016

[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 | Article

[Gra71] Gray, Stephen B. Local Properties of Binary Images in Two Dimensions, IEEE Trans. Comput., Volume C-20 (1971) no. 5, pp. 551-561 | Article | Zbl 0216.50101

[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 | Article | MR 2221466 | Zbl 1104.65012

[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 | Article | MR 3842168 | Zbl 1404.60034

[LR19] Lachièze-Rey, Raphaël Bicovariograms and Euler characteristic of random fields excursions, Stochastic Processes Appl., Volume 129 (2019) no. 11, pp. 4687-4703 | Article | MR 4013877 | Zbl 1448.60112

[Mat75] Matheron, Georges Random sets and integral geometry, Wiley series in probability and mathematical statistics: Probability and mathematical statistics, John Wiley & Sons, 1975 | MR 385969 | Zbl 0321.60009

[Pra07] Pratt, William K. Digital Image Processing: PIKS Scientific Inside, John Wiley & Sons, 2007 | Zbl 1303.68004

[PS15] Pausinger, Florian; Svane, Anne M. A Koksma–Hlawka inequality for general discrepancy systems, J. Complexity, Volume 31 (2015) no. 6, pp. 773-797 | Article | MR 3400993 | Zbl 1377.11084

[Ros79] Rosenfeld, Azriel Digital topology, Am. Math. Mon., Volume 86 (1979) no. 8, pp. 621-630 | Article | MR 546174 | Zbl 0432.68061

[Ser82] Serra, Jean Image Analysis and Mathematical Morphology, Academic Press Inc., 1982 | Zbl 0565.92001

[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 895588 | Zbl 0622.60019

[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 | Article | MR 3197345 | Zbl 1361.68297

[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 | Article | MR 3372113 | Zbl 1400.94030

[Thä08] Thäle, Christoph 50 years sets with positive reach—a survey, Surv. Math. Appl., Volume 3 (2008), pp. 123-165 | MR 2443192 | Zbl 1173.49039

[Ton90] Tong, Yung Liang The Multivariate Normal Distribution, Springer Series in Statistics, Springer, 1990 | Zbl 0689.62036

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