Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-28T18:40:29.430Z Has data issue: false hasContentIssue false

Random fields of bounded variation and computation of their variation intensity

Published online by Cambridge University Press:  11 January 2017

Bruno Galerne*
Affiliation:
Université Paris Descartes
*
* Postal address: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, 45 rue des Saints-Pères, 75006 Paris, France. Email address: bruno.galerne@parisdescartes.fr

Abstract

The main purpose of this paper is to define and characterize random fields of bounded variation, that is, random fields with sample paths in the space of functions of bounded variation, and to study their mean total variation. Simple formulas are obtained for the mean total directional variation of random fields, based on known formulas for the directional variation of deterministic functions. It is also shown that the mean variation of random fields with stationary increments is proportional to the Lebesgue measure, and an expression of the constant of proportionality, called the variation intensity, is established. This expression shows, in particular, that the variation intensity depends only on the family of two-dimensional distributions of the stationary increment random field. When restricting to random sets, the obtained results give generalizations of well-known formulas from stochastic geometry and mathematical morphology. The interest of these general results is illustrated by computing the variation intensities of several classical stationary random field and random set models, namely Gaussian random fields and excursion sets, Poisson shot noises, Boolean models, dead leaves models, and random tessellations.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adler, R. J. (1981). The Geometry of Random Fields, John Wiley, Chichester.Google Scholar
[2] Ambrosio, L., Fusco, N. and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press.CrossRefGoogle Scholar
[3] Aubert, G. and Kornprobst, G. (2006). Mathematical Problems in Image Processing (Appl. Math. Sci. 147), 2nd edn. Springer, New York.CrossRefGoogle Scholar
[4] Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
[5] Biermé, H. and Desolneux, A. (2016). On the perimeter of excursion sets of shot noise random fields. Ann. Prob. 44, 521543 .CrossRefGoogle Scholar
[6] Biermé, H., Meerschaert, M. M. and Scheffler, H.-P. (2007). Operator scaling stable random fields. Stoch. Process. Appl. 117, 312332.CrossRefGoogle Scholar
[7] Bordenave, C., Gousseau, Y. and Roueff, F. (2006). The dead leaves model: a general tessellation modeling occlusion. Adv. Appl. Prob. 38, 3146.CrossRefGoogle Scholar
[8] Chlebík, M. (1997). On variation of sets. Preprint. 44, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig.Google Scholar
[9] Cowan, R. and Tsang, A. K. L. (1994). The falling-leaves mosaic and its equilibrium properties. Adv. Appl. Prob. 26, 5462.CrossRefGoogle Scholar
[10] Galerne, B. (2011). Computation of the perimeter of measurable sets via their covariogram. Applications to random sets. Image Anal. Stereol. 30, 3951.CrossRefGoogle Scholar
[11] Galerne, B. and Gousseau, Y. (2012). The transparent dead leaves model. Adv. Appl. Prob. 44, 120.CrossRefGoogle Scholar
[12] Galerne, B. and Lachièze-Rey, R. (2015). Random measurable sets and covariogram realizability problems. Adv. Appl. Prob. 47, 611639.CrossRefGoogle Scholar
[13] Gikhman, I. I. and Skorokhod, A. V. (1974). The Theory of Stochastic Processes. I. Springer, Berlin.Google Scholar
[14] Hug, D., Last, G. and Weil, W. (2004). A local Steiner-type formula for general closed sets and applications. Math. Z. 246, 237272.CrossRefGoogle Scholar
[15] Ibragimov, I. A. (1995). Remarks on variations of random fields. J. Math. Sci. 75, 19311934.CrossRefGoogle Scholar
[16] Jeulin, D. (1997). Dead leaves models: from space tessellation to random functions. In Proceedings of the International Symposium on Advances in Theory and Applications of Random Sets, World Scientific, River Edge, NJ, pp.137156.CrossRefGoogle Scholar
[17] Kingman, J. F. C. (1993). Poisson Processes (Oxford Studies Prob. 3). Oxford University Press.Google Scholar
[18] Lantuéjoul, C. (2002). Geostatistical Simulation: Models and Algorithms. Springer, Berlin.CrossRefGoogle Scholar
[19] Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
[20] Meyer, Y. (2001). Oscillating Patterns in Image Processing and Nonlinear Evolution Equations (Univ. Lecture Ser. 22). American Mathematical Society, Providence, RI.Google Scholar
[21] Molchanov, I. (2005). Theory of Random Sets. Springer, London.Google Scholar
[22] Rataj, J. (2015). Random sets of finite perimeter. Math. Nachr. 288, 10471056.CrossRefGoogle Scholar
[23] Rudin, L. I., Osher, S. and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D 60, 259268.CrossRefGoogle Scholar
[24] Scheuerer, M. (2010). Regularity of the sample paths of a general second order random field. Stoch. Process. Appl. 120, 18791897.CrossRefGoogle Scholar
[25] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.CrossRefGoogle Scholar
[26] Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
[27] Serra, J. (ed.) (1988). Image Analysis and Mathematical Morphology, Vol. 2, Theoretical Advances. Academic Press, London.Google Scholar
[28] Stoyan, D. (1986). On generalized planar random tessellations. Math. Nachr. 128, 215219.CrossRefGoogle Scholar
[29] Stoyan, D., Kendall, W. S. and mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
[30] Villa, E. (2009). On the outer Minkowski content of sets. Ann. Mat. Pura Appl. (4) 188, 619630.CrossRefGoogle Scholar
[31] Villa, E. (2010). Mean densities and spherical contact distribution function of inhomogeneous Boolean models. Stoch. Anal. Appl. 28, 480504.CrossRefGoogle Scholar