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Image Analysis and Stereology 30 (2011) pages 39-51

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v1 (08-05-2010)

v2 (23-06-2010)

v3 (13-05-2011)

Computation of the Perimeter of Measurable Sets via their Covariogram. Applications to Random Sets

Bruno Galerne 1

(28/03/2011)

The covariogram of a measurable set $A\subset\R^d$ is the function $g_A$ which to each $y\in\R^d$ associates the Lebesgue measure of $A\cap (y+A)$. This paper proves two formulas. The first equates the directional derivatives at the origin of $g_A$ to the directional variations of $A$. The second equates the average directional derivative at the origin of $g_A$ to the perimeter of $A$. These formulas, previously known with restrictions, are proved for any measurable set. As a by-product, it is proved that the covariogram of a set $A$ is Lipschitz if and only if $A$ has finite perimeter, the Lipschitz constant being half the maximal directional variation. The two formulas have counterparts for mean covariogram of random sets. They also permit to compute the expected perimeter per unit volume of any stationary random closed set. As an illustration, the expected perimeter per unit volume of stationary Boolean models having any grain distribution is computed.

1 :	Centre de Mathématiques et de Leurs Applications (CMLA)
	CNRS : UMR8536 – École normale supérieure de Cachan - ENS Cachan

Domaine

:

Mathématiques/Analyse classique Mathématiques/Probabilités Mathématiques/Analyse fonctionnelle

Mots Clés : Boolean model – Covariogram – Directional variation – Random closed sets – Set covariance – Set of finite perimeter – Specific variation

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Contributeur : Bruno Galerne <>

Soumis le : Vendredi 13 Mai 2011, 10:14:28

Dernière modification le : Vendredi 13 Mai 2011, 12:48:24