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

The covariogram of a measurable set A ⊂ R d is the function g A which to each y ∈ R d associates the Lebesgue measure of A ∩ ( 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.


INTRODUCTION
The object of study of this paper is the covariogram g A of a measurable set A ⊂ R d defined for all y ∈ R d by g A (y) = L d (A ∩ (y + A)), where L d denotes the Lebesgue measure.Note that some authors prefer the terms set covariance or covariance function (Cabo and Janssen, 1994;Cabo and Baddeley, 1995;Rataj, 2004).
Given the covariogram g A of an unknown set A, a general inverse problem is to determine the geometric information on A that g A contains.As an important example, Averkov and Bianchi have recently established Matheron's conjecture: up to a translation and a reflection, convex bodies of R 2 , that is compact convex sets with non-empty interior, are fully determined by their covariogram (see (Averkov and Bianchi, 2009) and the references within).Contrary to the above mentioned result, this paper focuses on geometric information which is shown to be contained in the covariogram of any measurable set: the perimeter.
As our main results will demonstrate, the perimeter which can be computed from the covariogram is the one from the theory of functions of bounded variation (Ambrosio et al., 2000).In this framework, the perimeter of a set A is defined by and the directional variation in the direction u ∈ S d−1 of A is (Ambrosio et al., 2000, Section 3.11) where C 1 c R d , • denotes the set of continuously differentiable functions with compact support.The non-specialist reader may ask how the perimeter Per(A) is related to the (d − 1)-Hausdorff measure H d−1 of the topological boundary ∂ A, which one might consider to be the intuitive notion of surface area.Let us recall that if A is a compact set with Lipschitz boundary (e.g., A is a convex body), then Per(A) = H d−1 (∂ A), whereas in the general case we only have Per(A) ≤ H d−1 (∂ A) (Ambrosio et al., 2000, Proposition 3.62).More precisely, if one defines the essential boundary ∂ e A of A as the set of points of R d which are neither Lebesgue density points of A nor of the complementary of A, then ∂ e A ⊂ ∂ A and Per(A) = H d−1 (∂ e A) ≤ H d−1 (∂ A) (Ambrosio et al., 2000, Eq. 3.62).As shown in Chlebík (1997), the same conclusion holds for directional variations: defining the projection measure µ u in the direction u ∈ S d−1 by for all measurable subsets B ⊂ R d , one has V u (A) = µ u (∂ e A) ≤ µ u (∂ A).In particular, if A is a convex body then V u (A) = 2H d−1 (p u (A)), where p u denotes the orthogonal projection with direction u.
Results.We prove that for every measurable set A of finite Lebesgue measure, In addition, noting (g u A ) (0+) := lim r→0+ g A (ru) − g A (0) r the right directional derivatives at the origin of the covariogram, it is shown that Per where ω d−1 denotes the Lebesgue measure of the unit ball in R d−1 .Hence, for any measurable set A, the perimeter Per(A) can be computed from the directional derivatives at the origin of the covariogram g A .As a by-product, it is also shown that a measurable set A has finite perimeter if and only if its covariogram g A is Lipschitz, and in this case the Lipschitz constant is given by Previous work.Eq. 1 has already been proved for certain classes of sets.It was well-known by the mathematical morphology school (Matheron, 1965;Haas et al., 1967;Matheron, 1975;1986) that the right directional derivative at the origin of the covariogram g A of a convex body equals minus the surface area of the orthogonal projection of the set A. The convexity assumption was relaxed in (Rataj, 2004) where Rataj extends the result to compact sets in U PR satisfying a condition of full-dimensionality, U PR being the family of locally finite unions of sets with positive reach such that all their finite intersections also have positive reach 1 .In this more general framework, the surface area of the orthogonal projection is replaced by the total projection T P u (A) of A, the directional analogue of the (d − 1)-total curvature Φ d−1 (A) of A (Federer, 1959).Eq. 1 thus implies that if A is a full-dimensional compact U PR -set then V u (A) = 2T P u (A).This identity is the directional equivalent of a recent result due to Ambrosio, Colesanti, and Villa (2008): a fulldimensional compact set with positive reach A satisfies Per(A) = 2Φ d−1 (A) (Ambrosio et al., 2008, Theorem 9) (one could directly prove that V u (A) = 2T P u (A) by using the techniques developed in (Ambrosio et al., 2008) and (Rataj, 2004)).Since Eq. 1 is valid for any measurable set A such that L d (A) < +∞, one can argue that the directional variation is the relevant general concept when it comes to the derivative at the origin of the covariogram.
Eq. 2 has been widely stated in the mathematical morphology literature (Haas et al., 1967;Matheron, 1975;Serra, 1982;Lantuéjoul, 2002), under (more or less explicit) regularity assumptions on the set A. We rigorously show that it is valid for any measurable set A having finite Lebesgue measure, provided the perimeter Per(A) is understood as the variation of A.
The Lipschitz continuity of the covariogram seems to have received less attention in the literature.It is stated in (Matheron, 1986) that the covariogram of a compact convex set is Lipschitz and the upper bound of the Lipschitz constant given by Matheron is twice the actual value of this constant.
Applications.The covariogram is of particular importance in stochastic geometry when dealing with random closed sets (RACS) (Matheron, 1975;Stoyan et al., 1995;Molchanov, 2005;Schneider and Weil, 2008).In this context, one defines the mean covariogram of a RACS X as the function γ X (y) = E L d (X ∩ (y + X)) .The mean covariogram of a RACS X is related to the probability that two given points belong to X according to the following relation As a consequence the mean covariogram is systematically involved in second order statistics of classic germ-grain models, such as the Boolean model (Matheron, 1975;Stoyan et al., 1995;Schneider and Weil, 2008), the shot noise model (Rice, 1977;Heinrich and Schmidt, 1985), or the dead leaves model (Matheron, 1968;Jeulin, 1997;Lantuéjoul, 2002;Bordenave et al., 2006).
All the established properties of covariograms of deterministic sets extend to the case of mean covariograms of random closed sets.In particular, the stochastic equivalent of Eqs. 1 and 2 show that the expectations of the variations of a RACS X are proportional to the directional derivatives of its mean covariogram γ X (see property 8 of Proposition 16).
If X is any stationary RACS, then its mean covariogram only takes values in {0, +∞} and thus is always degenerate.Nevertheless Eqs. 1 and 2 also permit to study the mean variation of stationary RACS.Define the specific directional variation θ V u (X) of X as the mean amount of directional variation of X per unit volume (see Section "Specific variation of a stationary RACS" for a detailed definition).For any stationary RACS X, it is shown using Eq. 1 that Again, integrating over all directions u, one obtains an expression of the specific variation θ V (X) of X (i.e., the mean amount of variation of X per unit volume) As for Eq. 2, the above formula has been stated in the early works of Matheron (Matheron, 1967, p. 30) (Lantuéjoul, 2002, p. 26), but assumptions on the regularity of the RACS were not clearly formulated.It should be emphasized that the specific variation is well-defined for any stationary RACS, and that it can be easily computed as soon as one knows the probabilities P (ru ∈ X, 0 / ∈ X).As an illustration, the specific directional variations and the specific variation of stationary Boolean models are computed in this paper.The obtained expressions generalize known statistics of Boolean models with convex grains (Schneider and Weil, 2008).Because it is well-defined for any stationary RACS and easily computable, we claim that, when dealing with non negligible RACS, the specific variation is an interesting alternative to other extension of the usual specific surface area that derives from Steiner's formula (Schneider and Weil, 2008).
Contents.In Section "Covariogram of a measurable set" the covariogram g A of a Lebesgue measurable set A is defined and several properties of g A are recalled and established.In particular it is shown that as soon as A is non negligible its covariogram g A is a strictly positive-definite function.
The following section gathers several known results from the theory of functions of bounded directional variation.In Section "Directional variation, perimeter and covariogram of measurable sets", the main results relating both the derivative at the origin and the Lipschitz continuity of the covariogram of a set to its directional variations and its perimeter are stated.Finally, in the last section, applications of these results to the theory of random closed sets are discussed and illustrated.

COVARIOGRAM OF A MEASURABLE SET
As initially noted by Matheron (1965), the covariogram of A can be expressed as the convolution of the indicator functions of A and its symmetric Ǎ = {−x | x ∈ A }: As illustrated in the following proposition, this point of view is useful to establish some analytic properties of g A .
Proposition 2. Let A ⊂ R d be a L d -measurable set of finite Lebesgue measure and g A be its covariogram.Then Proof.The proofs of the three first properties are straightforward.Since 1 A and 1 Ǎ are in and Lacombe, 1999, Proposition 3.2 p. 171, for example).
It is well-known that the covariogram is a positivedefinite function (Matheron, 1965, p. 22;Lantuéjoul, 2002, p. 23).The next proposition improves slightly this result.In particular, it shows that for all x = 0, g A (x) < g A (0).

Proposition 3 (Strict positive-definiteness of the covariogram).
Let A be a L d -measurable set such that 0 < L d (A) < +∞.Then its covariogram g A is a strictly positive-definite function, that is, for all integers p ≥ 1, for all p-tuples (y 1 , . . ., y p ) of distinct vectors of R d , and for all (w 1 , . . ., The proof of Proposition 3 makes use of the following lemma. Lemma 4 (The translations of an integrable function are linearly independent).Let f be a non null function of L 1 R d and let y 1 , . . ., y p be p distinct vectors of Applying the Fourier transform we have Since f is non null and integrable, f is non null and continuous.Hence there exists an open ball B = B (ξ 0 , r) of center ξ 0 ∈ R d and radius r > 0 such that for all ξ ∈ B, f (ξ ) = 0, and thus for all ξ ∈ B, S(ξ ) := ∑ p j=1 w j e i ξ ,y j = 0.One easily shows that the sum S(ξ ) is null for all ξ ∈ R d in considering the one-dimensional restriction of S on the line containing ξ and ξ 0 : by the identity theorem, this one-dimensional function is null since it is analytic and null over an open interval.Applying the inverse generalized Fourier transform to S = 0 shows that ∑ p j=1 w j δ y j = 0.This implies that w 1 = • • • = w p = 0, since by hypothesis the vectors y j are distinct.
Proof of Proposition 3. By Lemma 4, the function Proof.First let us show that for all measurable sets A 1 , A 2 , and A 3 (3) We have Remark.
-The inequality of Proposition 5 shows that the Lipschitz continuity of the covariogram only depends on the behavior of the function in 0.

FACTS FROM THE THEORY OF FUNCTIONS OF BOUNDED DIRECTIONAL VARIATION
This section gathers necessary results from the theory of functions of bounded variation.For a general treatment of the subject we refer to the textbook of Ambrosio, Fusco, and Pallara (2000).When the enunciated properties of functions of bounded variation are not found in (Ambrosio et al., 2000), full proofs are given for the convenience of the reader.Let us add that these proofs can be skipped without impeding on the understanding of the main results of the paper that will be established in the next section.
The vector space of all functions of locally bounded variation in U is denoted by BV loc (U).The functions f ∈ BV loc (U) such that f ∈ L 1 (U) and |D f |(U) < +∞ are said to be functions of bounded variation in U and the corresponding function space is denoted by BV (U).
In what follows, S d−1 denotes the unit Euclidean sphere in for the directional derivative of ϕ in the direction u.
Definition 7 (Functions of bounded directional variation).Let U be an open set of R d and let u ∈ S d−1 .f ∈ L 1 loc (U) is a function of locally bounded directional variation in the direction u in U if the directional distributional derivative of f in the direction u is representable by a signed Radon measure, i.e., if there exists a signed Radon measure, noted D u f , such that for all The corresponding space is denoted by BV u,loc (U).
The functions f ∈ BV u,loc (U) such that f ∈ L 1 (U) and |D u f |(U) < +∞ are said to be functions of bounded directional variation in the direction u in U and the corresponding function space is denoted by BV u (U).
The variation in U of a function f ∈ L 1 loc (U) is defined by A fundamental result of the theory of function of bounded variation states that the variation et al., 2000).More precisely, Similarly, for all f ∈ L 1 loc (U) one defines the directional variation in the direction u of f by Besides, one writes V u (A,U) := V u (1 A ,U) for the directional variation of A in the direction u in U.In the case where U = R d , one simply writes and similarly for the variations of a set.
As shown by the next proposition, given all the directional variations V u ( f ,U), u ∈ S d−1 , one can compute the variation V ( f ,U).

Proposition 8 (Variation and directional variations).
Let U be an open set of R d and let f ∈ L 1 (U).Then, the three following assertions are equivalent: (iii) For all vectors e i of the canonical basis, f ∈ BV e i (U).
In addition, where ω d−1 denotes the Lebesgue measure of the unit ball in R d−1 .
The results of this proposition are mostly from (Chlebík, 1997).A proof is reproduced below for the convenience of the reader.First one needs the following lemma.
Proof.With the above notation one easily checks that D u f := ∑ j λ j D u j f is a signed Radon measure which represents the directional derivative of f in the direction u.Besides, by the triangle inequality Proof of Proposition 8. Clearly, Assertion (ii) implies Assertion (iii).Let us show that (i) implies (ii).Let f ∈ BV (U), let D f = (D 1 f , . . ., D d f ) be the Radon measure representing its distributional derivative, and let u ∈ S d−1 .Then D f , u := ∑ d i=1 u i D i f is a signed Radon measure which represents the directional derivative of f in the direction u, and by the Cauchy-Schwarz inequality Let us now show that (iii) implies (i).For all vectors e i of the canonical basis, f ∈ BV e i R d and there exists a signed Radon measure D e i f which represents the distributional partial derivatives of f .But then one easily checks that (D e 1 f , . . ., D e d f ) is a R d -valued Radon measure which represents the distributional derivative of f .In addition, from the definition of the variation and thus f ∈ BV (U).This concludes the proof of the announced equivalence as well as the proof of the inequalities since To finish let us show Eq. 5. First let us suppose that f / ∈ BV (U) and let us show that the right-hand side of Eq. 5 is equal to +∞.Remark that by Lemma 9 and the equivalence above the set of directions u ∈ S d−1 for which V u ( f ,U) < +∞ is contained in a linear subspace of dimension less than d − 1 (otherwise f would be in BV u (U) for all u and consequently f would be in BV (U)).Hence for H d−1 -all u in S d−1 , V u ( f ,U) = +∞, and thus the right-hand side of Eq. 5 equals +∞.Let us now suppose that f ∈ BV (U).By the polar decomposition theorem (Ambrosio et al., 2000, Corollary 1.29) Hence, by (Ambrosio et al., 2000, Proposition 1.23) For all ν ∈ S d−1 the following well-known identity holds Hence by Fubini's theorem The next proposition recalls fundamental properties related to the approximation of functions of bounded directional variation.For simplicity we restrict ourselves to the case U = R d .See (Ambrosio et al., 2000, Section 3.11) for the proofs.
-Variation of smooth functions: -Lower semi-continuity with respect to the L 1convergence: -Approximation by smooth functions: for every function f ∈ BV u R d , there exists a sequence of smooth functions One practical advantage of directional variations V u ( f ) over the non-directional variation V ( f ) is that it can be computed from the integrals of difference quotients, as the next proposition recalls.Although this is a standard result of BV functions theory2 , the author is not aware of any standard textbook which enunciates it.Consequently a proof is given for the convenience of the reader.
Proposition 11 (Directional variation and difference quotient).Let u ∈ S d−1 and let f ∈ L 1 R d be any integrable function.Then for all r = 0, and Proof.To prove the inequality we can suppose that Hence, using Fubini's theorem and the first point of Proposition 10, This inequality is shown to be valid for any f ∈ BV u R d by using approximation by smooth functions (see Proposition 10).
Let us now turn to the second part of the statement.Let f ∈ L 1 R d .Using the above inequality it is enough to show that Let us consider a family of mollifiers where the function ρ is even, non negative, C ∞ , with support contained in the unit ball, and such that Since Using the lower semi-continuity of the directional variation with respect to the L 1 -convergence we get the result.

DIRECTIONAL VARIATION, PERIMETER AND COVARIOGRAM OF MEASURABLE SETS
In this section, the main results of the paper are established (see Theorem 13 and Theorem 14).
Lemma 12 ( (Matheron, 1986)).Let A be a L dmeasurable set having finite Lebesgue measure and let g A be its covariogram.Then for all y The identity of Lemma 12, which is due to Matheron (1986), is the key point to apply the results from the theory of functions of bounded directional variations enunciated in the previous section.First, one establishes Eq. 1 and obtains a characterization of sets of finite directional variation.
Theorem 13 (Directional variation and covariogram of measurable sets).Let A be a L d -measurable set having finite Lebesgue measure, let g A be its covariogram, and let u ∈ S d−1 .The following assertions are equivalent: |r| exists and is finite.
(iii) The one-dimensional restriction of the covariogram g u A : r → g A (ru) is Lipschitz.In addition, the second equality being also valid when V u (A) = +∞.
Remark.Note that Assertion (ii) of Theorem 13 above can be replaced by "The right directional derivative g u A (0+) exists and is finite" since Proof.Since from Lemma 12, by applying Proposition 11 with f = 1 A one obtains the equivalence of (i) and (ii) as well as the formula Let us show that (i) implies (iii).By Proposition 5, for all r and s ∈ R Applying the inequality of Proposition 11 with Hence g u A is Lipschitz and Lip (g u A ) ≤ 1 2 V u (A).Let us now show that (iii) implies (i).For all r = 0 we have

APPLICATIONS TO RANDOM CLOSED SETS MEAN COVARIOGRAM OF A RANDOM CLOSED SET
A random closed set (RACS) X is a measurable map from a probability space (Ω, A , P) to the space F R d of closed subsets of R d endowed with the σ -algebra generated by the family (Matheron, 1975;Molchanov, 2005;Stoyan et al., 1995).
Definition 15 (Mean covariogram of a random closed set).Let X be a random closed set (RACS) of R d having finite mean Lebesgue measure, i.e.E L d (X) < +∞.The mean covariogram γ X of X is the expectation of the covariogram of X with respect to its distribution, that is γ As the next proposition will show, all the results relative to covariograms of deterministic measurable sets can be adapted for mean covariograms of RACS.However before stating these results, we need to introduce the notions of mean perimeter E(Per(X)) and mean directional variations E(V u (X)), u ∈ S d−1 , of a RACS X.
We say that a jointly measurable random field f : Ω × R d → R almost surely (a.s.) in L 1 loc R d has a.s.locally bounded variation if there exists a random R d -valued Radon measure3 D f which represents the distributional derivative of f , i.e., Eq. 4 holds a.s.

If in addition
loc R d has a.s.locally bounded directional variation in the direction u ∈ S d−1 if there exists a random signed Radon measure D u f representing the distributional directional derivative of f .If f ∈ L 1 R d a.s. and |D u f | R d < +∞ a.s., then one says that f has bounded directional variation in the direction u.
The mean variation E(V ( f )) and the mean directional variations E(V u ( f )), u ∈ S d−1 , of a random field f a.s. in L 1 R d are defined by and Since any RACS X defines a jointly measurable random field by (ω, x) → 1 X(ω) (x) (see (Molchanov, 2005, p. 59)), the mean perimeter E(Per(X)) and the mean directional variations E(V u (X)) of a RACS X are well-defined.
Proposition 16 (Properties of the mean covariogram of a RACS).Let X be a RACS of R d satisfying E L d (X) < +∞ and let γ X be its mean covariogram.Then 5. If E L d (X) > 0, then γ X is a strictly positivedefinite function.
8. We have The proofs are omitted since they mostly consist in integrating the results of the previous sections with respect to the distribution of the RACS X.When E (V u (X)) < +∞ and E (Per(X)) < +∞, both formulas of property 8 follows easily from the bounded convergence theorem.Using Fatou's lemma, one shows that these formulas are also valid when E (V u (X)) = +∞ and E (Per(X)) = +∞.
variation on an open set (Ambrosio et al., 2000).The second inequality is easily proved using the interpretation of the directional variation as the projection measure of the essential boundary mentioned in the introduction (Chlebík, 1997).
Proof of Theorem 17.First remark that Let B be any nonempty open ball.Since X is a stationary RACS As X ∩ (ru + X) is also a stationary RACS, we have In order to introduce the mean covariogram of the set X ∩ B, let us denote E r = (X ∩ B) ∩ (ru + (X ∩ B)).
Clearly we have the following inclusions This yields both an upper and a lower bound of P (ru ∈ X, 0 / ∈ X).We have .
By property 8 of Proposition 16 and the second inequality of Lemma 18, lim sup r→0 As for the lower bound, Again, by Proposition 16 and the first inequality of Lemma 18, we have lim inf The two established inequalities are true for any nonempty open ball B. Noting R the radius of B, The enunciated formula is obtained by letting R tends to +∞.

COMPUTATION OF THE SPECIFIC VARIATIONS OF BOOLEAN MODELS
In this section we apply Theorem 17 to compute the specific directional variations and the specific variation of any stationary Boolean model.The Boolean model (Stoyan et al., 1995;Schneider and Weil, 2008) with intensity λ and grain distribution P X is the stationary RACS Z defined by where {x i , i ∈ N} ⊂ R d is a stationary Poisson point process with intensity λ > 0 and (X i ) i∈N is a sequence of i.i.d.RACS with common distribution P X , independent of {x i , i ∈ N}.Moreover, the RACS (X i ) are supposed to have a finite mean Lebesgue measure (otherwise Z = R d a.s.).The avoiding functional of the Boolean model Z is well-known: for any compact K ⊂ R d we have where X denotes a RACS with distribution P X and X ⊕ Ǩ = {x − y, x ∈ X, y ∈ K} (see, e.g., Stoyan et al., 1995, p. 65 or Lantuéjoul, 2002, p. 164).Starting from the general expression of Eq. 8 (which determines the distribution of Z), let us compute the variogram ν Z of Z.For K = {0}, Eq. 8 becomes q := P (0 / ∈ Z) = exp −λ E L d (X) .
By Theorem 17 and property 8 of Proposition 16 we deduce Integrating this formula over all directions u we obtain θ V (Z).Our computation is summarized in the following statement.
Proposition 19 (Specific variations of a stationary Boolean model).Let Z be the Boolean model with Poisson intensity λ and grain distribution P X , let X be a RACS with distribution P X , and suppose that E L d (X) < +∞.Then for all u ∈ S d−1 , and Eq. 9 is valid for any grain distribution P X and generalizes known results for Boolean models with convex grains (Schneider and Weil, 2008, p. 386).Similar generalizations involving intensity of surface measures deriving from Steiner's formula have recently been established (Hug et al., 2004;Villa, 2010).As already stressed out, our result is similar but not identical since the outer Minkowski content of a set differs from its (variational) perimeter (Villa, 2009).
A promising direction for further works is to extend the notion of specific variation for non stationary RACS.In particular, following (Villa, 2010), one could try to derive local variation densities of certain non stationary Boolean models.
For any open subset U ⊂ R d , B(U) denotes the set of Borel subsets of U, and we writeV ⊂⊂ U if V ⊂ U is open and relatively compact in U.Definition 6 (Functions of bounded variation).Let U be an open set of R d .We say that f ∈ L 1 loc (U) is a function of locally bounded variation in U if the distributional derivative of f is representable by a R dvalued Radon measure, i.e., if there exists a R d -valued Radon measure, noted D f = (D 1 f , . . ., D d f ), such that for all ϕ by definition the perimeter of A in U is the variation of the indicator function 1 A in U and one notes Per(A,U) := V (1 A ,U).