Image Anal Stereol 2011;30:39-51
Original Research Paper
COMPUTATION OF THE PERIMETER OF MEASURABLE SETS VIA THEIR COVARIOGRAM. APPLICATIONS TO RANDOM SETS
Bruno Galerne
CMLA, ENS Cachan, CNRS, UniverSud, 61 Avenue du President Wilson, F-94230 Cachan e-mail: galeme@cmla.ens-cachan.fr
(Accepted February 16, 2011)
ABSTRACT
The covariogram of a measurable set /1C M'' is the function gA which to each yGM^ associates the Lebesgue measure of Ar\{y+A). This paper proves two formulas. The first equates the directional derivatives at the origin of gA to the directional variations of A. The second equates the average directional derivative at the origin of gA 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 stationaiy random closed set. As an illustration, the expected perimeter per unit volume of stationary Boolean models having any grain distribution is computed.
Keywords: Boolean model, covariogram, directional variation, random closed sets, set of finite perimeter, specific variation.
INTRODUCTION
The object of study of this paper is the covariogram gA of a measurable set C M'^ defined for all / G R'^ by gA{y) = {Af^{J+A)), where J^'^ 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 gA of an unknown set A, a general inverse problem is to determine the geometric information on A that gA 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^, 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 fianctions of bounded variation (Ambrosio et al, 2000). In this framework, the perimeter of a set A is defined by
Per(yl) = sup < / 1/1 (x) div (p (x) dx:
[jRd
and the directional variation in the direction u G ^ ofylis (Ambrosio etal, 2000, Section 3.11)
K(A) = sup I / ^ (V(p(x),u)dx:
< 1


id -nd
< 1
where (R'^, •) denotes the set of continuously differentiable fianctions with compact support. The non-specialist reader may ask how the perimeter Per(yl) is related to the (d - 1)-Hausdorff measure of the topological boundary d A, which one might consider to be the intuitive notion of surface area. Let us recall that if 4 is a compact set with Lipschitz boundary {e.g., is a convex body), then Per(yl) =	(dA), whereas in the general case
we only have Per(yl) <	(dA) (Ambrosio et al,
2000, Proposition 3.62). More precisely, if one defines the essential boundary d gA of A as the set of points of R'^ which are neither Lebesgue density points of A nor of the complementary of A, then dgA c dA and Per(y4) =	(d,A) < Jf'^-^ (dA) (Ambrosio etal.,
2000, Eq. 3.62). As shown in Chlebik (1997), the same conclusion holds for directional variations: defining the projection measure ßu in the direction u G S^^^ by
ßuiB) = !	(5n (x+Ru)) ^'^-i(dx),
for all measurable subsets B c R'^, one has = ßu{deA) < ßu{dA). In particular, if is a convex body then Vu{A) = IJf'^-^puiA)), where pu denotes the orthogonal projection with direction u.
Results. We prove that for every measurable set A of finite Lebesgue measure,
= (1)
r-^o	r	2
In addition, noting (^)' (0+) := lim
gA(ru)-gA(0)
r
the right directional derivatives at the origin of the covariogram, it is shown that
¥eriA) = -
1
COrf-l Jsd--^
(2)
where (Od-i denotes the Lebesgue measure of the unit ball in	Hence, for any measurable set A, the
perimeter Per(yl) can be computed from the directional derivatives at the origin of the covariogram ^/i. As a by-product, it is also shown that a measurable set A has finite perimeter if and only if its covariogram gA is Lipschitz, and in this case the Lipschitz constant is given by
Up{gA) = - sup Vu{A).
^ ueSd-i
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 etal, 1967; Matheron, 1975; 1986) that the right directional derivative at the origin of the covariogram gA 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 '^pj^ satisfying a condition of fiall-dimensionality, '^pp being the family of locally finite unions of sets with positive reach such that all their finite intersections also have positive reach^ In this more general framework, the surface area of the orthogonal projection is replaced by the total projection TPu{A) of A, the directional analogue of the (d- 1)-total curvature	of A (Federer,
1959). Eq. 1 thus imphes that if A is a fiall-dimensional compact '^Pie-set then Vu{A) = 2TPu{A). This identity is the directional equivalent of a recent result due to Ambrosio, Colesanti, and Villa (2008): a full-dimensional compact set with positive reach A satisfies Per(yl) =	(Ambrosio etal., 2008, Theorem
9) (one could directly prove that Vu{A) = 2TPu{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 Jf'^(A) < +0°, 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; Lantuejoul, 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(yl) 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 Sanction Yx{y) = The mean covariogram of a RACS X is related to the probability that two given points belong to A'according to the following relation
7xiy)= [ F{xeXandx+yeX)dx.
jRd
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 etal., 1995; Schneider and Weil, 2008), the shot noise model (Rice, 1977; Heinrich and Schmidt, 1985), or the dead leaves model (Matheron, 1968; Jeulin, 1997; Lantuejoul, 2002; Bordenave etal., 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 fx (see property 8 of Proposition 16).
If X is any stationary RACS, then its mean covariogram only takes values in {0, +00} and thus is always degenerate. Nevertheless Eqs. 1 and 2 also permit to study the mean variation of stationary RACS. Define the specißc directional variation Ov^iX) of Xas, the mean amount of directional variation of A'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


'We refer to (Federer, 1959) and (Rataj and Zähle, 2001) for definitions and results regarding sets with positive reach and 'Ž^p^j-sets respectively.
Again, integrating over all directions u, one obtains an expression of the specißc variation 6v{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) (Lantuejoul, 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(ruG 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 gA of a Lebesgue measurable set A is defined and several properties of gA are recalled and established. In particular it is shown that as soon as A is non negligible its covariogram gA is a strictly positive-definite fianction. The following section gathers several known results from the theory of fianctions 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
Definition 1 (Covariogram of a measurable set). Let
AdBß be a J^'^ -measurable set of ßnite Lebesgue measure. The covariogram of A is the function gA : R'^ ^ [0, +oo) defined for ally ^ R'^ by
gA{y)=J^'{Af^{J+Ä)) = f lA{x)lA{x-y)dx.
■JVß
As initially noted by Matheron (1965), the covariogram of A can be expressed as the convolution of the indicator fianctions of A and its symmetric A =
{-x\x(iA\.
gA = \A*^A-
As illustrated in the following proposition, this point of view is usefial to establish some analytic properties 0fS4-
Proposition 2. Let A (ZBß be a J^'^ -measurable set of finite Lebesgue measure and gA be its covariogram. Then
1.	For all yd R'^, 0 < gA{y) < gAiO) = J^'iA).
2.	gA is even: for all ye M'', gA{-y) = gA{y)-
3. f gA{y)&y=J^\A)\
■JVd
4. gA is uniformly continuous over R'^ lim gA{y) = 0.
and
Proof The proofs of the three first properties are straightforward. Since 1a and are in the fourth property is obtained in applying the LP-Z,^'-convolution theorem to gA = Ia* (see Hirsch and Lacombe, 1999, Proposition 3.2 p. 171, for example).	□
It is well-known that the covariogram is a positive-definite Sanction (Matheron, 1965, p. 22; Lantuejoul, 2002, p. 23). The next proposition improves slightly this result. In particular, it shows that for all x ^ 0,
gA{x)<gA{0).
Proposition 3 (Strict positive-definiteness of the covariogram). Let A be a J^'^ -measurable set such that 0 <	< +00. Then its covariogram gA is a strictly
positive-deßnite function, that is, for all integers p>l, for all p-tuples (/i,... ,yp) of distinct vectors ofW^,
and for all{wi,... ,Wp)	{0} we have
p
The proof of Proposition 3 following lemma.
makes use of the
Lemma 4 (The translations of an integrable function are linearly independent). Let f be a non null function of L^ (R'^) and let y\,... ,yp be p distinct vectors of R'^. Then the functions x ^ f{x+yj), J = I,..., p, are linearly independent in L^ (R'^) .
Proof. Let (ivi,..., iVp) G R^ be such that
p
^ wjf{x+yj) = 0 for a.e. xG I J=i
Applying the Fourier transform we have
£ wjS'yj^ VJ=I
=Oforall<^ g:
Since f is non null and integrable, / is non null and continuous. Hence there exists an open ball B =	of center g R'^ and radius r > 0
such that for all G	/ 0, and thus for
all ^ e B, := Yfj^^	= 0. One easily
shows that the sum 5(<^) is null for all G M"' in considering the one-dimensional restriction of 5 on the line containing ^ and : 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 5 = 0 shows that wßyj = 0. This implies that wi = ■■■ = Wp = Q, since by hypothesis the vectors yj are distinct. □
Proof of Proposition 3. By Lemma 4, the function
p
x^ X	yj) i® ^ot a.e. equal to 0. Hence
J=i
X ^j^kgA{yk-yj) j,i<=i
p	r
= X wj^k ^A{x)lA{x-yk+yj)dx j,k=i "'R
p	I-
= X wjWk 1a{x+yk)tA{x+yj)dx
f (' V
JR
u
Proposition 5. Let A (ZW' be a J^'^ -measurable set of ßnite Lebesgue measure and let gA be its covariogram. Then for ally z
\gA(y) -gA{z)\ < gAiO)-gAiy-z).
Proof First let us show that for all measurable sets Ai, A2, and
We have
< j^'^(AinA2) -j^'^iAi n/l2 n/13)
= j^'((AinA2)\(AinA2nA3)).
Now using that {Ai n A2) \ {Ai n >12 n yls) is included in the set \ ^3, Eq. 3 is proved. Applying Eq. 3 to the sets Ai = A, A2= y+ A and A3 = z+A we get
gA{y)-gA{z)=J^'{An{y+A))-J^'{An{z+A)) <	/1) - ((7+ /1) n (z+ /1))
= gA{0)-gA{z-y) .
□
Remark.
-	The weaker inequality
\gA(y)-gA(z)\<2(gA(0)-gA(y-z)) was estabhshed by Matheron (1986, p. 1).
-	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.
For any open subset U C M'^, [^{U) denotes the set of Borel subsets of U, and we write V dClUiiV d U is open and relatively compact in U.
Definition 6 (Functions of bounded variation). Let U
be an open set ofW^. We say that f G	^^
a function of locally bounded variation in U if the distributional derivative of f is representable by aW^-valued Radon measure, i.e., if there exists a W^-valued Radon measure, notedDf = (A /,..., L'rf /), such that for all(p = {(pu..., (pd) G {U, R'^)
< J^\A2\A3) = J^'{A2) -J^\A3nA2).
(3)	f fix)diY(pix)dx=-j;^ f (piix)Difidx). (4)
J U	J u
The vector space of all functions of locally bounded variation in U is denoted by BV\oi^{U). The functions
such that f Ü {U) and\Df\{U) < are said to be functions of bounded variation in U and the corresponding function space is denoted by BV{U).
In what follows, denotes the unit Euclidean sphere in R'^. If ^ G (/7, R) and u G S^-^, we write
and
d(p
(x) = (V(p(x),u}, xe:
for the directional derivative of cp in the direction u.
Definition 7 (Functions of bounded directional variation). Let U be an open set of R'^ and let u G	/ G	^^ ^ 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 Du f, such that for all G (R)
u
fix)
dcp
l{x)dx= -J^(p{x)Duf{dx) .
The corresponding space is denoted by BVu^\oc{U). The functions f G BVu,\o<,{U) such that feL^ (U) and \Duf\{U) < are said to be functions of bounded directional variation in the direction u in U and the corresponding function space is denoted byBVu{U).
The variation in of a function f G	is
defined by
Vi f,U) = sup <; fix) div(pix) dx:
(p G (U,
< 1
A fundamental result of the theory of function of bounded variation states that the variation V{ f, U) of feL\U) is finite if and only if / G BV{ U) (Ambrosio etal., 2000). More precisely,
V(f,U) =
1
\Df\{U) if feBV{U), +00 otherwise.
Similarly, for all f G	one defines the
directional variation in the direction u of /by
Vu{f,U) = snv^j^f{x)^{x)dx-.
Vu{m =
\
\Duf\{U)
if f(iBVu{U), otherwise.
If C R'^ is a ^'^-measurable set, by definition the perimeter of 4 in is the variation of the indicator function 1a in Uand one notes Per(yl, U) := V{1a,U). Besides, one writes Vu{A,U) := Vu{1a,U) for the directional variation of A in the direction u in U. In the case where U = R'^, one simply writes V{ f) = V (f, R'^) and Vu{f) = Vu{ f, R'^), and similarly for the variations of a set.
As shown by the next proposition, given all the directional variations Vu{f,U), uG	one can
compute the variation V{f,U).
Proposition 8 (Variation and directional variations).
Let U be an open set o/R'^ and let f^L^{U). Then, the three following assertions are equivalent:
(i) (W
(Hi)
feBViU).
f e BVui U) for all u e S^-K
For all vectors ey of the canonical basis, f^BVe,{U).
In addition.
\,V{f,U)<\^^Ve^{f,U)< sup Vu{f,U)<V{f,U) a	a
leSd-^
and
V{f,U) = ^ f	, (5)
where denotes the Lebesgue measure of the unit ballinM.'^-K
The results of this proposition are mostly from (Chlebik, 1997). A proof is reproduced below for the convenience of the reader First one needs the following lemma.
Lemma 9. Let / G L^{U) and let ui,..., U], e S^^^ such that f G BVuj{U) for all J = ..k. Then for all
u = YjXjUj G n span{ui,..., u^}, f e BVuiU).
Proof With the above notation one easily checks that Duf := Y.j^jDujf is a signed Radon measure which represents the directional derivative of f in the direction u. Besides, by the triangle inequality
\Duf\{U) < l'}\Xj\\Dujf\{U) < and thus /G BVuiU).	□
Proof of Proposition 8. Clearly, Assertion (ii) implies Assertion (iii). Let us show that (i) implies (ii). Let f G BV{U), let Df={Dif,...,Ddf) be the Radon measure representing its distributional derivative, and let	Then {Df, u) := Eti UiDJ is a signed
Radon measure which represents the directional derivative of f in the direction u, and by the Cauchy-Schwarz inequality
Vu{f,U) = \{Df,u)\{U) <
i=i
<\v\\Df\{U) = \Df\{U) = V{f,U)< .
Hence /G BVu{U) and Vu{ f,U) < V{ f,U). Let us now show that (iii) implies (i). For all vectors ey of the canonical basis, f G BVg. (M'^) and there exists a signed Radon measure Dg^ f which represents the distributional partial derivatives of f. But then one easily checks that {Dg^ f,---,Dgj f) is a R'^-valued Radon measure which represents the distributional derivative of f. In addition, from the definition of the variation
i=l
and thus f G BV{U). This concludes the proof of the announced equivalence as well as the proof of the inequalities since
V{f,U)<^ Ve,{f, U)<d SUV Vuif, U)<dV{f,U).
i=l	ueSd-^
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 G ^^^ for which f,U) < is contained in a linear subspace of dimension less than d — \ (otherwise f would be in BVu{U) for all u and consequently f would be in BV{U)). Hence for ^'^-i-all u in Vu{ f,U) = and thus the right-hand side of Eq. 5 equals +<=°. Let us now suppose that f G BV{U). By the polar decomposition theorem (Ambrosio et af, 2000, Corollary 1.29) there exists a unique IL'/l-integrable function o : U ^ S^-^ such that Df=o\Df\. With this notation, observe that for all u G ^^^ and A G
D,f{A) = {Df, u){A) = f {G{x), u)\Df]{dx) .
Ja
Hence, by (Ambrosio etaf, 2000, Proposition 1.23)
V,{f,U) = \D,f\{U)= [\{a{x),u)\\Df]{dx).
J u
^L. Ambrosio, personal communication.
For all V G ^ the following well-known identity holds
Jsd-^
Hence by Fubini's theorem

Sd-i
lu
= 2(0d-i\Df]{U)
□
The next proposition recalls fiandamental properties related to the approximation of Sanctions of bounded directional variation. For simplicity we restrict ourselves to the case U = R'^. See (Ambrosio et af, 2000, Section 3.11) for the proofs.
Proposition 10
approximation).
(Directional variation and
Variation of smooth
(R'^)nÜ (R'^) then
Vu{f) =
Sanctions: If f G
df
du

dx.
Lower semi-continuity with respect to the convergence: If fn converges towards f in Ü (R'^) then VJf) < liminfK,(4).
n-^+oo
Approximation by smooth fianctions: for every function f G BVu (R'^) , there exists a sequence of smooth functions fn G (R'') n BVu (R'') such that fn converges towards f in Ü (R'^) and lim Vu{ fn) = Vu{ f).
One practical advantage of directional variations Vu{ 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 fianctions theory^, 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 G and let f Ü (R'^) he any integrable function. Then for all r / 0,
\f{x+ru)-f{x)\
dx< K(f)
and
r-rohd	r	'
Proof To prove the inequality we can suppose that f G BVu (M'^). First suppose that / G (R'^) n BVu (M'^). Then
\f{x+ru)- f{x)\ =
<

/ r—{x+tru)dit Jo ou
{x+ tru)
d?.
Hence, using Fubini's theorem and the first point of Proposition 10,
\f{x+ru)-f{x)\
dx
<
\jRd
df
du
{x+ tru)
dx dt= Vu{f) .
y
This inequality is shown to be valid for any f G BVu (IK'^) by using approximation by smooth functions (see Proposition 10).
Let us now turn to the second part of the statement. Let / G (M'^) . Using the above inequality it is enough to show that
K,(/) <liminf /
\f{x+ru)-f{x)\
\r\
dx.
Let us consider a family of mollifiers {peJ^^Q, that is functions pe{x) := e^'^p{x/e) where the function p is even, non negative, with support contained in the unit ball, and such that /gd p{x)dx= 1. Define 4 = f* pe. Then 4 G (M'') n Ü (R'^) and /g converges towards /in Ü (R'^) as e tends to 0. By Fatou's lemma we have
K,(4)= I'
JRd
dfe
du

< lim inf
r-^O
dx
f,{x+ru)-f,{x)\
\r\
dx.
Since
\f,{- + ru)-f,{.)\\, = \\{f{. + ru)-f{.))*p,\\,
we deduce that for all e > 0
r-^0 .jRd	\r\
Using the lower semi-continuity of the directional variation with respect to the Ü -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 he a
measurable set having finite Lebesgue measure and let gA be its covariogram. Then for all ye R'^
gA{0)-gA{y) = \ f \lA{x+y)-lA{x)\ I JRf'
dx.
Proof
\lA{x+y)-lA{x)\dx
{lA{x+y)-lA{x)fdx 2{gA{0)-gA{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 J^'^ -measurable set having finite Lebesgue measure, let gA be its covariogram, and let u G S^^^. The following assertions are equivalent:
(i) A has finite directional variation Vu(A).
(a) lim
gA{0)-gA{ru)
exists and is finite.
+0 |r|
(iii) The one-dimensional restriction of the covariogram g^ : r^ gA{ru) is Lipschitz.
In addition,
the second equality being also valid when K, (yl) =
Remark. Note that Assertion (ii) of Theorem 13 above can be replaced by "The right directional derivative ^(0+) exists and is finite" since
^(0+) := hm
gA{ru) -gA{Qi)
-lim
r-^O
gA{^)-gA{ru)
\r\
Proof. Since from Lemma 12,
gA{^)-gA{ru) 1 r \lA{x+ru)-lA{x)\
2 Jvß
dx,
by applying Proposition 11 with f =1a 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 rand 5GR
\gA{ru) - gA{su) I <	- gA{{r-s) u)
2 jRd
Applying the inequality of Proposition 11 with f = 1a,
\gA{ru) -gA{su)\
\lA{x+{r-s)u)-lA{x)\
2	JR<

|r-5|
dx
Hence g^ is Lipschitz and Lip (g^) < ^ Vu{A).
Let us now show that (iii) implies (i). For all r / 0 we have
Lip(^) >
gA{0)-gA{ru)
1 f \lAix+ru)-lAix)\
dx.
Theorem 14 (Perimeter and covariogram of measurable sets). Let A be a J^'^ -measurable set having ßnite Lebesgue measure, and let gA be its covariogram. The following assertions are equivalent:
(i)	A has ßnite perimeter Per(yl).
(ii)	For all u G 5^-1,	:= gA{ru) -gA{())
lim
exists and is ßnite.
(iii) The covariogram gA is Lipschitz. In addition the following relations hold:
Lip(S4) = ^ sup Vu{A)<lFer{A)
ueSd-'^
and
¥eriA) = —
1
,, . (g"Ano+)jf'-\du), (6) 03dJsd-^
this last formula being also valid when Per(yl) = +0°.
Proof The equivalence of (i) and (ii) as well as the integral geometric formula of Eq. 6 derive from Proposition 8 and the identity
r	I
Let us now show that (i) implies (iii). Let 7, z G M'^. Denote by u the direction of such that y— z = \y— z\ u. By Proposition 5 and Theorem 13
\gAiy) -gAiz)\ < gA{0)-gA{y-z] /
\
<lv,{A)\y-z\<{l sup K(A) |7-z|.
^	V
Hence gA is Lipschitz and Lip(g4) < ^ sup^jVu{A). As for the converse implication and inequality, for all
ueS'-',
r-^o	|r|	2
Hence for all u e VuiA) < and Lip(^^) > i supy Vu{A). This concludes the proof	□
By Proposition 11 the right-hand side tends towards 5 Vu(A) as r tends to 0. Hence A has finite directional variation in the direction u and Lip (^) > ^ Vu{A). All in all we have shown that (i) and (iii) are equivalent andthatLip(^) = iK,(^).	□
Considering all the possible directions u G the results of the previous theorem lead to Eq. 2 (reproduced below as Eq. 6) and a characterization of sets of finite perimeter.
One natural question is whether Eq. 6 extends to the case of functions, that is if one can recover the variation V( f) of a fianction f from the directional derivatives of its covariogram gf{y) : = /gd f{x + y)f{}djdx. The answer to this question is negative. Indeed, if one considers a smooth function f G 'To} (R'^) , then its covariogram gf is well-defined and is differentiable in 0. But since gf is even, its derivative at the origin equals zero, and thus the variation of f is not equal to the integral of the directional derivatives of the covariogram gf.
APPLICATIONS TO RANDOM CLOSED SETS
MEAN COVARIOGRAM OF A RANDOM CLOSED SET
A random closed set (RACS) A' is a measurable map from a probability space (n,=s/,P) to the space (R'^) of closed subsets of R'^ endowed with the (T-algebra generated by the family {{Fg ^ (R'^),	0} , incompact} (Matheron,
1975; Molchanov, 2005; Stoyan et ah, 1995).
Definition 15 (Mean covariogram of a random closed set). Let X be a random closed set (RACS) of R'^ having ßnite mean Lebesgue measure, i.e. E i^Jsi"^ (X)) < +00. The mean covariogram jx of X is the expectation of the covariogram ofX with respect to its distribution, that is : R'^ ^ [0, is the function defined by
Yx{y) =ngx{y))
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(A^) and mean directional variations E{Vu{X)), u g of a RACS X
We say that a jointly measurable random field /: n X R'^ ^ R almost surely (a.s.) in (R'^) has a.s. locally bounded variation if there exists a random R'^-valued Radon measure^ Df which represents the distributional derivative of /, i.e., Eq. 4 holds a.s. If in addition f Ü (R'^) a.s. and \Df^ (R'^) is a.s. finite, then f is said to have a.s. bounded variation
field /a.s. in Ü
are defined by
m
Similarly, f : Q x
a.s. m
Li
loc
has a.s. locally bounded directional variation in the direction u G S^^^ if there exists a random signed Radon measure Duf representing the distributional directional derivative of /.If / G L^ (R'^) a.s. and \Duf \ (R"') < a.s., then one says that / has bounded directional variation in the direction u.
The mean variation E(l/(/)) and the mean directional variations E{Vu{f)), uG of a random
E(F(/)) =
and
E(K,(/)) =
E{\Df\ (R'^)) if /has a.s. bounded variation, otherwise,
Ei\Duf\ (R'^)) if /has a.s. bounded directional variation, otherwise.
Since any RACS X defines a jointly measurable random field by {(0,x) lx(o)){^) (see (Molchanov, 2005, p. 59)), the mean perimeter E(Per(Z)) and the mean directional variations E(	of a RACS Xare
well-defined.
Proposition 16 (Properties of the mean covariogram of a RACS). Let X be a RACS of R'^ satisfying E	< +00 and let Yx be its mean covariogram.
Then
1.	For all ye
2.	Yx is even.
3.	Yx{y) =
',0<Yx{y)<Yx{0)=E{j^'{X)).
F{xeX andx+yeX)dx.
4-	G [0,+H.
5.	//E {^'^{X)) > 0, then Yx is a strictly positive-definite function.
6.	For ally z
I rxiy) - Yx{z) I < Yx{0) - Yx{y- z).
7.
Yx is uniformly lim Yx{y) = 0.
continuous over R'^ and
8. We have

and, noting {Yx)' = lim
Yx{ru) - Yx{0)
1
(Od-l Jsd-^

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{Vu{X)) < and E(Per(Z)) < 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^{X)) = and E (Per(Z)) =
^We refer to (Horowitz, 1985) for definitions relative to random Radon measures and for a proof of the fact that the variation random Radon measure is a well-defined random positive Radon measure.
of a
SPECIFIC VARIATION OF A STATIONARY RACS
A RACS X is said to be stationary if for all 7 G R"', the translated RACS y+X has the same distribution as X. If a RACS X is stationary, one defines its variogram Vx as the fianction Vx{y) = P(7G A', 0 ^ Z) (see e.g., (Lantuejoul, 2002) for more details on variograms).
If a stationary RACS X has locally bounded variation, then one easily checks that its derivative Dlx, which is by definition a random R'^-valued Radon measure, is also stationary. Consequently, the variation measure \Dlx\ is a stationary positive Radon measure, and thus there exists a real number 6v{X) G [0, such that for all nonempty open sets U cc M'^,
E{V{X,U)) :=E{\mx\m = ev{X)J^\U) .
We choose to call this constant Ov(X) the specific variation of X or the variation intensity of X (see the discussion below). Similarly, for all u G there exists a real	G [0,+H such that
E{\D,lxm) = evSX)J^'{U). ev^X) IS called the specißc directional variation of X in the direction u (or also the directional variation intensity). As before, one extends the definition of the specific variation for stationary RACS A'which do not have a.s. locally bounded variation by setting Ov(X) = +0°, and similarly for the specific directional variation Ov^{X). In this context the integral-geometric formula of Eq. 5 gives
ev{X) = ^ f evSX)jf'-\du). 2(0^-1 Jsd-^
Theorem 17 (Specific variations and variogram). Let
X be a stationary RACS, let Vx be its variogram, and,
for all u^S'-K denote (v^)'(0+) := lim \x{ru).
r
Then for all u G S^^^ the specißc directional variation ev,{X) is given by
0K„(^=2(vl)'(O+)=21iml
r-^O r

fn other words, the specißc directional variation is twice the right directional derivative of the variogram at the origin. Integrating over all directions, one obtains the specißc variation ofX:
Ov{X). Eq. 7 is the formula given in (Lantuejoul, 2002, p. 26) and which originates from Matheron (1967, p. 30). In these references, the constant corresponding to the variation intensity 6v{X) is called the specißc {d — \)-volume of X {specißc perimeter if d = 2, specißc surface area if c/ = 3). However, in the later works of Matheron (1975) as well as on recent reference textbooks (Stoyan et al., 1995; Schneider and Weil, 2008), the specißc surface measure refers to the surface measure that derives from Steiner's formula. This measure has different names, depending on its normalization and the degree of generalization: intrinsic volume of index d — \ and Minkowski's content of index 1 for convex sets (Schneider and Weil, 2008), total curvature of index d- \ for sets with positive reach and '^Pie-sets (Federer, 1959; Rataj and Zähle, 2001), or also, in a more general setting, outer Minkowski content (Ambrosio et al, 2008; Villa, 2009; see also Hug et al, 2004). Even though the (variational) perimeter of a set and this notion of surface measure agree for convex sets (Ambrosio et al, 2008), the distinction is important. Indeed their extensions to non convex sets have different behaviors. For example, the outer Minkowski content counts twice the isolated fine parts of a set having a bounded and {d— l)-rectifiable topological boundary, whereas these fine parts have no influence on the perimeter (Villa, 2009, Proposition 4.1) (here "isolated fine parts" denotes the part of the boundary which has Lebesgue density 0). In order to make a clear distinction between the (variational) perimeter and the surface measure from Steiner's formula, the constant 6v{X) is named the specißc variation of X and not its "specific perimeter".
As mentioned in the introduction, one should notice that, contrary to the specific surface area (Schneider and Weil, 2008), the specific variation 6v{X) is well-defined for any stationary RACS. Besides, Theorem 17 shows that the specific directional variations 6v^{X) and the specific variation 6v{X) are easily computed as soon as one knows the variogram of X. This will be illustrated in the next section where the specific variations of stationary Boolean models are computed.
Let us now turn to the proof of Theorem 17 which uses the following intuitive lemma.
Lemma 18. Let A be a J^'^-measurable set and B be an open ball Then for all u G S^^^,
ev(X) =
1
(7)
(Od-l Jsd-l
Vu{A, B)<Vu	Vu{A, B) + Vu{B, R'').
Before proving this theorem let us discuss the terminology specißc variation of X for the constant
References for the proof The first inequality is immediate from the definition of the directional
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 (Chlebik, 1997). □
Proof of Theorem 17. First remark that
= P(OGZ)-P(OG^and - ru e X)
Let B be any nonempty open ball. Since A' is a stationary RACS


As Xf^ {ru+X) is also a stationary RACS, we have
FiOeXniru+X)) =

In order to introduce the mean covariogram of the set A'n B, let us denote E^ = {Xf^ B) n {ru+ {Xf^ B)). Clearly we have the following inclusions
Er(zXf^{ru+X)f^B
and
[Xf^{ru+X)f^B\\E,(zB\{Bf^{ru+B)) .
Noting that {B\ (5n {ru + B))) = ^5(0) - gB{ru), we obtain
jxnBirv) ^F^'{Xf^{ru+X)f^B))

^ Yxr\B{ru) ^ ^5(0) - gB{ru)


This yields both an upper and a lower bound of P(raG^,O^Z).Wehave

7X0^(0) - YxnB{ru)
By property 8 of Proposition 16 and the second inequality of Lemma 18,
As for the lower bound,
YxnßiO) - YxnB{ru) gB{0) - gB{ru)
>


Again, by Proposition 16 and the first inequality of Lemma 18, we have
r-^O |r| '
>
inVuiXr^B)) 1 Vu{B)
2 ^d^B) IJ^d^B)
The two established inequalities are true for any nonempty open ball B. Noting i?the radius of B,
Vu{B) (üö-iR'-' (Od-I 1

(Od R
The enunciated formula is obtained by letting i? 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 A and grain distribution Px is the stationary RACS Z defined by
Z=[jxi + Xi,
i'eN
where {xj, i G N} C M'^ is a stationary Poisson point process with intensity A > 0 and (A^Oy^fsj is a sequence of i.i.d. RACS with common distribution Px, independent of {xy,; G N}. Moreover, the RACS {Xi) are supposed to have a finite mean Lebesgue measure (otherwise Z a.s.). The avoiding functional of the Boolean model Z is well-known: for any compact C M'' we have
P(Zn = 0) = exp (-AE {X® K)
(8)
where X denotes a RACS with distribution Px and X®R= {x-y, x^X, y^K} (see, e.g., Stoyan et ah, 1995, p. 65 or Lantuejoul, 2002, p. 164). Starting from the general expression of Eq. 8 (which determines the distribution of Z), let us compute the variogram Vz of Z. For K = {0}, Eq. 8 becomes

^Z) =exp -AE ^
For K = {0, —ru}, with r / 0 and u G S'^ remark that we have
Hence in this case
E {X® K))=2E (X)) - rxiru) .
As a result the variogram v z is equal to (Stoyan et al, 1995, p. 68; Lantuejoul, 2002, p. 165)
vz{ru) = P(-ra G Zand 0 ^ Z)
= P(0 ^ Z) -P(Zn {0, -ru} = 0)
= g-exp (-A (2E {X)) - Yx{ru)
= q- gexp (-A {yxiO) - rxiru))) .
By Theorem 17 and property 8 of Proposition 16 we deduce
0K,(Z)=2(v|)'(O+)
= AE(K,(Z))exp f-AE (^'^{X)
Integrating this formula over all directions u we obtain 6v{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 A and grain distribution Px, let X be a RACS with distribution Px, and suppose that E {^''{X)) < +00. Then for allueS^-\
0i/„(Z) = AE(K,(Z))exp -AE
and
0i/(Z) =AE(Per(Z))exp -AE ^'^(Z)
(9)
Eq. 9 is valid for any grain distribution Px 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 fiarther 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.
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