Bicovariograms and Euler characteristic of regular sets
We establish an expression of the Euler characteristic of a -regular planar set in function of some variographic quantities. The usual framework is relaxed to a regularity assumption, generalising existing local formulas for the Euler characteristic. We give also general bounds on the number of connected components of a measurable set of in terms of local quantities. These results are then combined to yield a new expression of the mean Euler characteristic of a random regular set, depending solely on the third order marginals for arbitrarily close arguments. We derive results for level sets of some moving average processes and for the boolean model with non-connected polyrectangular grains in . Applications to excursions of smooth bivariate random fields are derived in the companion paper , and applied for instance to Gaussian fields, generalising standard results.
keywords: Euler characteristic, intrinsic volumes, shot noise processes, boolean model
2010 MSC classification: 52A22, 60D05, 28A75, 60G10
Physicists and biologists are always in search of numerical indicators reflecting the microscopic and macroscopic behaviour of tissue, foams, fluids, or other spatial structures. The Euler characteristic, also called Euler-Poincaré characteristic, is a favoured topological index because its additivity properties make it more manageable than connectivity indexes or Betti numbers. It is defined on a set by
It is more generally an indicator of the regularity of the set, as an irregular structure is more likely to be shredded in many small pieces, or pierced by many holes, which results in a large value for .
As an integer-valued quantity, the Euler characteristic can be easily measured and used in estimation and modelisation procedures. It is an important indicator of the porosity of a random media [7, 33, 19], it is used in brain imagery [23, 37], astronomy, [27, 31, 15], and many other disciplines. See also  for a general review of applied algebraic topology. In the study of parametric random media or graphs, a small value of indicates the proximity of the percolation threshold, when that makes sense. See , or  in the discrete setting.
The mathematical additivity property is expressed, for suitable sets and , by the formula , which applies recursively to finite such unions. In the validity domain of this formula, the Euler characteristic of a set can therefore be computed by summing local contributions. The Gauss-Bonnet theorem formalises this notion for manifolds, stating that the Euler characteristic of a smooth set is the integral along the boundary of its Gaussian curvature. Exploiting the local nature of the curvature in applications seems to be a geometric challenge, in the sense that it is not always clear how to express the mean Euler characteristic of a random set under the form
where only depends on , where is the ball with center and radius . We propose in this paper a new formula of the form above, based on variographic tools, and valid beyond the realm, and then apply it in a random setting. This paper is completely oriented towards probabilistic applications, it is not clear wether our formula has important implications in a purely geometric framework.
In stochastic geometry and stereology, an important body of literature is concerned with providing formulas for computing the Euler characteristic of random sets, see for instance [21, 32, 24, 29] and references therein. Defined to be for every convex body, it is extended by additivity as
for finite unions of such sets. Even though this formula seems highly non-local, it is possible to express it as a sum over local contributions using the Steiner formula, see (2.3) in , but it is difficult to apply it under this form. There has also been an intensive research around the Euler characteristic of random fields excursions [2, 8, 15, 16, 10, 37], based upon the works of Adler, Taylor, Sammorodnitsky, Worsley, and their co-authors, see the central monograph . We discuss in the companion paper  the application of the present results to level sets of random fields.
In this work, we give a relation between the Euler characteristic of a bounded subset of and some variographic quantities related to . Given any two orthogonal unit vectors , for sufficiently small,
where is the -dimensional Lebesgue measure. This formula is valid under the assumption that is , i.e. that is a submanifold of with Lipschitz normal and finitely many connected components. See Example 9 for the application of this formula to the unit disc.
In the context of a random closed set , call the right-hand member of (3). If is finite, the value of can be obtained as . The main asset of this formulation regarding classical approaches is that, to compute the mean Euler characteristic, one only needs to know the third-order marginal of , i.e. the value of
for arbitrarily close. We also give similar results for the intersection , where is a random regular closed set and is a rectangular (or poly-rectangular) observation window. This step is necessary to apply the results to a stationary set sampled on a bounded portion of the plane.
In the present paper, we apply the principles underlying these formulas to obtain the mean Euler characteristic for level sets of moving averages, also called shot noise processes, where the kernels are the indicator functions of random sets which geometry is adapted to the lattice approximation. Even though the geometry of moving averages level sets attracted interest in the recent literature [3, 12], no such result seemed to exist in the literature 222A more general result has now been derived by Biermé and Desolneux . As a by-product, the mean Euler characteristic of the associated boolean model is also obtained.
These formulas are successfully applied to excursions of smooth random fields in the companion paper . For instance, in the context of Gaussian fields excursions, one can pass (3) to expectations under the requirement that the underlying field is , i.e. in the context of bivariate functions with Lipschitz derivatives, plus additional moment conditions. This improves upon the classical theory  where fields have to be of class and satisfy a.s. Morse hypotheses. Here again, the resulting formulas only require the knowledge of the field’s third order marginals for arbitrarily close arguments.
Equality (3) gives in fact a direct relation between the Euler characteristic, also known as the Minkowski functional of order , and the function . We call the latter function bicovariogram of , or variogram of order , in reference to the covariogram of , defined by (see [26, 18] or  for more on covariograms). Let be the normalized Haar measure on the -dimensional circle . The formula
developped in the context of random sets by Galerne , and originating from the theory of functions of bounded variations , gives a direct relation between the first order variogram, and the perimeter of a measurable set , which is also the Minkowski functional of order in the vocabulary of convex geometry. Completing the picture with the fact that is at the same time the second-order Minkowski functional and the variogram of order , it seems that covariograms and Minkowski functionals are intrinsically linked. This unveils a new field of exploration, and raises the questions of extension to higher dimensions, with higher order variograms, and all Minkowski functionals.
The present work is limited to the dimension because, before engaging in a general theory, one must check that, at least in a particular case of interest, existing results are improved. In the present work and the companion paper, the results are oriented towards excursions of bivariate Gaussian fields, as they are of high interest in the literature. Despite the technicalities and difficulties, coming mainly from - 1 - the expression of topological estimates in terms of the regularity of the field, and - 2 - dealing with boundary effects, obtaining a formula valid for any random model, and relaxing the usual hypotheses to assumptions, provides a sufficiently strong motivation for pushing the theory further. Also, developing methods of proof and upper bounds in dimension will help developing them in more abstract spaces.
Another motivation of the present work is that the amount of information that can be retrieved from the variogram of a set is a central topic in the field of stereology, see for instance the recent work  completing the confirmation of Matheron’s conjecture. Through relation (3), the data of the bicovariogram function with arguments arbitrarily close to is sufficient to derive its Euler characteristic, and once again the extension to higher dimensions is a natural interrogation.
The paper is organised as follows. We give in Section 1 some tools of image analysis, and the framework for stating our main result, Theorem 7, which proves in particular (3). These results are used to derive the mean Euler characteristic of shot noise level sets and boolean model with polyrectangular grain. We then provide in Theorem 12 a uniform bound for the number of connected components of a digitalised set, useful for applying Lebesgue’s Theorem. In Section 2, we introduce random closed sets and the conditions under which the previous results give a convenient expression for the mean Euler characteristic. Theorem 15 states hypotheses and results for homogeneous random models.
1 Euler characteristic of regular sets
Given a measurable set of , denote by the family of bounded connected components of , i.e. the bounded equivalence classes of under the relation “ is connected to if there is a continuous path such that ”. We use the notation to indicate the image of such a path. Call the class of sets of such that and are finite. We call the sets of the admissible sets of , and define for ,
The present paper is restricted to the dimension , we therefore will not go further in the algebraic topology and homology theory underlying the definition of the Euler characteristic. The aim of this section is to provide a lattice approximation of for which has a tractable expression, and explore under what hypotheses on we have as .
For , call its components in the canonical basis. Also denote, for , by the segment delimited by , and .
1.1 Euler characteristic and image analysis
Practitioners compute the Euler characteristic of a set from a digital lattice approximation , where is close to . The computation of is based on a linear filtering with a patch containing pixels, see [29, 35, 36, 22]. Determining wether is a problem with a long history in image analysis and stochastic geometry.
For , call the square lattice with mesh , and say that two points of are neighbours if they are at distance (with the additional convention that a point is its own neighbour). Say that two points are connected if there is a finite path of connected points between them. If the context is ambiguous, we use the terms grid-neighbour,grid-connected, to not mistake it with the general connectivity. Call the class of finite (grid-)connected components of a set . We define in analogy with the continuous case, for bounded such that are finite,
where . Remark in particular that two connected components touching exclusively through a corner are not grid-connected.
Call the two canonical unit vectors or , and define for ,
Seeing also these functionals as discrete measures, define
The subscripts in and out refer to the fact that counts the number of vertices of pointing outwards towards North-East, and is the number of vertices pointing inwards towards South-West. Define for
The functional is intended to count the number of -configurations. Such configurations are a nuisance for obtaining the Euler characteristic by summing local contributions. Call the class of bounded such that , with .
It is well known that, viewing as a subgraph of , the Euler characteristic of can be computed as where is the number of vertices of , is its number of edges, and is the number of facets, i.e. of points such that . We therefore have
For each , the summand is in and can be computed in function of the configuration
. Enumerating all the possible configurations and noting that the configurations and do not occur due to the assumption , it yields that indeed only the configurations give and only the configurations give , which gives the conclusion. ∎
This formula amounts to a linear filtering of the set by a discrete patch, and is already known and used in image analysis and in physics on discrete images. Analogues of this formula [6, 29] exist also in higher dimensional grids, but the dimension seems to be the only one where an anisotropic form is valid, see  for a discussion on this topic. The anisotropy is not indispensable to the results discussed in this paper, but gives more generality and simplifies certain formulas. An isotropic formula can be obtained by averaging over the directions.
Given a subset of and , we are interested here in the topological properties of the Gauss digitalisation of , defined by . Define , and
is the Gauss reconstruction of based on . In some unambiguous cases, the notation is simplified to . In this paper, we refer to a pixel as a set , for not necessarily in .
We also use the notation, for , .
For , is connected in if is grid-connected in . The converse might not be true because of pixels touching through a corner, but this subtlety does not play any role in this paper, because sets with -configurations are systematically discarded. We also have if because connected components of (resp. ) can be uniquely associated to grid-connected components of (resp. ).
Most set operations commute with the operators . For any , and those properties are followed by the reconstructions
1.2 Variographic quantities
The question raised in the next section is wether, for , for sufficiently small, and the result depends crucially on the regularity of ’s boundary. A remarkable asset of formula (4) is its nice transcription in terms of variographic tools. Let us introduce the related notation. For , a measurable subset of , define the polyvariogram of order ,
The variogram of order is known as the covariogram of (see [26, Chap. 3.1]), and we designate here by bicovariogram of the polyvariogram of order . A polyvariogram of order can be written as a linear combination of variograms with orders for appropriate numbers . For instance for measurable with finite volume, we have
A similar notion can be defined on endowed with the counting measure: for ,
Lemma 1 directly yields for ,
We will see in the next section that for a sufficiently regular set , the analogue equality holds for small.
1.3 Euler characteristic of -regular sets
It is known in image morphology that the digital approximation of the Euler characteristic is in general badly behaved when the set possesses some inwards or outwards sharp angles, i.e. we don’t have as , the boolean model being the typical example of such a failure, see [34, Chap. XIII - B.6] or . Sets nicely behaved with respect to digitalisation are called morphologically open and closed (MOC), or -regular, see [34, Chap.V-C],.
Before giving the characterisation of such sets, let us introduce some morphological concepts, see for instance [26, 34] for a more detailed account of mathematical morphology. We state below results in because the arguments are based on purely metric considerations that apply identically in any dimension.
The ball with centre and radius in the -metric of is noted . The Euclidean ball with centre and radius is noted . For , define
We also note for resp. the topological boundary, closure, and interior of a set .
Note the unit circle in .
Say that a closed set has an inside rolling ball if for each , there is a closed Euclidean ball of radius contained in such that , and say that has an outside rolling ball if has an inside rolling ball.
A set has reach at least if for each point at distance from , there is a unique point such that . We note in this case . Call reach of the supremum of the such that has reach at least . The proposition below gathers some elementary facts about sets satisfying those rolling ball properties, the proof is left to the reader.
Let and be a closed set of with an inside and an outside rolling ball of radius . Then there is an outwards normal vector in each . For , resp. is the unique inside, resp. outside rolling ball in . Also, . Furthermore, and have reach at least for each .
We reproduce here partially the synthetic formulation of Blashke’s theorem by Walther , which gives a connection between rolling ball properties and the regularity of the set.
Theorem 3 (Blashke).
Let be a compact and connected subset of . Then for the following assertions are equivalent.
is a compact -dimensional submanifold of such that the mapping , which associates to its outward normal vector to , is -Lipschitz,
has inside and outside rolling ball of radius ,
Let be a compact set of . Assume that has finitely many connected components and satisfies either (i),(ii) or (iii) for some . Since the connected components of are at pairwise positive distance, each of them satisfies (i),(ii), and (iii), and therefore the whole set satisfies (i),(ii) and (iii) for some , which might be smaller than . Such a set is said to belong to Serra’s regular class, see the monograph of Serra . We will say that such a set is -regular, or simply regular.
An aim of the present paper is to advocate the power of covariograms for computing the Euler characteristic of a regular set in the plane, we therefore need to give results that can be compared with the literature. Since many applications are concerned with stationary random sets on the whole plane, we have to study the intersection of random regular sets with bounded windows, and assess the quality of the approximation.
To this end, call admissible rectangle of any set where and are closed (and possibly infinite) intervals of with non-emtpy interior, and note its corners, which number is between and . Then call polyrectangle a finite union where each is an admissible rectangle, and for . Call the class of admissible polyrectangles. For such , denote by the elements of that lie on . For , note the outward normal unit vector to in . We also call edge of a maximal segment of , i.e. a segment that is not strictly contained in another such segment of (in this case, ). Also call the total length of edges where the normal vector is collinear to , and remark that the total perimeter of is .
Using the considerations of Section 1.1, it is obvious that, for , is the cardinality of , which is formed by north-east outwards corners of , minus the cardinality of , the set of south-west inwards corners of . This remark can be used to compute the mean Euler characteristic of random sets living in .
Shot noise processes
Let be a probability measure on , and a probability measure on such that . Let be a Poisson measure on with intensity measure , where is Lebesgue measure on . Introduce the random field
To make sure that the process is well defined a.s., assume that
Then the law of is stationary, i.e. invariant under spatial translations.
This type of process is called a shot noise process, or moving average, and is used in image analysis, geostatistics, or many other fields. The geometry of their level sets are also the subject of a heavy literature, see for instance the recent works [3, 12], but no expression seems to exist yet for the mean Euler characteristic. We provide below such a formula under some weak technical assumptions, see Section 3.1 for a proof.
Let . Let , independent random variables with distribution respectively. Introduce the level set and assume that . Introduce
For instance, if is a Dirac mass in and is a Dirac mass in , a.s. and is a Poisson variable with parameter . The volumic part of the mean Euler characteristic (i.e. the one multiplied by ) is therefore
For and the Dirac mass in , has the law of the boolean model with random grain distributed as the and random germs given by , therefore formula (5) provides, with ,
Coming back to the Euler characteristic of smooth sets of , the following assumption needs to be in order for the restriction of a -regular set to a polyrectangle to be topologically well behaved.
Let be a -regular set, and . Assume that and that for , is not colinear with .
If a -regular set and a polyrectangle do not satisfy this assumption, might have an infinity of connected components, which makes the Euler characteristic not properly defined. Let us prove that the digitalisation is consistent if this assumption is in order.
Let be a -regular set of , satisfying Assumption 6 such that is bounded. Then and there is such that for ,
Also, for .
The proof is at Section 3.2.
The apparent anisotropy of (7)-(7) can be removed by averaging over all pairs of orthogonal unit vectors of . Even though (7) does not involve the discrete approximation, a direct proof not exploiting lattice approximation is not available yet, and such a proof might shed light on the nature of the relation between covariograms and Minkowski functionals.
The fact that the Euler characteristic of a regular set digitalisation converges to the right value is already known, see [36, Section 6] and references therein, but we reprove it in Lemma 18, under a slightly stronger form. One of the difficulties of the proof of Theorem 7 is to deal with the intersection points of and .
It is proved in Svane  that in higher dimensions, Euler characteristic and Minkowski functionals of order can be approximated through isotropic analogues of formula (4). The arguments of the proof of Lemma 18, treating the case , are purely metric and should be generalisable to higher dimensions. On the other hand, dealing with boundary effects in higher dimensions might be a headache.
Before giving the proof, let us give an elementary graphical illustration of (7) with in . Let . We note and . We should have for small
, and . The notation designate six distinct subsets (see Figure 1, below) such that . Symmetry arguments yield that , whence
The shape of is very close to that of a cube with diagonal length , i.e. with side length . Therefore , which confirms (rigorously proved by Theorem 7).
It should be possible to show that under the conditions of Theorem 7, and are homeomorphic, but we are only interested in the Euler characteristic in this paper.
It seems difficult to deal with manifolds that don’t have a Lipschitz boundary, in a general setting. Consider for instance in
Then is a embedded sub manifold of , but it has infinitely many connected components, which puts off the class of sets that we consider admissible for computing the Euler characteristic.
To have the convergence of Euler characteristic’s expectation for random regular sets, we need the domination provided by Theorem 12 in the next section.
1.4 Bounding the number of components
Taking the expectation in formula (7) and switching with the limit requires a uniform upper bound in on the right hand side. For small, (7) consists of a lot of positive and negative terms that cancel out. Since grouping them manually is quite intricate, this formula is not suitable for obtaining a general upper bound on . The most efficient way consists in bounding the number of components of and in terms of the regularity of the set.
The result derived below is intended to be applied to -regular sets, but we cannot make any assumption on the value of the regularity radius , because the bound must be valid for every realisation. We therefore give an upper bound on and valid for any measurable set .
The formula obtained bounds the number of connected components, which is a global quantity, in terms of occurrences of local configurations of the set, that we call entanglement points. Roughly, an entanglement occurs if two points of are close but separated by a tight portion of , see Figure 2. This might create disconnected components of in this region although is locally connected.
To formalise this notion, let grid neighbours. Introduce the closed square with side length such that and are the midpoints of two opposite sides. Denote , which has two connected components. Then is an entanglement pair of points of if and is connected. We call the family of such pairs of points.
For the boundary version, given we also consider grid points , on the same line or column of , such that
are within distance from one of the edges of (the same edge for and )
The family of such pairs of points is noted .
Even though and are not points but pairs of points of , for , we extend the notation , to indicate that the points of the pairs of are contained in (resp. the collection of pairs of points from where both points are contained in ), and idem for .
For and . Therefore . We have also .
Let be a bounded measurable set. Then
and for any ,
The proof is deferred to Section 3.3.
The boundary of a -regular set is a manifold, and can therefore be written under the form , and for some function such that on and is -Lipschitz on . Such a function is said to be of class , see . One can bound the right hand members of (8)-(9) by quantities depending solely on . For instance, it is proved in the companion paper  that in the context of Gaussian fields, and can be switched in (7) if the derivatives of are Lipschitz and their Lipschitz constants have a finite moment of order for some .
2 Random sets
Let be a complete probability space. Call the class of closed sets of , endowed with the -algebra generated by events , for open. A -measurable mapping is called a Random Closed Set (RACS). See  for more on RACS, and equivalent definitions. The functional is not properly defined, and therefore not measurable, on . We introduce the subclass of regular closed sets as defined in Definition 4, and endow with the trace topology and Borel -algebra, a random regular set being a RACS a.s. in . Taking the limit in in formula (7) entails that is measurable (the functionals and are also measurable). If a random regular set satisfies a.s. Assumption 6 with some , then and are also measurable quantities.
Introduce the support of a RACS as the smallest closed set such that . Mostly for simplification purpose, we will assume whenever relevant that is bounded.
It is easy to derive a result giving the mean Euler characteristic as the limit of the right hand side expectation in (7) by combining Theorems 7 and 12. We treat below the example of stationary random sets, i.e. which law is invariant under the action of the translation group. A non-trivial stationary RACS is a.s. unbounded, therefore we must consider the restriction of to a bounded window . The main issue is to handle boundary terms stemming from the intersection. They involve the perimeter of and the specific perimeter of . We introduce the square perimeter of a measurable set with finite Lebesgue measure by the following. Note the class of compactly supported functions of class on , and define , where