Modern Stochastics: Theory and Applications

The geometrical characterization of granular media (grains, pores, fibers, etc.) is an important issue in materials and process sciences. Indeed, several granular media can be modeled as random sets where the heterogeneity of the particles is studied with a probabilistic approach [12, 24]. In this context, the random closed sets (RACSs) have been particularly studied [29, 21, 8, 2] to get geometrical characteristics of such granular media. A RACS denotes a random variable defined on a probability space $(\varOmega ,\mathfrak{A},P)$ valued in $(\mathbb{F},\mathfrak{F})$, the family of closed subsets of ${\mathbb{R}}^{d}$ provided with the σ-algebra $\mathfrak{F}:=\sigma \{\{F\in \mathbb{F}\hspace{2.5pt}|F\cap X\ne \varnothing \}\hspace{0.1667em}X\in \mathfrak{K}\}$, where $\mathfrak{K}$ denotes the class of compact subsets on ${\mathbb{R}}^{d}$. In a probabilistic point of view, the distribution of a convex RACS is uniquely determined from the Choquet capacity functional [23, 16]. However, such a description is not suitable for explicitly determining the geometrical shape of the RACS. An alternative way is to describe a RACS by the probability distribution of real-valued geometrical characteristics (area, perimeter, diameters, etc.).

The aim of this paper is to show how such characteristics can be used to describe the geometrical shape of a convex random closed set in ${\mathbb{R}}^{2}$. It has already been shown [25] that the moments of the Feret diameter of a convex random closed set in ${\mathbb{R}}^{2}$ can be obtained by the area measures on morphological transforms of it. A Feret diameter (also known as caliper diameter) is a measure of a set size along a specified direction. It can be defined as the distance between the two parallel planes restricting the set perpendicular to that direction.

A set $X\in {\mathbb{R}}^{2}$ is said to be central symmetric or, more simply, symmetric if it is equal to the set $\breve{X}:=-X$. Note that the Feret diameter is not sensitive to such a central symmetrization [22]. Indeed, for a nonempty compact convex set $X\subset {\mathbb{R}}^{2}$, its symmetrized set $\frac{1}{2}(X\oplus \breve{X})$ (see [21, 17]) has the same Feret diameter as X. Consequently, the Feret diameter of a convex set X is not enough to fully reconstruct X (but only its symmetrized set). However, the Feret diameter is still useful to describe the shape of convex sets for two reasons. Firstly, a convex set X and its symmetrized set $\frac{1}{2}(X\oplus \breve{X})$ share a lot of common geometrical descriptors (perimeter, eccentricity, etc.). Secondly, there are many applications in which symmetric convex particles are considered. In this way, the reported work is focused on the symmetric convex sets (i.e., $X=\frac{1}{2}(X\oplus \breve{X})$). By abusing the notation, the conditions “nonempty and compact” will be often omitted in this paper. In other words, without explicit mentioning of the contrary, a convex set will refer to a nonempty compact convex set.

In this paper, we show that the Feret diameter of a random symmetric convex set can be used to define some approximations of it as random zonotopes. The polygonal approximation of a deterministic convex set has already been studied several times [18, 14, 4, 7]. However, in most cases, the approximation is made by using the support function, which is not available in most of the geometric stochastic models. Random polygons have already been studied several times [10, 3, 20]. However, they are defined in different ways and for other objectives, and they are not characterized from their Feret diameters. In our point of view, a zonotope (which is a Minkowski sum of line segments) is described by its faces (direction and length) and can be characterized by its Feret diameter. We will show that the Feret diameter of a symmetric convex set evaluated on a finite number of directions $N>1$ can be used to define some approximations of it as zonotope. Such zonotope approximations will be generalized to the random symmetric convex sets. Therefore, a random symmetric convex set will be approximated by a random zonotope, and such approximations will be characterized from the Feret diameter of the random symmetric convex set. The considered random zonotope will be uniquely determined by the lengths of its faces, and their directions will be assumed to be known. The approximations considered are consistent as $N\to \infty $ with respect to the Hausdorff distance.

This work is a preliminary study in order to describe the geometrical characteristics of a population of convex particles in the context of image analysis. Indeed, such images of population of convex particles can be modeled by stochastic geometric models. In such a model, the projection of a particle represented by a random convex set. Consequently, this work can be used to get information on such convex particles. In addition, when the particles are supposed to be symmetric, they have a symmetric 2-D projection that can be fully characterized by the Feret diameter. Such a symmetric hypothesis is suitable in several industrial applications in chemical engineering (gas absorption, distillation, liquid–liquid extraction, petroleum processes, crystallization, etc.).

An area of application is the gas–liquid reactions. Indeed in a such process, the gas bubbles can be modeled as an ellipsoid the 2-D projections of which are ellipses (see [32, 30, 5]). The main area of application is crystals manufacturing. Indeed, many crystals are 3-D zonohedrons, and their 2-D projections are zonotopes. For example, the crystals of oxalate ammonium [26, 1], the crystals of calcium oxalate dihydrate [31], or the (L)-glutamic acid [6]. In such applications, the considered approximations coincide with the real data.

The paper is organized as follows. The first part is devoted to the case of a deterministic symmetric convex set X. Some properties of the Feret diameter are first recalled, and then for any integer $N>1$, an approximation ${X_{0}^{(N)}}$ of X as a zonotope [11] is described. It is shown that this approximation is consistent as $N\to \infty $ with respect to the Hausdorff distance [27]. A more accurate zonotope approximation ${\tilde{X}_{0}^{(N)}}$ of X that is invariant up to a rotation is also defined with the consistency also satisfied. This approximation is particularly interesting to describe the geometrical shape of X.

The second part is devoted to a characterization of the random zonotopes. First, we explore some properties of the random process associated with the Feret diameter. Then we study the random zonotopes, define some their classes, and discuss their descriptions by their faces. Finally, we study the characterization of some random zonotopes from their Feret diameters random process.

In the last part, we study a random symmetric convex set X. We show that it can still be described as precisely as we want by a random zonotope ${X_{0}^{(N)}}$ and up to a rotation by a random zonotope ${X_{\infty }^{(N)}}$ with respect to the Hausdorff distance. We give the properties of these approximates and show that they can be characterized from the Feret diameter random process of X. In particular, the mean and autocovariance of the Feret diameter random process of X can be used to get the mean and the variances of the random vectors composed by the face lengths of their zonotope approximations.

The support function allows us to fully characterize a convex set. Indeed, any positive homogeneous convex real-valued function on ${\mathbb{R}}^{2}$ is the support function of a convex set [27]. In the following, we give some important properties of the support function. The proofs are omitted since they can be found in the literature [13, 27].

Items 1 and 2 relate the convexity of the support function, and expression (1) allows the reconstruction of a convex set from its support function. Note that the positive homogeneity of the support function involves that it can be completely determined on the Euclidean unit sphere. We adopt the following representation for the support function of X: which is a continuous and $2\pi $-periodic function.

Note that the Feret diameter of a convex set X, denoted $H_{X}$, can be expressed by the support function as where $\breve{X}$ is the usual notation for the symmetric set $-X$. It is easy to see that the Feret diameter of X coincides with the support function of $X\oplus \breve{X}$, where ⊕ denotes the Minkowski sum. Therefore, the functional $H_{X}$ is sufficient to fully characterize the symmetrized body $\frac{1}{2}(X\oplus \breve{X})$. Note that if X is already symmetric, then $H_{X}$ fully characterizes X. We recall some important properties of the Feret diameter.

The Feret diameter can also be related to the mixed area [27] by using a line segment as a structural element. Indeed, using the Steiner formula [27] with two convex sets X and Y, we have where $W(X,Y)$ denotes the mixed area between X and Y. The mixed area functional $W(\cdot ,\cdot )$ is a symmetric mapping, which is also homogeneous in its two variables (see [19, 27] for details). It is often used to describe some morphological characteristics of a convex set X by using different structuring elements. For instance, if X is a bounded convex set and B is the unit disk, then $W(X,B)=\frac{1}{2}U(X)$, where $U(X)$ denotes the perimeter of X. Let X be a bounded convex set, and $S_{\theta }$ be a unit line segment directed by θ. Then

The proof is omitted since it consists in a simple drawing and can be found in the literature [25, 19].

Now we give some properties of the Feret diameter of a convex set and its connection with the mixed area. Here the zonotope will be defined and particularly the class of the 0-regular zonotopes, some properties of the zonotopes will be discussed. In particular, we will show how a symmetric convex set can be approximated by a 0-regular zonotope as precisely as we want.

Let $\mathcal{C}$ denote the class of all symmetric convex sets of ${\mathbb{R}}^{2}$, where the symmetry is given in the sense of Minkowski: $X=\frac{1}{2}(X\oplus \breve{X})$. Let $S_{0}$ be the unit line segment $[-\frac{1}{2},\frac{1}{2}]$, and $S_{t}$ its rotation by the angle $t\in [0,\pi [$. Consider now the convex set X such that Note that X is a compact convex symmetric polygon with at most $2n$ faces, where for all $i=1,\dots ,n$, $\alpha _{i}$ is the length of the two faces of X oriented by $\theta _{i}$. It is easy to see that every compact convex symmetric polygon has an even number of faces and can be represented as (6) up to a translation. Furthermore, note that X has a nonempty interior if and only if $n>1$.

Several geometric characteristics and properties of zonotopes can be easily expressed from representation (6).

In the following, we use a regular subdivision θ. We will show that if the subdivision step is sufficiently small, then any symmetric convex set can be approximated by a zonotope as precisely as we want.

Note that ${\mathcal{C}_{0}^{(N)}}\subset {\mathcal{C}}^{(N)}$ and ${\mathcal{C}_{0}^{(N_{1})}}\subset {\mathcal{C}_{0}^{(N_{2})}}$ if and only if $N_{1}$ is a splitter of $N_{2}$. In addition, ${\mathcal{C}_{0}^{(N)}}$ can be identified to ${\mathbb{R}_{+}^{N}}$ by the mapping $\alpha \to X=({\bigoplus _{i=1}^{N}}\alpha _{i}S_{\theta _{i}})$, which is an isomorphism between the semigroups $({\mathbb{R}_{+}^{N}},+)$ and $({\mathcal{C}_{0}^{(N)}},\oplus )$. That is, this mapping is a bijection, and

This theorem shows that a symmetric body can be always approximated by a 0-regular zonotope as close as we want. Note that the choice of the sequence ${X_{0}^{(N)}}$ is not the best one. Indeed, by taking $\frac{\operatorname{diam}(X)}{\operatorname{diam}({X_{0}^{(N)}})}{X_{0}^{(N)}}$ there is a finer approximation with respect to the Hausdorff distance. However, the sequence ${X_{0}^{(N)}}$ presents some important advantages: it always contains X, the approximation of a Minkowski sum is the Minkowski sum of the approximations, and its face length vector is expressed only from a linear combination of the Feret diameter of X. Furthermore, if there exists $M>1$ such that $X\in {\mathcal{C}_{0}^{(M)}}$, then ${X_{0}^{(M)}}=X$, and X is an adhesion value of the sequence ${X_{0}^{(N)}}$.

Of course, the ${\mathcal{C}_{0}^{(N)}}$-approximation is sensitive to rotations (see Fig. 1). Obviously, it can be problematic to describe the geometry of sets. Let us consider the following example of an ellipse.

For the reasons mentioned, it can be interesting to have an isometric invariant approximation. Fortunately, for a symmetric convex set X, the better ${\mathcal{C}_{0}^{(N)}}$-approximation (in terms of the Hausdorff’s distance) of the family of rotations of X can be used to define such an isometric invariant approximation.

We have shown previously how a symmetric convex set X can be approximated in the class of 0-regular zonotopes. Such an approximation is sensitive to the rotations. However, in order to study convex sets, there is sometimes a need to have isometric invariant tools. Therefore, we will define here an approximation that is invariant up to a rotation. To meet this goal, there is a need to perform the approximation on a class larger than ${\mathcal{C}_{0}^{(N)}}$, namely the class of regular zonotopes.

All the properties of ${\mathcal{C}_{0}^{(N)}}$ cited before are also true for ${\mathcal{C}_{t}^{(N)}}$, $t\in \mathbb{R}$. Therefore, we will define an approximation in ${\mathcal{C}_{\infty }^{(N)}}$.

The theorem gives important information. The ${\mathcal{C}_{\infty }^{(N)}}$-approximation of a symmetric convex set X is the best regular zonotope with at most $2N$ faces containing X. It is always a better approximation than the ${\mathcal{C}_{0}^{(N)}}$-approximation. This approximation can be used for not so large N. For example, for $N=2$, the ${\mathcal{C}_{0}^{(2)}}$-approximation of an ellipse depends on the orientation of the ellipse, but its ${\mathcal{C}_{\infty }^{(2)}}$-approximation is the best way to put the ellipse inside a rectangle (see Fig. 3). An illustration of the approximations of that ellipse for higher values of N is represented Fig. 4.

The accuracy of the ${\mathcal{C}_{0}^{(N)}}$-approximation is presented in Fig. 2, and we remark that the best orientation corresponds to the ${\mathcal{C}_{\infty }^{(N)}}$-approximation. Then, for the considered ellipse, the accuracy of the ${\mathcal{C}_{\infty }^{(N)}}$-approximation in function of the number of faces N is represented in Fig. 2. However, the accuracy of the ${\mathcal{C}_{\infty }^{(N)}}$-approximation depends on both shape and size of the symmetric convex set X.

We have studied two different approximations of a symmetric convex set X. The first one is an approximation of X as a 0-regular zonotope, and the second as a regular zonotope. These approximations have been characterized from the Feret diameter of X. The next objective is to study these approximations when X becomes a random symmetric body, and then how they can be characterized from the Feret diameter of X. In order to do this, we need to study some properties of the random zonotopes, which lead us to the following section.

Let X be a random convex set, that is, a random closed set that is almost surely a convex set. In this subsection, we state some properties of the random process [9] corresponding to the Feret diameter of X.

The trajectories of $H_{X}$ are the Feret diameter of the realizations of X. The properties in Proposition 2 are also true for these trajectories, in particular, the continuity and π-periodicity. We can also notice that the Feret diameter random process characterizes the symmetric convex sets.

Let ${X^{\prime }}$ be a random symmetric body, and X be an isotropized set of it. Then X and ${X^{\prime }}$ have the same shape distribution and the same zonotope rotational approximations (see Theorem 2).

In the following, we will show that the Feret diameter random process $H_{{X^{\prime }}}$ of ${X^{\prime }}$ can be expressed from that of X. We will use this property to show that a random symmetric convex set can be described up to a rotation by an isotropic random zonotope.

Let us recall that the Feret diameter random process $H_{{X^{\prime }}}$ of ${X^{\prime }}$ is sufficient to characterize ${X^{\prime }}$. Then, for any $\theta \in \mathbb{R}$, the Feret diameter $H_{{X^{\prime }}}$ of ${X^{\prime }}$ can be expressed as Let B be a Borel subset of $\mathbb{R}$. Because of the uniformity of η and its independence from ${X^{\prime }}$, it follows that Furthermore, by using the π-periodicity of the Feret diameter the distribution of $H_{X}(\theta )$ can be expressed as Consequently, the moments of the Feret diameter process of the set ${X^{\prime }}$ and the isotropized set X are related. Of course, we need to ensure their existence, but we will treat this later.

We have shown some properties of the Feret diameter random process. Let us discuss now the random zonotopes, that is, the random sets almost surely valued in ${\mathcal{C}}^{(N)}$.

Here we will define some classes of random zonotopes, in particular, the class of the random zonotopes almost surely valued in ${\mathcal{C}_{0}^{(N)}}$ and the class of those almost surely valued in ${\mathcal{C}_{\infty }^{(N)}}$. We will study several properties of the random zonotopes. In particular, we will show how a random zonotope can be described by a random vector corresponding to its faces.

Such a random set can be described almost surely as The distribution of the random vector $(\alpha ,\beta )$ characterizes X. The random vector α is called a face length vector of X.

According to Proposition 3, for any face length vector α of X, some geometrical characteristics (Feret diameter, perimeter, area) of X can be expressed as:

A 0-regular random zonotope X can be almost surely expressed as where $\theta _{i}$, $i=1,\dots ,N$, denotes the regular subdivision on $[0,\pi ]$.

The distribution of the face length vector α characterizes the distribution of X. In addition, this relation is bijective; in other word, the distribution of α is uniquely defined and is called the face length distribution.

Of course, the 0-regular random zonotopes can be used to approximate the random symmetric convex sets as $N\to \infty $ (see Section 4.1). However, it is not the best way to model a random symmetric convex set. Indeed, notice that a 0-regular random zonotope cannot be isotropic. For instance, we need to use a large N in order to describe a random set built as an isotropic random square; see Example 3. This is the reason for using a larger class of random zonotopes.

This proposition shows that an isotropic regular random zonotope can always be described as in (24). Such a zonotope is consequently defined by its random face length vector α. However, different distributions of α can lead to the same distribution of X, as mentioned in the following proposition.

Notice that a central random face length vector has all components identically distributed. Furthermore, its distribution has many interesting properties.

We have shown that the distribution of an isotropic random zonotope X can be described by its central face length distribution and studied the properties of such distributions. Here we will show how its characteristics can be connected to the geometrical characteristics of the random zonotope. In particular, we will give formulae that allow us to connect the first- and second-order moments of the Feret diameter of X to those of the central face length distribution.

Let X be an isotropic random zonotope represented by its face length vector α. Let us recall that X can be almost surely expressed as where η is a uniform random variable independent of $\alpha \ge 0$. Suppose that the condition $\mathbb{E}(U{(X)}^{2})<\infty $ is satisfied. Then, according to Proposition 6, $\alpha \in {L}^{2}({\mathbb{R}_{+}^{N}})$, and the mean and autocovariance of $H_{X}$ exist.

According to Proposition 3, for any representative α of X, some geometrical characteristics of X can be expressed as Therefore, by considering $\alpha \in {L}^{2}({\mathbb{R}_{+}^{N}})$ and the independence of α and η, their expectation can be computed by integration with respect to the uniform distribution of η: Note that $k_{S}$ is a π-periodic function and can be expressed on $[0,\pi ]$ as

Using Eq. (31) and the stationarity of $H_{X}$, we have and by introducing the functional it follows that

Suppose now that α is a central representative of X. We will show that the first- and second-order moments of the central distribution can be easily expressed from the Feret diameter process.

This theorem gives the first- and second-order moments of the central face length distribution from those of the Feret diameter. Note that $V[\alpha ]$ and $V[{H_{X}^{(N)}}]$ satisfy the properties of symmetry. Indeed, by denoting $m=\lfloor \frac{N}{2}\rfloor $ we have shown that if N is an even integer, then $V[\alpha ]={}^{t}(v_{0},\dots ,v_{m-1},v_{m},v_{m-1},\dots ,v_{1})$ and if N is an odd integer, then $V[\alpha ]={}^{t}(v_{0},\dots ,v_{m},v_{m},\dots ,v_{1})$, where $v_{k}=\mathbb{E}[\alpha _{1}\alpha _{1+k}]$, $k=0,\dots ,m$. The vector $V[{H_{X}^{(N)}}]$ can be expressed in the same way: for $i=1,\dots ,N$, $\pi -\theta _{i}=\frac{N-i+2-1}{N}\pi =\theta _{N-i+1[N]+1}$, and Therefore, if N is an even integer, then $V[{H_{X}^{(N)}}]={}^{t}(c_{0},\dots ,c_{m-1},c_{m},c_{m-1},\dots ,v_{1})$, and if N is an odd integer, then $V[{H_{X}^{(N)}}]={}^{t}(c_{0},\dots ,c_{m},c_{m},\dots ,c_{1})$, where $c_{k}=V[{H_{X}^{(N)}}]_{k+1}$, $k=0,\dots ,m$. In practice, the vector $V[\alpha ]$ can be computed by the knowledge of the $m+1$ first components of $V[{H_{X}^{(N)}}]$, and the linear problem (42) can be rewritten and solved as a linear problem of size $m+1$.

We have discussed some properties of the random zonotopes. The 0-regular random zonotopes and the regular random zonotope were defined and studied. We have shown that a 0-regular random zonotope can be describes by a unique face length distribution. Such a distribution can be easily related to the Feret diameter of the 0-regular random zonotope by the relations established in Section 1.

We have studied different face length distributions of a regular random zonotope. We have shown that, among them, one can be identified, the central face length distribution. Finally, we have given some formulae that allow us to compute the first- and second-order moments of the central face length distribution from those of the Feret diameter of the regular random zonotope. The following section is devoted to a description of a random symmetric convex set as a 0-regular random zonotope and as a regular random zonotope.

Here we investigate the approximation of a random symmetric convex set X by a 0-regular random zonotope. We show that the random set ${X_{0}^{(N)}}$ valued in ${\mathcal{C}_{0}^{(N)}}$ that is defined as the ${\mathcal{C}_{0}^{(N)}}$-approximation of realizations of X can be characterized from the Feret diameter of X. Finally, we give some formulas that allow us to compute the moments of the random vector of the faces of ${X_{0}^{(N)}}$.

According to relation (16), $\mathbb{E}[\operatorname{diam}(X)]$ can be replaced by $\frac{1}{2}\mathbb{E}[U(X)]$.

Let X be a random symmetric convex set, and ${X_{0}^{(N)}}$ be its ${\mathcal{C}_{0}^{(N)}}$-approximation. The, in the same way as in the deterministic case, the face length distribution can be related to the Feret diameter of X.

In order to describe the shape of a random symmetric convex set as a zonotope with a small number of faces, we need to have an approximation insensitive to the rotations. This leads us to the following approximation.

Previously, we have shown that a random symmetric convex set can be approximated as a random 0-regular zonotope. However, we have also shown that such an approximation can be problematic for small values of N. The aim of this section is to define and characterize an approximation in ${\mathcal{C}_{\infty }^{(N)}}$ that is invariant up to a rotation and that can be used for not so large N. For this objective, we give the approximation for an isotropized set of X instead of X. We will show that a random symmetric convex set can be approximated up to a rotation by an isotropic random regular zonotope.

Let us note $Y=R_{z}(X)$ the isotropized set of X with z an independent uniform variable on $[0,\pi ]$. Let ${X_{\infty }^{(N)}}$ be a ${\mathcal{C}_{\infty }^{(N)}}$-approximation of X. Then According to the definition of the ${\mathcal{C}_{\infty }^{(N)}}$-approximation, the random set ${Y_{\infty }^{(N)}}=R_{z}({X_{\infty }^{(N)}})$ is a ${\mathcal{C}_{\infty }^{(N)}}$-approximation of Y. Consequently, ${Y_{\infty }^{(N)}}=R_{z+\tau }({\tilde{X}_{0}^{(N)}})$. Because of the independence of η and X, by the property of addition modulo π the random variable $\eta =z+\tau $ is a uniform random variable on $[0,\pi ]$ independent of X. Then ${Y_{\infty }^{(N)}}$ is an isotropic regular zonotope. We will use such a random regular zonotope as the approximation of X up to a rotation.

In order to describe the shape of X, the best way would be to characterize the central face length distribution of ${\tilde{X}_{\infty }^{(N)}}$ from information available on X. Unfortunately, there is no way to compute the characteristics of the random process $H_{{\tilde{X}_{\infty }^{(N)}}}$ from those of $H_{X}$. However, the approximation of the first- and second-order moments of $H_{{\tilde{X}_{\infty }^{(N)}}}$ can be estimated from those of the Feret diameters of an isotropized set of X (i.e., $H_{Y}$, where Y is an isotropized set of X).