Stereology for Cellular Materials. Part 2. Spatial Pattern - CiteSeerX

Stereology for Cellular Materials. Part 2. Spatial Pattern

Dvoralai Wulfsohn Institute of Mechanical Engineering Aalborg University Pontoppidanstraede 101 9220 Aalborg East, Denmark

Task D: Stereology and Image Analysis

Structurally Graded Polymeric Materials and Filled Polymers – an Integrated Approach

November 2000

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Preface The first part of this literature review presented methods for estimating global mean characteristics of 3-D geometry, without consideration of the spatial distribution of microstructural elements other than allowing that the material may be heterogeneous and anisotropic (Wulfsohn, 1999). This report continues where the previous left off, and is a review of the theoretical and applied spatial statistics literature relevant for the characterization of spatial arrangement of inclusions in a material. Properties or stereological terms that were defined in the first part of this report will be shown in sans-serif typeface. As in Part 1, the focus is on the characterization of cellular materials (specifically disordered cellular materials such as foams), but the methods are general enough to be applicable to many multiphase (composite) materials. Most practically oriented spatial statistics texts deal exclusively with planar processes; however, the theory is applicable for d-dimensions (Daley & VereJones, 1988; Stoyan et al, 1995; Jensen, 1998; Møller, 1999a). Because we are interested in what are three-dimensional (‘spatial’) structures, we will present models for the spatial case (d = 3) and focus on the literature which deals with characterization of 3-D structures. Nevertheless, to illustrate concepts, most figures show 2-D examples. There is a large body of applied literature addressing 2-D structures (or often, 3-D materials observed on single planar sections and treated using 2-D analyses), but these will be largely neglected. Astronomical physicists have developed similar methods for the 3-D analysis of galaxies as spatial point processes. There are several monographs which can be consulted for a survey of the applied literature as well as for spatial analysis of 2-D data sets (e.g., Diggle, 1983; Ripley, 1981, 1988; Cressie, 1991; Stoyan et al, 1995; Stoyan & Stoyan, 1994).

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Contents Preface............................................................................................................................ i Introduction.................................................................................................................. 1 S3—Stochastic Geometry, Stereology and Spatial Statistics ..................................... 4 Some terminology and definitions .............................................................................. 9 Representation of Cells.............................................................................................. 15 Spatial Sampling ........................................................................................................ 19 Sampling gradient structures ................................................................................... 19 Typical point of a process........................................................................................ 19 Edge effects ............................................................................................................. 20 Parametric Models for Spatial Point Processes....................................................... 23 Poisson point processes as null models ................................................................... 23 Models for cluster point processes .......................................................................... 24 Hard-core models .................................................................................................... 25 Gibbs processes (Markov point processes) ............................................................. 27 Second-Order Statistics for Point Processes............................................................ 28 Nearest-neighbour distance distributions ................................................................ 29 K-function................................................................................................................ 31 L-Function ............................................................................................................... 32 Radial distribution function ..................................................................................... 32 Product density ........................................................................................................ 32 Pair correlation function .......................................................................................... 33 Estimation of Second-order Property Functions .................................................... 35 Intensity ................................................................................................................... 35 Distance Functions and Second Moment Measures ................................................ 36

iii Performance of Estimators....................................................................................... 48 Stereological estimators ........................................................................................... 51 Generalization for Random Sets...............................................................................53 Covariance function................................................................................................. 53 n-point correlation functions.................................................................................... 55 Generalized K-functions .......................................................................................... 58 Random Spatial Tessellations ...................................................................................60 Poisson-Voronoi tessellation ................................................................................... 64 Johnson-Mehl tessellations ...................................................................................... 65 Moments of Voronoi tessellations ........................................................................... 70 Stereology for spatial Voronoi tessellations ............................................................ 74 Stereology and Non-Destructive Imaging ................................................................81 Virtual spatial sampling ........................................................................................... 81 Three-dimensional reconstruction ........................................................................... 82 Concluding Remarks .................................................................................................84 Acknowledgments ......................................................................................................85 References ...................................................................................................................86

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Introduction A microstructure is a space filling arrangement of geometric features (volumes, surfaces, lines or points) in three-dimensional space (DeHoff, 2000). Typical parameters to describe geometric properties of spatial structures are the global stereological intensities (VV, SV, LV, NV, χV) and the related means of size distributions (e.g. v N , vV ). These quantities have the advantage of being unambiguous as well as being possible to estimate stereologically without bias for all practical purposes (Wulfsohn, 1999). As global averages these are statistical first moments of spatially distributed quantities, denoted ‘first-order’ quantities (Cruz-Orive, 1989). But two microstructures with the same volume fraction VV, say, may possess quite different arrangements of features in space (spatial pattern). Suffice to say, the macroscopic material properties and failure characteristics of these two microstructures can differ dramatically (OstojaStarzewski et al, 1994; Torquato, 1998). So-called second order properties partially characterise the arrangement of features and are used to predict the variance of stereological estimators of first-order properties (Cruz-Orive, 1989). Thus, first order and second order properties play roles similar to the mean and variance of a random variable. Second order stereology refers to stereological methods of determining second order property functions. Geometrical disorder causes discontinuities in local properties and spatial fluctuations of microscopic stresses and deformations. There have been several approaches to dealing with spatial heterogeneity. One is to replace a real disordered material with a homogeneous one, where local material properties are averages over the classical representative volume element (RVE) used in classical continuum mechanics theories (e.g. Drugan & Willis, 1996; OstojaStarzewski, 1998; Nemat-Nasser & Hori, 1993). Typically, the selected RVE would be some 10-100 times the scale of the heterogeneity. The main difficulty

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with this approach is in identifying, even defining, the interaction scale between neighbouring components of a given microstructure, which can vary considerably depending on the material property of interest (cf. Bunge et al, 2000; Bustawros et al, 2000; Ostoja-Starzewski, 1994, 1998; Pyrz, 1999). Lu & Torquato (1990) proposed using the range of variation of the ‘coarseness’ of a microstructure to calculate the minimum length scale over which random fluctuations of phase volume fraction can be considered sufficiently small. The coarseness is a parameter related to the volume fraction and the covariance (a second-order property) of the phase of interest. In many materials science applications it is not possible to view through the opaque material and estimates must be made from planar sections or projections of thin sections. There is a significant loss of information, both qualitative and quantitative on a section through a spatial structure.1 It has been argued that differences in spatial variability on planar sections can reflect qualitative differences in the three-dimensional structures (Stoyan & Schnabel, 1990). Where connectivity of phases is important, even this cannot be assured – connected phases in 3-D may not appear connected on planar sections. One might say that it is necessary to capture that composite materials are composed of both connected (matrix) and disconnected (particle) phases. In general, parameters obtained from single sections do not adequately quantify the 3-D structure. (The only exception is for a ‘completely random’ spatial structure.) Thus, we have a stereological problem. Where a material can be non-invasively sectioned (i.e., optically), we are still sampling geometrically and the problem of quantification is a stereological one. Furthermore, the depth of the material that can be optically probed by available technology may be smaller than the range of interparticle interactions.

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Various types of spatial interactions can be identified. One is due to heterogeneity (‘trend’) over the extent of a graded material and depends on the scale of observation (fig 1). In a weakly disordered material, irregularity of the spatial arrangement of microstructural elements disappears as one samples at progressively larger or smaller scales, whereas, a strongly disordered material will exhibit disorder at all scale lengths (Pyrz & Bochenek, 1998).

z

(a)

(b)

Figure 1–Scales of disorder over a graded random material. (a) Due to a gradient of temperature and gas concentration during formation of this polycarbonate microcellular foam there is a gradient of pore size in the z-direction, from the outer solid skin toward the centre of the sheet. (b) Observation windows placed at two ‘points’ in a ‘homogeneous’ domain of the material, located in the interior of the foam sheet, far away from the skin. Clearly, structural properties such as void fraction and cell intensity fluctuate from point to point.

Within a ‘homogeneous’ sub-region of a material (as established for a given scale), different types of spatial relations can exist between discrete components of the microstructure (fig 2). There may be a tendency for clustering or

1

The terms spatial and 3-D will be used for processes in 3, and planar or 2-D for processes restricted to the plane, 2.

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regularity of particle arrangement. There may be a relation between particle size, number of faces and position in the material. There may attraction or inhibition between components of a microstructure such as small cells forming around larger particles (fig 2b) or around inclusions embedded in a matrix (fig 2c). The geometric structure of dry foams, such as soap froth, come close to that predicted by the ‘ideal model’ of a foam in equilibrium (Weaire et al, 1999; Kraynik et al, 1999). Cell faces are well defined, curved thin films. Exactly three faces meet at 120º each vertex (triple lines); four edges (the Plateau borders) meet at 109.5º in symmetric tetrahedral vertices. Wet foams, such as the aluminium foam shown in fig 2d and polymer foams, disobey the Plateau rules. As liquid fraction increases and/or viscosity decreases the Plateau borders at the vertices contain more material and the cell faces shrink and may even disappear to form an open-celled structure. The solidified structure can have junctions with four or more Plateau borders meeting at a vertex, violating minimum surface energy conditions. The spatial intensity distribution of Plateau borders can provide a measure of the stability of a foam.

S3—Stochastic Geometry, Stereology and Spatial Statistics The mathematical theory of second-order stereology is a synthesis of sampling theory, integral geometry and stochastic geometry (Stoyan et al, 1995; cf. Baddeley, 1999a). From a practical perspective, sampling considerations are paramount to eliminate (or at least, minimise) estimation biases and to control variances. From a theoretical perspective, the development of unbiased stereological estimators has drawn upon the estimation theory developed by Horvitz & Thompson (1952) in a sampling setting (cf. Jensen, 1998).

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(a)

(c)

(b)

(d)

Figure 2–Examples of spatial patterns in cellular materials and crystallizable polymers: (a) relatively few large (~10 µm) cells dispersed amongst smaller ( 0 is the mean number of random points per unit volume (called NV in stereological notation) and is a constant independent of W. It is always assumed that λ is finite (Stoyan et al, 1995). More generally, it is usually reasonable to assume that the intensity measure has a smooth density with respect to Lebesgue measure,4 the intensity function λ(x),

Λ(W) =

∫ λ ( x)dx .

(3)

W

Note that the probability that there is a point from the point process Ν in an infinitesimal volume dV located at x is approximately λ(x)·dV. A map of λ(·) can be a most informative description for a heterogeneous point pattern (Ripley, 1988a). If we want to examine interaction of points of a pattern, then we can consider, respectively, the variance and the (signed) covariance measure of N(W),

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var[N(W)] = E[N(W)2]– Λ(W)2

(4)

cov[N(W1), N(W2)] = E[N(W1)⋅N(W2)] – Λ(W1)Λ(W2)

(5)

The Lebesgue measure is the d-dimensional volume of elementary geometric objects. For d = 3, it is simply the volume measure V(·).

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where W1 and W2 are arbitrary regions in ℜ. Both the variance and covariance of a spatial point process can be expressed in terms of the second moment measure of Ν, defined by

µ(2)(W1 × W2) = E[N(W1)⋅N(W2)]

(6)

where W1 × W2 denotes the Cartesian product5 of two arbitrary regions W1 and W2. To understand this expression, consider the point process to consist of all pairs of points (x1, x2) from the original observed point pattern (allowing repetitions). Then eqn. 6 denotes the mean number of such point pairs in W1 × W2. Higher-order moments can also be defined (Daley & Vera-Jones, pp. 129-134). In general, the kth moment measure µ(k) is given by the expected number of ordered k-tuples (x1, x2, …, xk) of points from the point pattern (allowing repetitions),

µ(k)(W1 × ··· × Wk) = E[N(W1)N(W2)···N(Wk)], k = 1, 2, …

(7)

The moment measures are global in character; their form is not influenced by the nature of the observation regions, e.g. if two observation regions overlap, the moment measures coincide over the intersection (Daley & Vera Jones, 1988). To avoid redundancy of information, the kth factorial moment measures α(k) are often preferred. They are defined in a similar manner to the kth moment measures, except that the new process consists of all k-tuples of distinct points

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An important example of a Cartesian product is that of 3-D space: 3 =  ×  ×  = {(x1, x2, x3): x1, x2, x3 ∈ }.

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from the pattern (still distinguishing order within the k-tuple, but not allowing repetitions). Thus, the 2nd factorial moment measure evaluated at W1 × W2, α(2) (W1 × W2), is the mean number of point pairs (x1, x2) such that x1 ≠ x2, x1 ∈ W1 and x2 ∈ W2. Clearly,

µ(2)(W1 × W2) = α(2)(W1 × W2) + Λ(W1 ∩ W2).

(8)

The second factorial moment measure can be written in terms of its density with respect to Lebesgue measure, the second-order product density ·(2)(x1, x2), as

α(2)(W1 × W2) =

∫∫·

(2)

(x1, x2) dx2 dx1

(9)

W1W2

The product density may be explained as follows. Take two disjoint infinitesimal elements of volumes dV1 and dV2 with respective midpoints at x1 and x2. Then Pr[point in dV1 and one in dV2] ≈ · (2)(x1, x2) dV1 dV2

(10)

(Stoyan & Stoyan, 1994; Stoyan et al, 1995). If the point process is stationary then · (2)(x1, x2) depends only on the relative position of the points, r = x2 – x1, and is conventionally written · (2)(r). If the process is isotropic, then the direction of r is unimportant, and the product density is written simply as · (2)

(r), and describes the relative frequency of inter-point distances of length r = 7

x2 – x1 7 in the point process. A normalized version of the second order product density, the pair correlation function g(r), is widely used to examine local (i.e., small distance) properties of a point process (fig 4). When the process is stationary, reduced kth-order measures κ

(k)

are often used.

These are moment measures that have been normalized with respect to

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Lebesgue measure. For k = 1, the reduced moment measure equals the intensity,

λ (Daley & Vere-Jones, 1988). In the case of stationarity and isotropy, the reduced second moment function K(r) = κ (b(x, r)) is a key quantity for (2)

describing a point process. The value of λK(r) can be interpreted as the mean number of extra points of the point process in a ball of radius r centered at a point of the process x (fig 4).

d(O, Ν)

N(W), V(W)

+

dV, dN r

r + dr

x

Figure 4—Popular summary statistics for a point pattern Ν. The ratio of the number of points N(W) in a window W of volume V(W) provides an unbiased estimate of the intensity λ of the process. The mean number of points N in the ball of radius r centered at a ‘typical’ point x of the pattern (not counting x) defines the reduced second moment function K(r) by N = λ·K(r). The mean number of further points dN in the distance interval (r, r + dr) around a point defines the pair correlation function g(r) by dN = λ·g(r)·dV. The distance from a point O to the nearest point of the process, denoted d(O, Ν), is used to determine the distance distribution functions G(r) and F(r) when O is respectively, a point x of the process or an arbitrary point o in space.

In summary, we use first moments to detect and describe heterogeneity and second moments for interaction in point patterns. Estimation of λ, F(r), K(r),

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g(r) and related second-order functions will be reviewed in some detail in this report.

Representation of Cells A cellular or polycrystalline material can be visualized as a subdivision of the 3D reference space into bounded domains, the ‘cells’. These cells may be as disparate as isolated voids (including their walls) – ‘particles’ – in closed-cell foams (fig 1) to the interconnected networks of joints and struts of an openedcell structure (fig 2d). The random positions of observed individual elements can be represented as a set of distinct points, Ν = {x1, x2, …, xn} in 3. The order of labelling the points is irrelevant so {xi} is an unordered set (Daley & Vera-Jones, 1988; Ripley, 1988a). There are several ways to introduce points to represent the individual cells of a cellular material. The main ones are: (1) To represent each cell by some arbitrarily assigned unique point such as the cell centroid or the left-most vertex of a polyhedral cell; or (2) by the cell vertices.6 The use of the centroid may be preferable for closed-cell materials in that corresponding results are easier to interpret; however, in practical applications such data may not be as easily or objectively obtained, especially in 3 and for space-filling structures. A representation of objects with finite size by a point process is acceptable providing the objects are small compared with the distances between them and compared to the extent of the region of the material under study (Ripley, 1981; Okabe et al, 1992). In many materials, this condition is not satisfied.

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Marked point processes can be used to refine the representation of spatial architecture, by attaching one or more properties to each point or ‘germ’. The so-called ‘marks’ can be qualitative or quantitative and can be 0-, 1-, 2- or 3dimensional. This may be a convenient way to describe a non space-filling process. For example, cell or vertex position is a natural choice for the germ along with properties such as cell volume, total surface area, mean cell wall thickness, number of sides, the mean thickness or number of Plateau borders meeting at a vertex, local stress concentration or ‘local energy’ (e.g. Pyrz, 1994; Pyrz & Bochenek, 1998; Stoyan & Grabarnik, 1991). Random sets provide very general models for complicated microstructures. A special case of a random set is to represent a spatial structure as random closed sets (RACS) in space, {Xj} = {mj(xj): xj ∈ Ν}.

(11)

where the ‘germs’ xj ∈ 3 are points of a random point process and the ‘grains’ mj are random compact sets located at the xi (fig 5). This is known as a generalised Boolean model (Serra, 1982; Cressie, 1991) or a germ-grain model (Stoyan & Stoyan, 1994; Stoyan et al, 1995). If the germ points are Poisson distributed and the grains are independently and identically distributed (i.i.d.), then we have what is known as a Boolean or Poisson germ-grain model. A marked point process can be considered a special case of a RACS where the mj represent real random variables (i.e. the ‘marks’). Similarly, a point process is also a random set with marks all equal to “1”. Stoyan & Stoyan (1994) and

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All points in 3-D space at which four cells meet (‘quadruple points’, McNutt, 1968; DeHoff, 2000) or where the Plateau borders of an open-celled structure intersect (Note that for a 2-D tessellation in ordinary equilibrium, three cells meet at a vertex).

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Stoyan et al (1995, chapter 6) review mathematical definitions, basic properties and statistics of random closed sets. An intuitive approach for space-filling cellular structures, where distances between cells are generally similar to their sizes (e.g. fig 2), is to represent the structure as a random packing of polyhedra (fig 6). The thickness of cell walls (as boundaries of the polyhedra) is simply ignored. The cells (and their boundaries) generated by such a tiling of space are random closed sets (Cressie, 1991). The Voronoi tessellation is often the model of physical interest. Here the space is divided into convex regions based on a generating point process. Each cell consists of those locations in space that are nearer to a given point than to all other points of the process. For many cellular materials only the cells (or cell edges) can be observed in practice, but not the nucleating sites or cell centres (in general, the generating point of a Voronoi cell is not its centroid). Fortunately for purposes of analysis the ‘cell nucleus’ may be any unique observable point of the cell (Møller, 1999b). This approach may prove to particularly useful because it incorporates the distribution of cell faces as well as of vertices. For example, Thiele et al (1999) studied the evolution of a 2-D collagen cellular system. They distinguished two local equilibria: a vertex equilibrium of soap froth; and, an edge equilibrium of the coalescing dewetting network.

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xj O

xj + mj = Xj

mj

mj

xj O

xj + mj = Xj

closed-cell structure with isolated voids

opened-cell structure of interconnected struts Figure 5—A cellular structure {Xj} may be represented with great flexibility as a random closed set {mj(xj): x ∈ Ν} where the ‘germs’ {xj}∈ 3 represent cell (or Plateau vertex) positions and the ‘grains’ mj(xj) are random polyhedra (i.e. having random sides and angles). The germ points are a mathematical convenience and are not restricted to be the ‘cell’ centroid, but may be any unique ‘associated point’ that can be assigned exclusively to a cell, e.g. the extreme point of a cell in an assigned direction or the circumcentre of a sectional cell.

Figure 6—Representation of a piece of a cellular material as a space-filling random tessellation of polyhedra.

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Spatial Sampling Because sampling is such an important part of stereological methods, it merits some further discussion. Sampling gradient structures

Most early analyses of point processes were developed under the assumption of motion invariance of the pattern (e.g. Diggle, 1983) with a few exceptions in Ripley (1981) of anisotropy and Upton & Fingleton (1985) of heterogeneity. The assumption has been that data will be subdivided into sufficiently small units with obvious trends removed, so that homogeneity or isotropy can be invoked (Ripley, 1981). Jensen & Nielsen (2000) show that some classes of heterogeneous point processes may be treated as homogeneous after applying a suitable parametric transformation to the original pattern (the challenge is to identify the transformation). A common practice when dealing with graded materials where a strong gradient exists near the natural border has been to exclude these regions from the analysis and sample only from the homogeneous interior. An alternative approach has been to take a series of samples along the direction of the gradient, and assume homogeneity in each sample. In other words the material is considered as made up of homogeneous layers. Typical point of a process

Any point generated by a random spatial process is termed a ‘typical’ point to distinguish it from an ‘arbitrary’ position in space. If the process, X, is stationary in 3 then any arbitrarily selected location that falls in X, is a typical point of X. In practice this approach is impractical for surface or point processes because the probability of an arbitrary point hitting the process is low

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or zero. Cruz-Orive (1989) explains how to sample a typical point of a structure using m-dimensional probes and both model-based and design-based approaches. If X is a stationary spatial surface process (q = 2, d = 3), then the intersections of an arbitrarily placed test line (m = 1) with X is a set of typical points. If X is not isotropic, then it is sufficient that the test line(s) be isotropic, e.g. sine-weighted line(s) superimposed on a vertical section (Baddeley et al, 1986). In the design-based approach where X is considered non-random and fixed, then the probes need to be placed uniformly randomly. Analogously, if X is a point pattern Ν (q = 0) and the test system an arbitrarily placed window (m = 2 or 3) then the collection of points intersected by the test window is a set of typical points of Ν. Again, the only difference between the design-based versus the model-based approach is that the window(s) must be placed uniformly randomly rather than just arbitrarily. Edge effects

The act of sampling a spatial pattern by observing it in a bounded window complicates the estimation of geometric characteristics by loss of information due to edge effects. Baddeley (1999b) identifies two main types of edge effects: ‘sampling bias’ where the probability of observing an object depends on its size or shape, and ‘censoring effects’ where the full extent of the object to be measured cannot be observed within the window (fig 7). The bias introduced in the estimation of the second-order characteristics due to edge effects can be severe, and become increasingly dominant as the dimension of the domain increases (Ripley, 1981). For example, points that lie within 0.1 units from the boundary of a unit square occupy over 1/3 of the area. For a three-dimensional unit cube they occupy almost half its volume. The use of ‘naive estimators’ (i.e. without edge correction) is therefore strongly discouraged. In contrast, edge effects are usually ignored in hypothesis testing; uncorrected empirical

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distribution functions are compared with simulated theoretical distributions also without edge correction (see Ripley, 1981, 1988b; Diggle, 1983). The influence of edge effects is the major difficulty with techniques to characterise processes based on distances between points. The problem manifests itself in two ways (Doguwa & Upton, 1990): (1) the ball of radius r centred at a sampling point is certain to overlap the boundary of the observation window as r increases; and (2) points lying near a boundary may have a ‘nearest neighbour’ lying outside the window (see fig 7b).

W

(a)

(b)

Figure 7—Edge effects. (a) ‘Minus sampling’ samples only those objects (shown shaded) that lie completely within the (rectangular) frame, thereby introducing a sampling bias in favour of smaller objects. It is impossible, for example, to sample any ‘cell’ larger than the window. (b) Edge effect for distances in point processes can be considered analogous to a censoring effect. An observation is censored if the reference point (either a fixed point in W or a point of the pattern Ν) is closer to the boundary of the observation window W than to the nearest point of the process.

There have been two general approaches to dealing with sampling bias (Baddeley, 1999b):

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(1) The use of unbiased sampling rules. These generally require some information from outside the sampling window, e.g. the 2-D ‘unbiased counting frame’ (Gundersen, 1977), the ‘disector’ (Sterio, 1984), and the ‘associated point rule’ (Miles, 1978); and, (2) The use of weighted sampling methods. Usually these are of the spatial Horvitz-Thomson type, where spatial sampling bias is corrected by weighting each sampled object by the reciprocal of the sampling probability (Jensen, 1998). Examples include several of the ‘edge corrected’ estimators for point processes (e.g. Hanisch, 1984; Fiksel, 1988) and stereological estimators of Kfunctions (e.g. Jensen et al, 1990). Another solution is to modify the statistic of interest to one that has the property of being an ‘additive functional’ of the process and therefore not susceptible to edge effects. Objects may potentially be counted more than once, but each object is assigned a fractional weight to compensate for this. For example, the domain can be tessellated with copies of the sampling window and then each sampled object is weighted by the reciprocal of the number of windows that intersect it. More recently, methods of survival analysis have been applied to tackle censoring effects, i.e. Kaplan-Meier estimators for point processes (Baddeley & Gill, 1997; Baddeley, 1999b). Unbiasedness via edge correction has its price, typically a large estimator variance. Furthermore, many estimators of distribution characteristics are in the form of a ratio where the numerator is an edge corrected estimator and the denominator is an unbiased estimator of the intensity λ or its square. The ratio estimators generally have some bias as well as considerable variances (Stoyan & Stoyan, 1998).

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Several edge-corrected estimators for distribution functions of spatial patterns (i.e. point processes, random closed sets) will be reviewed in later sections of this report as relevant. See Baddeley (1999b) for a more comprehensive overview and a mathematical treatment of edge effects for spatial processes, including sections focussing on point patterns (with bibliographies). Surveys of methods to reduce or minimize errors due to sampling bias for point patterns can also be found in Diggle (1983, chapter 5), Ripley (1988b, chapter 3), Upton & Fingleton (1985, section 1.5), Cressie (1991, chapter 8), Baddeley (1993) and Stoyan et al (1995, pp. 133-139).

Parametric Models for Spatial Point Processes Poisson point processes as null models

A (inhomogeneous) Poisson process with mean intensity Λ is characterized by two properties: The number of points in k disjoint bounded regions of space form k independent random variables, and for any bounded arbitrary region W, the number of points N(W) has a Poisson distribution,

Pr[N (W ) = n] = e

− Λ (W )

(Λ (W ) )n , n!

n = 0, 1, 2, …; Λ > 0

(12)

The homogeneous Poisson process is the special case of eqn. 12 where the intensity measure is constant, Λ = λ·V(W) where λ > 0 is a constant for all W (cf. eqn. 2). The homogeneous Poisson distribution is traditionally used as the model for complete spatial randomness (the absence of structure) of point patterns to which other models are compared and as the basis for Monte Carlo simulations for significance tests in hypothesis testing. It is one of the only point processes for which all moments are known analytically. Of particular

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note, for a Poisson process var[N(W)] = E[N(W)] and cov[N(W1), N(W2)] = 0 when W1 and W2 are disjoint. The Poisson process is therefore a convenient model, despite the fact that it does not occur in practice (although spatial patterns can come close). Patterns often exhibit non-stationarity (and sometimes anisotropy). The inhomogenous Poisson process may be used as the null model for general nonstationarity (Stoyan et al, 1995, p. 42). Other general alternative null hypotheses are also possible (see Baddeley et al, 1993, p.645 for references). The graded microcellular foam shown in fig 1a presents an example of heterogeneity in one direction, z (‘horizontal stationarity’). A non-uniform Poisson process with (unspecified) intensity function λ(x, y, z) = λ(z) may be suitable as a null hypothesis for such a structure. Models for cluster point processes

There are essentially two different mechanisms used to construct aggregated point processes. Doubly stochastic processes and cluster processes. Several of the more important models will be mentioned here. More information, including first and second order characteristics of various processes can be found in Diggle (1983), Ripley (1991, 1988), Stoyan & Stoyan (1994), Stoyan et al (1995), Cressie (1991) and Daley & Vere-Jones (1988). Doubly stochastic Poisson point processes, also called Cox processes, are constructed in a two-step random mechanism. In the first step a random intensity measure Λ(x) is chosen according to some distribution, and in the second step a Poisson point process is formed conditional on Λ(x). One example given by Stoyan et al (1995, p. 155) is of a Cox process formed by

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randomly scattering points with some intensity NL (the number of points per unit length) along a random fibre process. Neyman-Scott point processes are a popular class of Poisson cluster processes (cf. Fig 3c). In the first step, a homogeneous Poisson process of parent points is generated. In the second step a cluster of a daughter points is dispersed around each parent point. Each cluster of daughter points has a random number of points that are scattered i.i.d. around the parent with some distribution. Only the daughter points are included in the final point process. One special realisation is given by Matérn’s cluster process. Poisson-distributed numbers of daughter points are uniformly and independently distributed in balls of radius ρc. around each parent. Many point processes can be formed by either mechanism. For example, the negative binomial distribution may be constructed using logarithmically distributed clusters or by considering the intensity to be a gamma-distributed random variable. Because the number of points per cluster has a Poisson distribution, Matérn’s cluster process is also a Cox process (Stoyan et al, 1995). Hard-core models

A hard-core or simple inhibition point process has the property that no pair of points can be closer than some minimum distance. Hard-core point processes are therefore used as models for point processes underlying patterns exhibiting repulsion between elements, such as systems of non-overlapping particles. Diggle (1983) describes a hard-core process, which is referred to as the simple sequential inhibition process (SSI). It is usually simulated in some finite region W as follows. In the first step, a point x1 is chosen uniformly randomly in W.

26

Then a ball b(x1, ρh) of radius ρh is centred at x1. A second independent uniform random point is chosen from W. If it lies in b(x1, ρh) then it is rejected and another point is chosen independently uniformly randomly. This continues until a point outside of b(x1, ρh) is obtained. This point is denoted x2 and allocated its ball b(x2, ρh). This process can be repeated sequentially until a point pattern with some desired number of points is obtained (fig 3b). No two points of the process will be closer than ρh. Several hard-core models have been suggested based on the work of Matérn (1960). Matérn’s aim was to model the position of trees in a forest. One of Matérn’s models starts with a Poisson process ΝP in d with intensity λ, which is then thinned as follows. Each of the points of the process are given an independent mark m uniformly distributed on [0, 1] (the ‘time of birth’). A typical point x with mark m(x) is eliminated if and only if there is another point, with a mark smaller (‘older’) than m(x), located within the ball b(x, ρh) centered at x with radius ρh. The result is a hard-core process with minimal interpoint distance ρh. Stoyan & Stoyan (1985) proposed two generalisations of Matérn’s hard-core model. The ‘type I Matérn hard-core’ model now independently marks the points with positive marks m that have some (non-uniform) distribution function, e.g. exponentially distributed. The ‘type II Matérn hardcore’ model is a further generalisation where now the points of the Poisson process ΝP are independently marked by two independent marks m and r, each having some distribution. The thinning process involves eliminating a point x with marks m(x) and r(x) when the ball b(x, r(x)), contains a further point with mark m smaller than m(x). This type of model with variable radii is also called a ‘soft core’ process by some workers. Models I and II can also be considered as marked point processes where the points are the points of the thinned process

27

and the marks are the marks of the original independent marking of ΝP (Stoyan & Stoyan, 1985). Gibbs processes (Markov point processes)

Gibbs processes (or random Markov point processes with interaction) can be used to construct models for many types of point patterns. They are particularly useful for describing regular point processes with moderate to even pronounced inhibition, but can also be used to generate moderately aggregated patterns. Gibbs processes were originally developed in the field of statistical physics based on considering the equilibrium states of finite subsystems of ‘infinitely’ large closed physical systems (Stoyan et al, 1995; Møller, 1999a). More generally, interactions between points are studied through the modelling of an associated potential function (Stoyan et al, 1995). A Gibbs point process is defined by starting from a base process Ν 0 and then generating a new process Ν by giving the probability density φ of Ν relative to

Ν 0. Ripley (1981) presents the example of a hard-core model, in which no pair of points is closer than ρh, as a Gibbs process. The hard-core process (Ν) is defined by excluding all realizations of a Poisson point process (Ν 0) in which two points are closer than ρh. It is also clear that the process can be defined this way only within a bounded region; in the whole of 3, the probability that no two points will lie closer to each other than R is zero. (In practice this is not a problem, as observations are always in bounded regions.) Some of the attractive features of Gibbs processes are that they have relatively few parameters and they can be simulated using spatial birth-and-death processes and so by Markov Chain Monte Carlo (MCMC) methods (Glözl, 1981; Ripley, 1977; Møller, 1999a). On the other hand, Gibbs processes can be

28

very expensive to generate because the ‘acceptance’ probabilities for accepting a new point can become very small. It is also difficult to determine parameters, but for special cases (Stoyan & Grabarnik, 1991). Pairwise interaction processes and Strauss processes are important examples of Gibbs processes (Stoyan et al, 1995). ‘Nearest neighbour’ Gibbs point processes were introduced by Baddeley & Møller (1989). Local energy is energy required to add new points to an existing point process configuration.

Second-Order Statistics for Point Processes The exploratory data analysis for point patterns normally involves the determination of the intensity and several different summary statistics of second order moments. Functions take on known forms for the homogeneous Poisson point process, which serves as the model of complete spatial randomness. Deviations from these forms are traditionally used to indicate aggregated versus inhibited alternatives (Diggle, 1983). Second order moment functions also serve as a tool in constructing parametric models for point process data (Diggle, 1983). They have also been used to generate model microstructures that have the same statistical properties as a real disordered material. Effective properties evaluated from direct simulation are then compared to experimentally measured properties (Torquato, 1998). Baddeley et al (1993) argue that no one of the second-order functions to be presented here is necessarily ‘better’ than another. Second order properties do not characterize a point process. Several examples have been given in the literature of point processes with different distributions but the same second order characteristic (Baddeley & Silverman, 1984; Bedford & van den Berg, 1997; Stoyan, 1991; Stoyan & Stoyan, 1996). Therefore, the statistical analysis

29

of a spatial point pattern normally includes several measures of interaction. Many of the methods can also be extended to marked point processes. Nearest-neighbour distance distributions

Two types of nearest neighbour distance distributions can be defined (Diggle, 1979, 1983), both of which are usually determined when nearest neighbour methods are used. The conventional “nearest-neighbour distribution”, G(r), is the distribution of the distance from a typical point (i.e. one selected at random from the point process) to the nearest other point from the process (Roder, 1975; Diggle, 1979), G(r) = Pr[the distance between a typical point and its nearest neighbour is less than or equal to r] = Pr[d(x, Ν) ≤ r | x ∈ Ν]

(13a)

where d(x, Ν) denotes the shortest distance from x to (another point in) the process Ν, and x is a typical point of Ν. The function G can be defined equivalently as G(r) = Pr[the sphere of radius r, centred on a typical point, contains at least one other point] =Pr[N(b(x, r)) > 1 | x ∈ Ν].

(13b)

For a homogeneous Poisson point process of intensity λ, GP(r) = 1 – exp[– 4 λπr3], 3

noting that the factor 4π/3 is volume of the unit sphere in 3. Values of

G(r) smaller than the Poisson value GP(r) are taken to indicate regularity in the point pattern; greater values are interpreted as clustering.

30

A related distribution, the point-to-nearest-event distance distribution function or “empty space function” is the probability of there being one or more points of the process within a distance r from an arbitrary point o, F(r) = Pr[N(b(o, r)) > 0]

(14)

Suppose that there are no points of the process within a distance r from o. This implies that if we were to place a ball of radius r centred at each point of the pattern, b(x, r), then not one of these spheres would overlap o. This observation leads to an alternative interpretation of F(r) as the proportion of the volume occupied by the spheres (Upton & Fingleton, 1985). For a homogeneous Poisson point process, FP(r) = 1 – exp[– (4/3) λ π r3]. Values of F(r) smaller than the value for a Poisson process are taken to indicate aggregation in the pattern; greater values suggest inhibition.7 van Lieshout & Baddeley (1996) combined the F and G functions into the ‘Jfunction’

J (r ) =

1 − G (r ) 1 − F (r )

(15)

The J-function is equal to 1 for a Poisson point process, and values of J(r) greater than or less than 1 are used to suggest inhibition or aggregation, respectively.8

7

See fig 9b. Clearly, the more clustered a point process, the more overlap there will be of the shaded circles, and the smaller the value of F(r). 8 It is possible to find non-Poisson point processes that also have J(r) = 1 (Bedford & van den Berg, 1997).

31

K-function

Ripley (1976, 1977) suggested using the second reduced moment function, or “K-function” K(r), to describe the second-order behaviour of a stationary, isotropic point pattern. In the case of motion invariance, λ and K(r) completely determine the second-order properties of a point process (Stoyan et al, 1995). The K-function can be defined in several ways, including (1) λK(r) is the mean number of further points within distance r of a typical point of the process,

λK(r) = E[N(b(x, r)) – 1 | x ∈ Ν],

(16)

where b(x, r) is the ball with radius r, with its centre at a typical point x, which itself is not counted; (2) λ2K(r) is the mean number of pairs of distinct points in a pattern less than distance r apart, with the first point in a given region of unit volume. For a stationary spatial Poisson point process, KP(r) =

4π 3

r 3 . Values of K(r)

greater than KP(r) for a range of r, indicate aggregation. Conversely, K(r) < KP(r) indicates inhibition. Analogues of K-function have also been proposed for the analysis of heterogeneous point patterns. Baddeley et al (2000) ‘normalize’ the K-function with respect to the intensity function λ(x) of the process. Anisotropic point processes can be treated as a special case of heterogeneity.

32

Other, related, functions are often used to describe the second-order statistics of a motion invariant point process. Which function is preferred depends mainly on convenience. L-Function

The L-function is a linearised version of the K-function. For a motion invariant point process in 3 it is defined by

L(r) =

3

3K ( r )

4π

for r ≥ 0.

(17)

For a homogenous Poisson point process, LP(r) = r. The L-function has often been preferred over K(r) when examining empirical point patterns because linearity simplifies its interpretation and presentation. Radial distribution function

The radial distribution function, R(r) of motion invariant Ν is given by ⎡number of points in a spherical shell with ⎤ R(r ) ⋅ dr = λ ⋅ dK (r ) = E ⎢ ⎥ (18) radii r and r + dr , centred at typical point x ⎣ ⎦ Product density

The product density was defined earlier (eqns. 9-10). The K-function and second order product density for a spatial point process are related as follows:

· (2)(r) = λ2

dK (r ) 1 , r>0 dr 4πr 2

(19)

33

(Stoyan et al, 1995). For a stationary Poisson point process, the second order product density is ·(2)P(r) = λ2. Pair correlation function

The pair-correlation function g(r) of a motion-invariant point process is a normalized version of the second order product density ·(2)(r), g(r) = · (2)(r) / λ2.

(20)

Thus, g(r) is related to R(r) and K(r) by, g (r ) =

R (r ) ⋅ dr

λ ⋅ 4πr 2 ⋅ dr

=

dK (r ) 4πr 2 dr 1

(21)

Note that the term 4πr2dr is the volume of a spherical shell with radii r and r+dr. The pair-correlation function can be interpreted as the ratio of the expected number of points in a spherical shell of radii r and r + dr centred at a typical point, to the expected number of points in a spherical shell of identical size but centred at a sampling point (Cruz-Orive, 1989; Hermann, 1991). The shape of the pair-correlation function provides important information about local interactions in a point process. The use of g(r) for exploratory data analysis has been consistently promoted by Dietrich Stoyan (numerous references) since it is possible to relate characteristics of the function to meaningful distances associated with structural arrangement (fig 8a). The form of g(r) has been determined for many theoretical models (Stoyan et al, 1995; Hermann, 1991; and fig 8b). For the homogeneous Poisson point process in any dimension, gP(r) is constant and independent of the intensity λ, and equal to one. For very large r relative to λ-1, the value of g(r) tends to one for any

34

homogeneous point pattern. Values greater than one indicate that the corresponding interpoint distances occur more frequently than under complete randomness, and conversely. The first maximum, if one exists, is often a good characterization of the minimum distance between points, and is of relevance for models simulating ‘repulsion’ between points in a process (‘hard core’ models). Maxima of g occurring in specific intervals of r, indicate frequent occurrences of interpoint distances at such values. Similarly, minima of g indicate inhibition at these distances; if g(r) = 0, then interpoint distance r is impossible. The sharpness and abundance of peaks and the rate at which g(r) → 1 also characterize the order of the point process. The order of the pole of g(r) at r = 0 (if it exists) was used by Ogata & Katsura (1990) to estimate fractal dimension. Stoyan & Stoyan (1994, section 14.4.2) present and interpret a range of empirical pair correlation functions for planar point processes. Similar interpretations apply to pair correlation functions for spatial point patterns. Stoyan & Schnabel (1990) used a shape parameter for the pair correlation function to quantify the variability of alumina ceramic structure, where the vertices of grain profiles on a section establish a point process. The shape parameter s was calculated as the difference between the peak value of g(r) and the minimum value occurring after this peak for r less than 4 l , where l is the mean chord length. Small values of s thus correspond to homogeneity and

randomness, whereas large values indicate aggregation or inhibition. They analysed 12 alumina microstructures, all having similar chord length distributions but different spatial arrangements of grains. The parameter s was found to correlate with mean bending strength better than did the first-order parameter l or the closely related specific surface density, SV.

35 g(r)

g(r)

hard-core

1

dense packing of spheres with radius R

cluster

1 Poisson

r0

r1

r2

r3

r4

r

2R

4R

r

Figure 8—(a) Interpretation of a pair-correlation function: r0 = hard-core distance, r1 = most frequent distance to first order neighbouring points, r2 = distance of the gap between first and second order neighbouring points, r4 = correlation width. (b) Typical pair-correlation functions for various point process models.

Estimation of Second-order Property Functions Intensity

The intensity is a parameter common to all spatial distribution models. A general unbiased estimator for the intensity (cf. eqn. 2) is simply

λˆ = N (W ) / V (W )

(22)

where N(W) is the number of points of Ν falling in an arbitrarily positioned and shaped observation window W of volume V(W) > 0 (fig 4). The quantity λˆ converges to the mean intensity λ with increasing V(W). (To increase efficiency, and to evade influences of local heterogeneity, the intensity may be estimated using systematically positioned windows.) Stoyan & Stoyan (1998) provide several examples of surface-weighted and volume-weighted estimators of λ. Alternative approaches to estimating λ use interpoint distances (Diggle, 1983; Ripley, 1981).

36

It is sometimes necessary to estimate λ2 as a term in estimators for second-order distribution functions. Stoyan & Stoyan (1998) recommend against the popular method of simply squaring the λ-estimator (eqn. 22), as generally introducing bias and considerable estimation errors. They proposed the use of weighting functions to construct unbiased λ2-estimators. For a motion invariant process the estimators take the form ∧ 2

λ =

∑ ∑

x1 ∈Ν x 2 ∈Ν x1 ≠ x 2

w( x1 ) w( x 2 ) g (r )γ (r )

(23)

where w = weighting function, γ = global geometry normalizing factor, g(r) = pair correlation function, r = 7 x1 – x2 7, and the sum is over all distinct point pairs of Ν. The weighting functions need to be estimated from the data. The price of unbiasedness is substantially increased computation. The estimation of a smooth intensity function λ(x) is one of the most important tasks when analysing heterogeneous point patterns. It is not a straightforward task because of the difficulty in objectively distinguishing between heterogeneity and clustering in many cases. Parameteric (i.e. maximum likelihood method) and non-parametric (i.e. the use of kernel functions) approaches have been proposed (e.g. Baddeley et al, 2000; Berman & Turner, 1992; Diggle, 1983, 1985; Ripley, 1989; Silverman, 1986; Stoyan & Stoyan, 1994). Distance Functions and Second Moment Measures

Empirical analogues of the definitions of F and G give rise to the following naive estimators for a point process:

37

Fˆ ( r ) =

where Ν (+r) =

[

V Ν (+ r ) ∩ W V (W )

U b ( x, r )

] = V [o ∈W : (d (o, Ν ) ≤ r )] V (W )

(24)

denotes the dilation of the process Ν by centering a

x∈Ν

ball of radius r at each point, and # (d ( x, Ν ) ≤ r ) Gˆ (r ) = N (W )

(25)

where #( · ) should be read as the “number of points in the point process for which” and N(W) is the number of points of Ν that lie in the window W.

The numerator of eqn. 24 is the volume made up by the region within the window W for which the closest point of the random point process is within a distance r. To estimate F(r) on the basis of eqn. 24, a fine point grid is placed in the sampling window W, the distance from each grid point o to the nearest observed point of the process x ∈ Ν is computed, and then the number of grid points is counted for which d(o, Ν) ≤ r, ΣP. The volume in the numerator can be unbiasedly estimated as v(o)⋅ΣP where v(o) is the volume associated with each grid point. Estimates of the K-function are made either as the ratio of estimates of λK(r) and λ or of λ2K(r) and the intensity squared. From eqn. 16, a naive estimate of

λK(r) is given by

λK (r ) V (W ) = N (W ) λˆ

∑ N (b( x, r ) \ x )

x∈N ∩W

N (W )

(26)

38

=

(

)

V (W ) # x − x j ≤ r , x ∈ Ν ∩W , (x j ∈ Ν ) ≠ x N (W ) 2

where \ is the set difference operator. Here, for each point x ∈ Ν ∩ W the intensity estimate typically used is that determined from the same W. The product density ·(2) and the pair-correlation function g(r) can be estimated by applying numerical differentiation to an estimate of λ2K(r) (cf. eqn. 19). Based on eqn 20, g(r) is often estimated as the ratio of estimates of the product density and the intensity or the squared intensity. Analogous to the estimation of probability density functions, kernel estimators can be used to estimate the product density (Stoyan & Stoyan, 1994). A naive estimate of ·(2)(r) is given by

·^ (2)(r) =

1

1 ∑ 4πr 2 V (W ) x∈Ν ∩W

∑

kh ( y∈Ν ∩W x≠ y

x − y − r) , r > 0

(27)

where kh is a kernel function with band-width h. The choice of bandwidth is somewhat arbitrary, but it depends on λ. A large value of h will tend to reduce estimator variance at the expense of introducing bias. Silverman (1986) provides guidelines on choosing the bandwidth in kernel methods. König et al (1991) recommended using the Epanechnikov kernel with bandwidth h = 0.05 5 λ−1/3 or 0.1 5 λ−1/3 for 3-D point patterns.

The major difficulty with distance techniques in practice is due to edge effects. In some cases, information from outside the observation window is not available in which case the above estimators are simply not feasible.9 For

9

This problem generally does not occur in the plane when using computer-assisted stereological systems based on live microscope images, but may exist in the third

39

example, bias is introduced in the estimation of F(r) when one cannot search for nearest neighbour points outside W. Similarly, eqn. 26 is in theory a ratiounbiased estimator of λK(r), but this unbiasedness depends on complete observations inside the balls b(x, r) (Jensen, 1998, p. 210).

b(x, r)

W W(-r)

W W(-r)

r

(a)

(b)

Figure 9—2-D illustration of the ‘border method’ for edge corrected estimation of: (a) the empty space function FB(r), as the volume (area) fraction of the shaded region to that of the eroded window; and, (b) for empirical nearest neighbour , Gˆ B (r ) , second moment reduced, Kˆ B (r ) , and related functions, reference points must lie within W(-r). Measurements are made in an observation window, W(-r), resulting from the erosion of the sampling window W by a ‘guard’ distance equal to r (after Baddeley, 1999b).

One way to circumvent edge effects is to simply remove the edge. The ‘border method’ (Diggle, 1979; Ripley, 1977, 1981, 1988b; Upton & Fingleton, 1985; Baddeley, 1999b), an application of minus sampling (Miles, 1974), is probably the simplest way to perform edge correction estimation of F(r), G(r), ·(2)(r) and K(r). This approach involves restricting sampling to an inner region of the observation window, which is surrounded by a ‘guard zone’ of width r. Points in the guard zone are not sampled in the analyses, but are allowed to be the

direction (along the focal axis) and is also a feature of digitized images. Even where an effectively ‘infinite’ domain of the material is available in the focus plane, as r

40

nearest neighbours of sample points (fig 9). Using the border method, eqs. 24-

26 are modified to provide unbiased estimators for F(r), G(r) and λK(r) as follows:

Fˆ B ( r ) =

Gˆ B (r ) =

[

V Ν ( + r ) ∩ W( − r )

]

(

# d ( x, Ν ∩ W( − r ) ) ≤ r

V (W ) Kˆ B (r ) = N (W )

(28)

V (W( − r ) )

)

(29)

N (W( − r ) )

∑ N (b( x, r ) \ x)

x∈Ν ∩W( − r )

(30)

N (W( − r ) )

Similarly, the estimator 27 for ·(2)(r) becomes,

·^ (2)B(r) =

1

1 ∑ 4πr 2 V (W( − r ) ) x∈Ν ∩W( − r )

∑ kh ( x − y

− r)

(31)

y∈Ν ∩W x≠ y

Clearly, as r increases the size of the eroded window decreases. Because much of the data is discarded the border method is extremely wasteful, especially in 3D and especially for estimation of G(r). The border method appears to be the only unbiased estimator of F that is in use. Diggle (1983) ignores edge effects for F, and uses the uncorrected empirical distribution of observed distances in Monte Carlo based hypothesis testing, e.g. versus the null hypothesis of a homogeneous Poisson distribution. Baddeley et al (1993) concluded that for spatial point patterns this kind of approach was preferable to edge corrected estimation of G.

increases the practical efficiency of these estimators will suffer.

41

Another simple approach, which can be used when the pattern is spatially homogeneous and the observation window is rectangular, is the ‘toroidal method’ or ‘periodic continuation’ (Ripley, 1977). An edgeless representation of the process is produced by imagining that the study region is wrapped around a toroid, so that points on opposite edges are considered are close. In practice this is achieved by tessellating the space around the window with copies of the pattern observed in the observation window (fig 10). Little is known about the performance characteristics of the toroidal method for analysis of point processes and it does not seem to be much in use (at least in the spatial statistics community) now that a variety of specialised edge-corrected estimators are available. It is plausible that for highly regular patterns there will be a bias in favour of reduced interpoint interactions. Consider for example, a hard core process where any two points in the pattern must lie some minimum distance apart. This distance constraint may be violated if we tessellate the plane with copies of the window. Stoyan & Stoyan (1994) simply state that if W is large enough, there is no danger of undesirable correlation between the points at the edges of W but provide no specific guidelines for choosing the size of W. So called ‘edge corrected estimators’ (Baddeley, 1999b) that weight the observed distances between pairs of points by the reciprocal of the ‘probability’ of observing this distance (usually under assumptions of stationarity) are generally an improvement on the border method for G and K functions. Ripley (1988b, section 3.2) described two edge-corrected estimators of K(r) for planar point processes using ‘translation’ and ‘isotropic’ corrections. These have been adapted to three dimensions by Baddeley et al (1987, 1993). The translation-corrected estimator for K becomes

42

W

Figure 10—Illustration of a toroidal edge correction for a rectangular region W.

Kˆ T (r ) =

V (W ) 2 N (W )

2

∑ N (b( x, r ) \ x)

x∈Ν ∩W

γ W (x − x j )

, ( x j ≠ x) ∈ Ν : 0 < x − x j ≤ r

(32)

where γW(h) = V{o ∈ W: o + h ∈ W} is the ‘set covariance’ function, i.e., the volume of the intersection between W and W shifted by h, and may be estimated stereologically using a uniformly randomly placed point grid. If the process is known to be isotropic, then the isotropic corrected estimator can be used,

Kˆ I (r ) =

V (W ) N (W )

2

∑ N (b( x, r ) \ x)

x∈Ν ∩W

wW sW

(33)

43

where wW is an edge correction weighting factor and sW is a global geometry correction factor. Analytical expressions for wW and sW are given by Baddeley et al (1993, Appendix A) for a 3-D rectangular box. König et al (1990, Appendix 2) also adapted Ripley’s translation corrected estimator for 3-D. Hanisch (1984) introduced the idea of using only those points to points x ∈ Ν in the sampling window for which the nearest neighbour distance is closer than the boundary ∂W (so that the nearest neighbour is known to be in W). Hanisch proposed two estimators for G(r), # (d ( x, Ν ) ≤ r ∧ d ( x, ∂W ) ) # (d ( x, Ν ∩ W( − d ) ) ≤ r ) = Gˆ H (r ) = # (d ( x, Ν ) ≤ d ( x, ∂W ) ) N ( Ν ∩ W( − d ) )

(34)

where W(-d) is the sampling window eroded by a ball of diameter d(x, Ν) and a∧b denotes min{a, b}, and

λˆGˆ HW (r ) =

# ( d ( x , Ν ∩ W( − d ) ) ≤ r ) V (W( − d ) )

,

(35)

a weighted version of Gˆ H . The weighting function for a distance h is the reciprocal of the volume of W eroded by a ball of radius h, which for a box of dimension a×b×c is equal to {(a – 2h)(b – 2h)(c – 2h)}-1. Chiu & Stoyan (1998) proposed a Hanisch type for the empty space function. Doguwa & Upton (1990) introduced ad-hoc improvements on the border and Hanisch edge corrected estimators for G(r). They realized that it is not always necessary to find the nearest neighbour; it is sufficient to determine whether or not a further point exists within r (cf. eqn. 13b). The resulting estimator uses all points in a window and the known probability that a point will lie within

44

distance r of x, given that x is the only point lying in an observable region, under the assumption that the points are Poisson distributed. They claimed that the estimator has only small biases for non-Poisson point process models. Reed and Howard (1997) used a similar idea to develop an estimator of G for 3-D point patterns, which they dubbed the ‘expectation estimator’. Other edge-corrections for distance-based functions have been described by Vere-Jones (1978), Ohser (1983), Fiksel (1988), Doguwa & Upton (1989) and Floresroux & Stein (1996). Some of these are modifications of presented estimators. Edge-corrected kernel density estimators for (2) are also available for both isotropic and not necessarily isotropic point processes (Doguwa, 1990; Fiksel, 1988; König et al, 1991; Ohser, 1983; Stoyan et al, 1995; Stoyan & Stoyan, 1996, 1998). The L-function is usually calculated from eqn. 17 using estimates of K(r) (Doguwa & Upton, 1989; Ohser, 1983; Discussion by Besag in Ripley, 1977; Stoyan et al, 1995; Stoyan & Stoyan, 1998; Vere-Jones, 1978). Stoyan & Stoyan (1998b) suggested several possible new estimators for the K, L and g(r) functions based on some ideas pursued by Hamilton (1993), Landy & Szalay (1993) and Picka (1997) on improving the estimation of correlation functions g(r) of point processes and random sets in 3. The basic idea is to use estimators of the intensity (or intensity squared, eqn. 23) that are adapted to the numerator term of the ratio estimators for these functions. The resulting correlation between numerator and denominator of the ratio should decrease the bias and variance of the estimator. The interested reader is referred to the report by Stoyan & Stoyan (1998) for more detailed descriptions of the proposed estimators. This class of estimator will be referred to as ‘adapted intensity estimators’.

45

Baddeley & Gill (1994, 1997) and Baddeley (1999b) described new estimators for F(r), G(r) and K(r) by considering the interpoint distances in a sampling window to be censored by the distance to the nearest boundary, in analogy with censoring times in survival analysis. Basically, the distance from a given reference point to the nearest point of the point process is censored by its distance to the boundary of W. The set of reference points which are at least r away from both the nearest point of the point process and the border of W, {y: d(y, Ν) ∧ d(y, ∂W) ≥ r}, are considered to be ‘at risk of failure’ (where ‘failure’ means that the nearest neighbour from Ν has been identified – really a ‘success’!). The set of reference points which have a nearest neighbour from Ν at distance r which is closer to y than the boundary, {y: d(y, Ν) = r ≤ d(y, ∂W)}, are considered to be ‘observed failures’. The “Kaplan-Meier” estimator for F is given by

(

)

⎛ r S W( − s ) ∩ ∂ ( Ν ( + s ) ) ⎞ KM ˆ ds ⎟ (r ) = 1 − exp⎜ − ∫ F ⎜ ⎟ ⎝ 0 V W( − s ) \ Ν ( + s ) ⎠

(

)

(36)

(see fig 11). In practice, one would not compute the integral in (39), but instead discretise W by superimposing a rectangular lattice of points. For each point o of the grid in W one calculates the censored distance d(o, Ν) ∧ d(o, ∂W) and the number of points which have a nearest neighbour from Ν closer than the boundary. Then one calculates, ⎛ # (d (o, Ν ) ≤ s ≤ d (o, ∂W ) ) ⎞ ⎟⎟ . Fˆ KM (r ) = 1 − ∏ ⎜⎜1 − ( ) ∧ ∂ ≥ # d ( o , Ν ) d ( o , W ) s ⎠ s≤r ⎝

(37)

46

As the point grid gets finer, the empirical function Fˆ KM converges to a smooth function. Chiu & Stoyan (1998) showed that the border, Hanisch and KaplanMeier estimators of F(r) are all closely related.

W

r

Figure 11—Geometry of the Kaplan-Meier estimator of F(r). Points o ∈ W ‘at risk’ are shaded, and ‘observed failures’ constitute the curved boundary of the shaded region (after Baddeley, 1999b).

A discretised form of the Kaplan-Meier estimator for G(r) is given by k # (r < d ( x, Ν ) ≤ r KM i i +1 ∧ d ( x, ∂W ) ) ˆ G (rk ) = 1 − ∏ # (d ( x, Ν ) ∧ d ( x, ∂W ) ≥ rk ) i =1

(38)

where x ∈ Ν ∩ W, rk = k·∆ (k = 1,…, n) and ∆ is the increment of distance used. Baddeley et al (1993) exploit replication and nesting in the sampling design to develop more efficient estimators of second-order functions. They exploited the fact that the estimators for second-order functions are ratio estimators (where

47

the denominator involves the variable volume of the sampling window) to use a ratio regression approach for pooling replicates (fig 12). Karlsson & Liljeborg (1994) demonstrated the application of several of the above estimators for the 3-D coordinates of pore centroids in a translucent alumina obtained using confocal scanning laser microscopy. They determined Gˆ H (r ) and Kˆ I ( r ) from coordinates of centroids obtained using computerassisted image analysis. Estimates of the functions L(r) and g(r) were calculated from estimates of K(r). Data collected from various positions in the material were pooled. Reed et al (1997) used fluorescence confocal microscopy to acquire a time sequence of 3-D images of the evolving structure of liquid foam. As a liquid

Poisson F

r

(a)

(b)

Figure 12—‘Population’ estimates of (a) empty space function (border method) and (b) K-function (isotropic correction), pooled across all windows in a replicated 3-D point process (Baddeley et al, 1993) (—— estimated; ··········· Poisson; - - - - - 95% confidence bounds on estimate). Note that Fˆ B(r) is plotted versus the theoretical Poisson FP while K(r) is plotted versus r. The Poisson F lies everywhere outside or on the boundary of the confidence intervals of Fˆ B(r) indicating an inhibited (regular) pattern. The empirical K-function shows a definite dip relative to the theoretical Poisson curve, suggesting regularity of the pattern, in the range 15-35 µm and a recovery beyond 35 µm.

48

foam ages, initially spherical bubbles (froth) burst, coalesce and reorder themselves, resulting in an increasingly coarse polygonal foam structure. The cartesian coordinates of the foam vertices were extracted from the confocal images. Empirical G(r), K(r) and L(r) functions were derived for the resulting spatial point patterns. The results revealed a clear hard-core region (representing the smallest length scale of the foam) followed by regularity and then a region of clustering at larger length scales – a combination of regular and aggregated behaviour. They cited an unpublished study by Reed & Vandenbroeck (1995) where similar qualitative behaviour was reported for a flexible polyurethane foam structure. Baddeley & Turner (2000) describe and illustrate the use of maximum pseudolikelihood functions for estimating the parameters of a wide variety of Gibbs point processes. The approach can accommodate spatial inhomogeneity, interaction between points, and marked point processes. The procedure relies heavily on the use of Monte Carlo simulations. Performance of Estimators

Reed & Howard (1997) compared several edge-correction methods for G(r) with respect to the edge-effect bias introduced for replicated samples of points in a three-dimensional brick of fixed volume from Monte-Carlo simulations of a spatial Poisson point process. The estimators considered were ‘plus sampling’ (Miles, 1974), ‘minus sampling’ (i.e. the ‘border method’), GH and GHW, Doguwa-Upton, the GKM and their expectation estimator GE. They concluded that for 800-1000 total sample points (i.e. replicates × points per replicate) the GE, Doguwa-Upton, GKM and GHW estimators all performed very well. In practice the labour involved in sampling this number of points from spatial distributions can be prohibitive. For smaller total sample sizes (400 or less) the

49

GE and the related Doguwa-Upton performed best. ‘Plus sampling’, the GH estimator and a naive estimator all performed poorly. Baddeley et al (1993) found that both the border method GB and the GH estimators for G(r) gave unreasonable results because for their data set they ended up with too few points in both the numerator and denominators of the estimators. They suggest that kernel smoothing techniques do not solve the edge problem, but only obscure it. They had better results from hypothesis testing of uncorrected G(r) estimates compared with a Poisson null model. Stoyan & Stoyan (1998) carried out Monte Carlo simulation studies to evaluate the biases and variances of various proposed adapted-intensity estimators for g, K, L. The three-dimensional case was investigated by simulating several point process models – including Poisson, hard-core and cluster point processes – in the unit cube. The reader is referred to the report for further details. One reason that the L-function has been promoted for the description of planar point processes, is that variances are stabilized compared to K(r) estimates. There is some disagreement as to whether or not this happens in the three-dimensional case (König et al, 1991; Baddeley et al, 1993). Based on this study, Stoyan & Stoyan (1998) concluded that it did. The performance of model-based estimators can only be as good as the validity of the assumptions upon which they are based. Considerable bias may result from the application of estimators assuming homogeneity or anisotropy to point processes not exhibiting these properties and vice versa (Stoyan, 1991). Doguwa & Upton (1989) found that the Ohser–Stoyan (1981) estimator for K(r) of an anisotropic process performed poorly when applied to isotropic point patterns. Another challenge when estimating summary statistic functions, is identifying the features of the functions for small r values (e.g. Stoyan & Stoyan, 1996).

50

The estimators for spatial point processes described so far, all require that the coordinates of the points in 3-D space are accessible for large enough r in all possible orientations. Methods to obtain quantitative data have included reconstruction from physical sections (e.g. Bjaalie and Diggle, 1990; Braengaard & Gundersen, 1986; Diggle et al, 1991) or coordinates of points in stacks of serial optical sections (bricks) obtained using confocal scanning laser microscopy (e.g. Baddeley et al, 1987, 1993; Karlsson & Lileborg, 1994; König et al, 1991; Rigaut et al, 1988). Baddeley et al (1993) captured about 800 points in 40 rectangular bricks of dimensions 82 µm × 100 µm × h µm (h ranging 30 – 115 µm, approximately). After edge corrections, only a fraction of these (about 50 in the worst case of minus sampling) were available for analysis. König et al (1991) captured 50–165 cells/brick in bricks of dimensions 85 µm × 85 µm × 36µm, reduced to about 79 µm × 79 µm × 30 µm after allowing for a guard region. Karlsson & Liljeborg (1994) also discarded a significant portion of their data using the border method for G estimation. Three-dimensional reconstruction from thin sections obtained using conventional (non-confocal) light microscopy is subject to large biases due to projection effects. Furthermore the depth of material in a ‘thick’ section that can be optically scanned using conventional light microscopy may be inadequate. For example, using light microscopy it is possible to obtain high quality optical sections through a focal depth of no more than about 2–4 cell diameters of a microcellular polycarbonate polymer foam, depending on the foam relative density. For practical application of random process statistics to spatial structures there is a real need for unbiased stereological methods to determine second order characteristic functions from limited sets of thin physical or optical sections.

51

Stereological estimators

Most of the published stereological methods to estimate second order distribution functions for particle systems from information on a single section required making assumptions about the shape of grains for practical application. It has been common to assume that they are convex (typically, with constant shape) in which case the profiles of the germs observed on an arbitrary section also form a germ-grain model with convex grains (Stoyan et al, 1995). In general, these model-based estimators have large biases. Recently, developments in ‘local stereology’ have been extended to propose stereological methods for estimating the K-function from information contained in a slice having some thickness (unlike true planar sections) centred at each of a sample of reference points, without any assumption about the shape of particles (Jensen, 1998). These estimators for general random models will be discussed later. For a spatial point process the proposed estimator is given by

λK (r ) =

1 ∑ N (W ) x∈Ν ∩W

∑

1

⎛ slice of width ⎞ η y − x ,t

(39)

y∈Ν ∩(b ( x ,r ) \ x )∩⎜ ⎟ ⎝ t centred at x ⎠

where N(W) is the number of reference points x sampled in W, and ηy-x,t is the probability of sampling a point in a slice of width t. The estimator can be considered as an ‘edge corrected’ estimator in the sense defined by Baddeley (1999b) and relies on isotropy of the process for unbiasedness. The weighting function in eqn. 39 is given by

η y − x ,t

t ⎧ ⎪⎪ 2 y − x if t ≤ y − x =⎨ ⎪ 1 if t > y − x ⎪⎩ 2

(40a)

52

for isotropic conditions, and

η y − x ,t

t ⎧1 -1 sin ⎪⎪ d ⊥ ( x, y ) = ⎨π ⎪1 ⎪⎩ 2

if t ≤ d ⊥ ( x, y ) (40b)

if t > d ⊥ ( x, y )

when used with vertical sections, where d⊥(x, y) is the orthogonal distance from point y to an axis, oriented in the vertical direction, through the sampling point x (Jensen et al, 1990). The estimator becomes practical when the material can be sectioned optically, and requires the user to measure linear distances, without the need to determine 3-D angles (fig 12).

b(x, r) r x

t

W Figure 12—Illustration of a sampling design to estimate the K-function of an isotropic spatial point process based on optical sectioning in 3-D and eqns. 39 and 40a (after Jensen, 1998). A pair of parallel lines a distance t apart (the 2-D projection of a slice or disector) is centred at each ‘typical’ point x captured in the sampling frame W, from which distances to all other xi contained in the slice are measured. Providing the optical plane is very thin (e.g. obtained using a high numerical aperture objective), then distances measured between points on the screen will be close to the 3-D spatial distance. Note too that this design can be implemented using a virtual global spatial sampling design and that W itself may be an optical section through a disector of known height.

53

Generalization for Random Sets Up to this point, many ideas and methods of spatial statistics have been reviewed in the context of spatial point patterns. It may be preferable to treat cells of a cellular microstructure as particles with positive volume without defining real or imaginary ‘points’. The various second-order property functions that have been described can all be naturally extended to general spatial patterns – random closed sets. Some of these will be presented now. Covariance function

The covariance of a volume process (e.g., of regions occupied by one phase of a composite material) is the second-order product density of the volume measure (Stoyan et al, 1995). The covariance C(r) of a volume process X is the conditional probability that two points separated by r both hit the process, given that both points hit the reference space (the matrix material10) Y C(r) = Pr[x ∈ X, (x + r) ∈ X | x ∈ Y, (x + r) ∈ Y]

(41)

By definition, C(0) = VV. For large r, C(r) → VV2. Under conditions of stationarity and isotropy C(r) can be estimated stereologically without bias from planar sections as the volume fraction of X dilated by r (a form of minus sampling) (Gerlach & Stoyan, 1986; Stoyan et al, 1995, pp. 214-215). Line grids constructed of pairs of grid points a distance r apart may be superimposed onto a section as shown in fig 13. For each distance r the number of point pairs r apart falling within X, Pr(X), is counted and the number of point pairs falling entirely within the reference space, Pr(Y), is counted. An unbiased estimate of C(r) is given by the ratio 10

If Y is ‘infinite’ in extent, C(r) can be defined more simply (cf. Stoyan et al, 1995).

54

C(r) := Pr(X)/Pr(Y)

(42)

Gerlach & Stoyan (1986) assumed motion invariance of the volume process and then estimated C(r) from single arbitratry planar sections. In practice it can be extremely time consuming to use a series of line grids of different fixed lengths. One practical alternative when using automatic image analysis is to use a regular point grid and record the coordinates of all points hitting phase X for further analysis. The pattern of point hits from a quadratic grid with interpoint spacing ∆ can be used to determine the covariance for r = ∆, √2·∆, 2∆, … etc. Where motion invariance cannot be assumed, (42) can be used with equal validity in a design-based approach using isotropic uniformly randomly sampled sections. Reed & Howard (2000) used linear dipoles with sine weighted directions on vertical sections.



r







































Figure 13—An estimate of the value of the covariance function for distance r, Cˆ ( r ) , is given by the ratio of point pairs separated by r [i.e., the endpoints of a grid of lines, coined linear dipole probes by Reed & Howard (1999)] lying inside the primary (shaded) phase X to the number of point pairs falling within the reference space (the entire section). The line joining the endpoints is for convenience. It does not have to be fully within X. Edge effects are avoided by using minus sampling; if one or both points of a pair do not hit the reference space, they are simply not counted. For the case illustrated, an estimate of C(r) for the matrix phase of a cellular material is, Cˆ ( r ) = 1/8 = 0.125. Either the material must be isotropic or sections must be IUR for the estimate to be unbiased.

55

Mattfeldt et al (1993) used automatic image analysis and stereology to estimate the covariance function and volume fractions of benign and malignant cancerous cells in an essentially isotropic tissue. To investigate the 3-D spatial arrangement they then determined empirical K(r), g(r) and R(r) functions from the estimates of C(r) and volume ‘intensity’ λ = VV. They concluded that C(r) and R(r) alone did not permit safe distinctions between experimental groups because these functions depended on spatial pattern as well as intensity. Definite non-Poisson spatial interactions between cells were detected using g(r) (in particular) and K(r), which are not influenced by VV. Wiencek & Stoyan (1993) also strongly caution that use of the covariance alone as a measure of spatial dispersion can lead to erroneous conclusions. This is true for all the second order characteristics we have considered. Torquato et al (1988) write the covariance as the sum of two terms: the probability that o and r belong to the different particles C1(r); and, the probability that the points belong to the same particle, the ‘two-point cluster function’ C2(r). They argue that C2(r) is a substantially better parameter than C(r) for describing a microstructure. Stoyan et al (1995) suggest that the decomposition of C(r) terms is also useful for interpreting empirical covariances. n-point correlation functions

Torquato (1986) introduced the very general ‘n-point distribution function’, Hn(xs; xm; xq) of a two-phase material, defined as the probability that s random points{xs} lie on certain interfaces ℑ and m random locations {xm} lie in an eroded sub-region ℘ of the complementary phase Y, given that phase X consists of q inclusions located at {xq}. The region ℘ ⊂ Y is that formed by dilating X by a ball of diameter db; the surface ℑ is then the interface between region ℘

56

and its complement space ℘* (fig 14). In the limit as db → 0, ℘* is X, and ℑ the interface between X and Y, ∂X. The material is thus represented by a random closed set, specifically a generalized Boolean model. The q grains may have a distribution of radii and may overlap to form a wide range of microstructures.

ℑ

db

℘

Figure 14—Subdivision of two-phase material made up of (shaded) particles of phase X randomly dispersed in matrix Y, into two complementary regions: (hatched) ℘ is the erosion of Y formed by dilating X by a ball of diameter db; and (bold line) interface ℑ = ∂℘.

Torquato (1991) presented a series representation of Hn for a stationary Boolean model with spherical grains. By considering the limit as db → 0 and certain restrictions on the {xs} and {xm}, analytical expressions for a variety of n-point correlation functions were derived. These included various second-order properties. The ‘two-point/q-particle function’ G2(r) corresponds to the empty space function F(r). The nearest neighbour distribution function can also be expressed as well as the ‘n-point probability functions’ Sn(xn). The latter describe the probability of finding one point at each of n positions x1, …, xn in one of the phases (Brown, 1955; cited in Torquato, 1991). The Sn(xn) are closely related to the nth-order product density and nth-order factorial moment measure (cf. eqns. 9-10). More precisely, S1 = VV and the ‘void-void correlation function’ Fvv = S2(r) is the covariance C(r) of the ‘void’ phase. Similarly,

57

surface-surface FSS(r) and surface-void FSV(r) correlation functions are defined as the probabilities that given two random points separated by r with one point lying on the interface, the second point will lie on the interface or in the void phase, respectively.11 For large r, FSV → SVVV and FSS → SV2. The third-order correlation function S3(x1, x2, x3) can be considered the probability that the triangle with uniformly randomly positioned vertices falls in the phase of interest. These functions have been calculated for a variety of (generalized) Boolean models (Torquato, 1991). Two-point and higher order correlation functions have been used to characterise the microstructure of stationary two-phase composite and porous media for purposes of determining bounds on effective material properties of both homogeneous and heterogeneous materials (for references, see the reviews by Torquato, 1991, 1998, 2000; Pyrz, 1999). The use of the third-order correlation functions has substantially improved predictions of bounds for effective properties of two-phase materials compared to use of second-order moments. It may be worth noting that the empirical estimation of interphase correlation functions should be based on proper sampling procedures. For example ‘typical interface points’ used in the estimation of the surface-surface and surface-void correlation functions can be sampled using line grids, as described earlier. Edge effects must be accounted for. Chiu & Stoyan (1998) present edge-corrected estimators of F(r) for a stationary random closed set.

11

Clearly, the choice specifying the ‘void’ phase versus the complementary ‘particle’ phase of a two-phase material will depend on the property and material of interest. The terminology used by Torquato and others, arises from the fact that this approach to micromechanics was originally applied to transport in porous media.

58

Generalized K-functions

The concept of K-functions can also be naturally extended for random sets (Cruz-Orive, 1987, 1989; Jensen, 1991, 1998; Jensen & Kiêu, 1992). Let the q-dimensional intensity λq, (0 ≤ q ≤ d), be the expected measure (the mean ‘amount’) of a random q-dimensional set Y per unit volume (Mecke, 1981; Cruz-Orive, 1989; Stoyan et al, 1995). For example in 3, we can interpret the intensity in stereological notation as NV, LV, SV, and VV for q = 0, 1, 2 and 3, respectively. We will restrict ourselves to d = 3. The q-dimensional K-function for X in 3 is now defined as (Cruz-Orive, 1989)

λ q K q (r ) = [Mean measure of particles contained in a sphere of radius r, centred at a ‘typical point’ of Y] How to sample typical points of a structure was examined earlier in the section on Spatial Sampling. Similarly q-dimensional analogous to the related secondorder functions – e.g. gq(r), Rq(r) – can also be defined (Cruz-Orive, 1989). Cruz-Orive (1987, 1989) proposed stereological methods to estimate Kq(r) and gq(r) for volume processes in 3-D, with edge corrections, using test systems of points on isotropic section planes through the structure. Jensen et al (1990) and Jensen & Kiêu (1992) described the development of a general estimator of Kq for random sets. The approach is model-based (it requires isotropy to be unbiased) but draws upon developments in design-based ‘local stereology’. In the first step a set of reference points {xi} is sampled, e.g. in a window W. Each reference point is then treated as an origin O. The next step is to determine the q-dimensional content (the ‘q-volume’) Vq (i.e. number

59

for q = 0, length for q = 1, etc.) of each ‘particle’ y ∈ Y that lies at a distance no more than r from O,

(λ K ) = q

q

(

1 ∑∑Vq y j : xi − y j ≤ r N (W ) x∈W y

)

(43)

where N(W) is the number of reference points x sampled in W. They propose estimating the q-volume of each sampled particle based on measurements made in a slice centred on each O. These are Horvitz-Thompson type estimators where the measured geometric quantity is weighted by the inverse sampling probability. Jensen et al (1992) and Jensen (1998) present tables of sampling probabilities for different sampling schemes. Eqn. 39 and fig 12 presented the special case for a spatial point process (q = 0, d = 3) with sampling probabilities given by eqn. 40. Other than the earlier study of Evans & Gundersen (1989) who estimated the spatial distribution of glial cells around neurons in biological tissue, it appears that these new stereological estimators of 3-D spatial distribution have not yet been implemented in empirical studies. Hahn et al (1999) presented model-based stereological methods to estimate the continuous variation of stereological intensities VV(z), SV(z), LV(z) and NV(z) of horizontally stationary structures using measurements made on vertical sections. They presented several applications of their methods, including the estimation of VV(z) and SV(z) and particle diameter distribution for a Boolean model with a gradient (fig 15).

60

d(z), mm

NV(z), mm-1 Figure 15—Left: Vertical section of a gradient bronze sinter filter. Right: Estimated spatial intensity NV(z) and mean particle diameter d(z) (Hahn et al, 1999).

Random Spatial Tessellations A random ‘tessellation’ (or ‘mosaic’) is a random partitioning of space (d, d ≥ 2) into d-dimensional polytopes, the ‘cells’.12 Cells are non-overlapping and, together with their edges, space filling (figs 16-18). One way to construct a random spatial tessellation uses a locally finite random point pattern in 3, Ν = {xi}, as a set of ‘nuclei’, ‘germs’ or ‘seeds’. Each nucleus xi generates a bounded, convex Voronoi polyhedron of all points in 3 which have xi as its nearest neighbour (Okabe et al, 1992; Stoyan et al, 1995; Møller, 1999b). The 12

This approach to subdividing space is called by a plethora of names. ‘Voronoi diagram’ or ‘Voronoi tessellation’ (after Voronoi, 1908) is probably the most extensively used, and was the name adopted by Okabe et al (1992) in their classic monograph. The cells may also be called ‘polygons/ polyhedra’, ‘crystals’, point

61

resulting tessellation is called a Voronoi tessellation, also known as a Theissen or Dirichlet tessellation (solid lines in fig 16). The ‘dual tessellation’, widely coined a Delauney tessellation, may be constructed by joining points whose Voronoi cells share a planar facet (dashed lines in fig 16). The Delauney tessellation is mathematically useful because some distributional properties of the edges and vertices of Voronoi tessellations can be derived by consideration of the Delauney tessellation (Muche, 1998). A comprehensive treatment of Voronoi and Delauney tessellations, their analysis and application is given by Okabe et al (1992). This monograph summarizes and synthesises knowledge from many disciplines, and includes tables of properties of spatial and sectional tessellations. Korneenko (1991) provides an extensive bibliography of literature on Voronoi diagrams including applications. Stoyan et al (1995, chapter 10, section 11.5) also review random tessellations. Voronoi tessellations have been successfully used to describe space-filling mosaic structures arising from growth processes. The Voronoi tessellation, which does not satisfy Plateau’s laws of equilibrium, has been used as the basis of simulations of 3-D closed-cell foam structure (Kraynik et al, 1999). Modified tessellations, relaxed so that they do obey Plateau’s laws, have been used to simulate dry foam structures (Kraynik et al, 1999; Sullivan, 1999). Quite remarkably, the statistical properties during foam evolution appear to be independent of scale, e.g. the average number of sides of polygonal cells do not change as a result of evolution (Stavans, 1999). This kind of behaviour is also reflected in properties of the Voronoi tessellation. Kurtz & Capay (1980a,b) carried out a very extensive study of many aspects of the topological evolution of grain growth in metals, including what is seen on a sectional tessellation through the structure.

‘neighbourhoods’ or ‘domains’, depending on the application. See Okabe et al

62

Figure 16—A planar Voronoi tessellation (solid lines) and its dual Delaunay tessellation (dashed lines).

Figure 17—A spatial Voronoi cell (Hahn & Lorz, 1994).

(1992) for a history of the development of the concept of Voronoi diagrams.

63

Figure 18—(left) Simulations of (a) ‘regular’ (i.e. Matérn hard-core), (b) completely random (i.e. Poisson), and (c) ‘clustered’ (i.e. Matérn cluster) 3-D Voronoi tessellations and (right) random plane sections through these (Hahn & Lorz, 1994).

Voronoi tessellations have also been used as the basis of engineering models for material behaviour. The Voronoi finite element method (VFEM) uses a Voronoi tessellation to discretise the material domain into a mesh of finite elements (Ghosh & Mukhopadhyay, 1993; Ghosh & Mallett, 1994; Kumar & Kurtz,

64

1994; Moorthy & Ghosh, 1996; Vejen & Pyrz, 1999). In this way, the effect of local geometrical features of the microstructure, such as size, shape, orientation and location of particles can be incorporated. The Voronoi tessellation can be used to analyse point processes (e.g. Baddeley & Møller, 1989; Gavrikov et al, 1993; Murphy & Selkow, 1990; Okabe et al, 1992 chapter 7; Pyrz & Bochenek, 1998; Sibson, 1980). Baddeley & Møller (1989) used ‘nearest neighbour’ relations established from the Voronoi tessellation in the generation of Markov marked point processes where new points are added to an existing point process configuration conditional on neighbourhood relationships of the points. Poisson-Voronoi tessellation

If the generating nuclei constitute a homogenous Poisson point process then the associated tessellation is called a Poisson-Voronoi tessellation (PVT). Figure

19 shows the growth of a PVT (called the ‘cell model’ by Meijering, 1953). Spatial Poisson-Voronoi cells are convex polyhedra with planar faces and straight edges. Knowledge of the single parameter, the intensity λ of germs, completely describes the properties of the PVT. Note that λ-1 is the mean volume of cells, v N .

Just as the homogeneous Poisson point process is used as a null-model for point patterns, the Poisson-Voronoi model is used as a normative model against which tessellations are evaluated. Poisson-Voronoi tessellations have very well defined properties (tables 1, 3). Some of these properties are inherited from the properties of the underlying point process (e.g. stationarity, isotropy) while others are common to all random tessellations.

65

Figure 19—Computer generated time-lapse sequence showing development of a Poisson-Voronoi microstructure from simultaneously activated Poisson-distributed germs as seen on a planar section through the transforming body (Mahin et al, 1980).

Non-homogeneous Poisson-Voronoi and anisotropic ‘Voronoi-G’ tessellations can also be generated (fig 20). Johnson-Mehl tessellations

A homogeneous Johnson-Mehl tessellation (JMT) (Johnson & Mehl, 1939) results from allowing constant net rate growth of Poisson distributed germs to initiate randomly at Poisson distributed times (fig 21). Growth stops along a ray when the cell impinges on another. The final attained tessellation has curved faces and edges, with the curvature being convex towards the cell which nucleated first. The spatial JMT is topologically different than the spatial

66

Voronoi tessellation and cells may be non-convex (fig 22). Some theoretically determined characteristics of the typical JMT cell are summarized in tables 2-3. On 2-D sections through the structure, ‘caps’ (one- and two-sided profiles) appear when the section cuts through the nose of the hyperboloid surface separating two cells. Thiele et al (1999) hypothesized that the statistical distribution of cells of a 2-D collagen foam, would resemble distributions of the JMT. The JMT model was found to be acceptable only for qualitative comparisons because the rate of growth of the collagen cells was not constant for all times, but decreased with increasing viscosity as the solvent evaporated.

(a)

(b)

Figure 20—Examples of non-stationary and non-isotropic tessellations: (a) Vertical section through a simulated gradient Poisson-Voronoi tessellation with intensity function λ(z) = (0.05 + 0.2z)-3, 0 ≤ z ≤ 2 (Hahn et al, 1999); (b) Anisotropic tessellation generated from Poisson distributed germs with cells growing two times more rapidly in the vertical-direction (Scheike, 1994).

67

Table 1. Some characteristics of the typical spatial Poisson–Voronoi cell. Characteristic

Expected value

Variance

In 3: Number of vertices P

27.071

44.5

Number of edges E

40.606

100.1

Number of faces F

15.535

11.125

Total edge length B

17.496 λ-1/3

13.618 λ-2/3

Mean height (caliper diameter) H

1.458 λ-1/3

0.0305†

Surface area S

5.821 λ-2/3

2.192 λ-4/3

λ-1

Volume V Randomly selected dihedral angle α

2π/3

0.179 λ-2

π2

18

−3

8

As seen on a random planar section:

Number of vertices n0

6

2.863

Perimeter l1

3.136 λ-1/3

1.477 λ-1/3

Area a

0.686 λ-2/3

0.227 λ-4/3

Randomly selected interior angle β

2.000†

0.351†

Sources: Meijering (1953), Miles (1972), Lorz (1991), Møller (1989), Muche (1998) † Determined from simulations. * i.e. the angle between two faces emanating from a randomly selected edge of a typical cell.

68

Figure 21—Computer generated time-lapse sequence showing development of a Johnson-Mehl (JMT) microstructure as seen on a planar section through the transforming body (Mahin et al, 1980).

Figure 22 Examples of spatial Johnson-Mehl cells (Møller, 1995). The cell to the left is simulated 3-D ‘lense’ consisting of two faces and one edge (a closed curve) and no vertices.

69

Table 2. Some characteristics of the typical spatial Johnson-Mehl cell. The inequalities arise because of the possible presence on a planar section of caps fully surrounded by the planar section of an adjoining cell.

Characteristic

Expected value

In 3: Number of vertices P

22.559

Number of edges E

>33.839

Number of faces F

>13.280

Total edge length B

17.496 λ-1/3

Mean height H

>1.226 λ-1/3

Surface area S

5.143 λ-2/3

Volume V

λ-1

As seen on a random planar section:

Number of vertices n0

1.238 λ2/3 †

LV = 2 PA

PA = 2.916 λ2/3

2.476 λ2/3 †

SV = π4 B A = 2 I L

BA = 2.286 λ1/3

2.019 λ1/3

Derived relations* Mean edge length of sectional cell

BA l =2 Q A n0

0.523 λ−1/3