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10 Stereology Adrian BacT6eley Centre for Mathen~atics anti Computer Science ~ O. ~ Introduction Stereology is a spatial version of sampling theory. It was initially devel- opecl in biology and materials science as a quick way of analyzing three- dimensional solid materials (such as rock, living tissue, and metals) from information visible on a two-dimensional plane section through the material. It now embraces all geometrical sampling operations, such as clipping a two- dimensional image inside a window, taking one-dimensional linear probes, or sampling a spatial pattern at the points of a rectangular grid. Applications include anatomy, cell biology and pathology; materials science, mineralogy and metallurgy; botany, ecology and forestry; geology and petrology; and image processing and computer graphics. It is not the aim of stereology to reconstruct an entire three-dimensional object. Typically, only a few sections or samples are taken, and their spa- tial position is not recorded. Further it is usually impossible to model the three-climensional structure explicitly. Instead, stereology uses simple non- parametric techniques to estimate "geometrical parameters" such as volume and surface area. Simplicity is the key word; the estimation relies only on fundamental geometric facts and classical sampling theory. As a result, stereological methods are almost "assumption free," and are applicable in many different sciences. Applications and general concepts are described in 10.1. Section 10.2 is a more detailed statistical treatment. Section 10.3 describes newer discov- eries and research problems. 181

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82 lO.2 Concepts and Applications 10.2.1 Information from Lower-Dimensional Samples In 1847 the French mineralogist Delesse published a revolutionary method for measuring the mineral content in a sample of rock [223. Instead of crush- ing the rock to separate the different minerals, one simply cuts a plane section through it. Delesse had realized that the proportion by volume of a particular mineral can be estimated from its proportion by area visible in the section. Model the rock as a set X C R3 containing a subset Y C X, the mineral phase of interest. The objective is to estimate the volume fraction V(Y) where Vt ~ denotes volume. Let T denote a plane in three dimensions, so that X n T is the plane section of the rock, and Y n T is that part of the section occupied by the mineral phase. Delesse's method estimates Vv from the area fraction AA = A(Y n T) A(XnT) ' where A(-) denotes area in the two-dimensional section. This is like a survey sampling problem: X represents the "population" and X n T the "sample" from which we want to estimate a population parameter Vv. Astoundingly, AA is an unbiased estimator VV = EAA (10.1) (under the right sampling conditions), without any assumptions about the shape of X and Y. This follows from the basic geometrical fact that the volume of a three-dimensional object is the integral of the areas of its two- dimensional plane slices. Here E denotes expectation with respect to a suitable random sampling design (not the most obvious one); we give details in 10.2. Delesse's method was later simplified [74] by placing a grid of parallel lines over the plane section, with the aid of a transparent sheet. Then area fractions AA can be estimated from length fractions At,, i.e., the relative lengths of the mineral phases on the line grid. This was simplified even further by Glagoleff [23] who showed that if we superimpose a rectangular grid of points over the section plane, the area fraction AA can be estimated

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183 from the proportion Pp of grid points that "hit" (lie over) the mineral phase. In both cases the estimators are unbiased. Demonstrate this with a "party trick." Take a sheet of graph paper ruled with (say) thin lines every 1 mm and thick lines every 5 mm. Cut out an arbitrary shape. Ask someone to determine the area of the cutout by counting all the ~ mm squares. Meanwhile estimate the area stereologically by counting the 5 mm crossing points that are visible on the paper, and multiplying by 25. The result wiD be unbiased, typically accurate to about To, and is 25 times as fast to compute. Similar tricks exist for estimating other geometrical quantities. The length of a plane curve can be estimated from the number of crossing points between the curve and a grid of paraHel lines. The surface area of a curved surface in three-dimensional space can be estimated from the length of its trace on a plane section [824. The length of a curve in space can be es- timated from the number of points where the curve hits a section plane. Certain quantities related to curvature can also be estimated [9,214. TABLE 10.1: Standard Notation for Geometrical Quantities Space dimension n set X Letter Meaning 3 solid domain V volume curved surface S (surface) area space curve ~ curve length finite set of objects N number of objects curved surface M,K integral of mean curvature 2 plane domain A area curve [,B curve length finite set of points I,P number of points finite set of objects N,Q number of objects curve C total curvature These methods are summarized in Table 10.2 with notation listed in Table 10.1. Each quantity in Table 10.2 is an unbiased estimator of the quantity to its left (following the arrow). The table is valid only under very strict assumptions of "uniform sampling" (see 10.2) but with very minimal geometrical assumptions, because it relies only on fundamental relationships between volume, area, and length.

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184 TABLE 10.2: Classical Stereological Formulas 3 Vv Sv LV Dimension of Space 2 1 0 ~ up AA or BA 2QA LL ( 2IL Plate 10.1 (preceding page 71) shows an optical microscope image field from a plane section of the lung of a gazelle (magnification x1500. A stereologica] test grid has been superimposed on the image, consisting of 40 test points (circled) and line segments totalling 42 cm in length. Since 7 out of 40 test points hit the tissue (rather than the empty airway), we estimate the volume fraction of tissue as Vv = AA = 7/40 = 17.5%. There are 16 positions where a line segment crosses the tissue-airway boundary, so the surface area of lung/air interface per unit volume of lung is estimated at Sv = 21~ = 2 x i6/~42/1500) = 1143 cm-. Thus, a cubic centimeter of gazelle Jung contains about 1100 cm2 of lung/air interface. 10.2.2 Stereology is Classical Sampling Theory Results like (10.1) were known as early as 1733 with the celebrated needle problem of Button [~] and its successors in integral geometry and geomet- rical probability [84,30,4S,75,76,804. However, the first rigorous statistical foundation was laid out only in 1976 by Miles and Davy t20,61,625. Unbiased estimation, rather than maximum likelihood or minimum mean squared error estimation, is emphasized for several reasons. The distribution of any statistic is difficult to compute because of geometrical complications, and to do so requires severe assumptions about shape (e.g., assuming that X and Y are spheres). One of the beauties of the estimators above is that they are known to be unbiased without geometrical assumptions: they are effectively nonparametric moment estimators. A simple test grid requires only a few decisions ("hit" or "not hit") on any image. This is convenient in some applications where it is laborious or difficult to recognize boundaries or identify the objects of interest. Yet it appears to throw away most of the information in the image. This is

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185 ~ . ~ . . ~ . ~ in fact desirable, for stereological experiments usually generate hundreds of images; it is not efficient (statistically or economically) to analyze a single image in great detail. There is usually enough replication (sections from deferent parts of the sampling material, windows from different parts of a section) to dramatically reduce the overall sampling variance. In biological applications, the variance contributions associated with variation between animals, and between parts of the same animal, are usually far greater than the variance due to stereological sampling [17,244. One of the main stereological discoveries of the 1980s was the pervasive importance of systematic sampling. Recall that for a finite population of n individuals. ordered arbitrarily and numbered 1, . . ., n, a systematic sample 7 ~ _, . ~ ~ - ~ . ~ . ~ ~ ~ with Inverse sampling traction k Is generated by choosing a random number m uniformly distributed in {1, . . ., k) and taking the individuals numbered m, m + k, m + 2k, .... The sample has random size, but can be said to consist of a fixed fraction of the population. The population total of some variable pi associated with each individual, Z=~zi i=1 can be estimated unbiasedly by taking k times the sample total, Z=k ~Zm+jk, see [11]. The approach is similar for a "continuous population": to estimate an integral ~ = If(X)/iX, the numerical integral ~ = ~ it, f (U + A/\) (10.2) is an unbiased estimator of ~ when U is uniformly distributed over [0, A]. Stereological estimates based on grids of points, lines, and the like, are essentially systematic sampling estimates. A point grid is a two-dimensional systematic sample of the continuous two-dimensional plane. Estimators based on systematic samples are indeed quite efficient. The estimator of the area of a plane set using a point grid is known to have asymptotic variance ~ ];a3 as a ~ O. where a is the distance between grid points and ~ is the perimeter length of the set. This is of order n-3/2 rather than net, where n is the expected number of points counted. Negative correlation in systematic samples tends to make them more efficient than independent random samples (depending on the structure of the sampling population).

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186 10.2.3 The Particle Problem Now the bad news. Suppose that our sampling material contains iclentifiable individual objects -call them "particles" such as biological cells, crystal grains in a mineral, or holes in a porous rock. We want to regard these particles as individuals forming a population, and make sampling inferences about them: number of particles, average volume. and so on. UsualIv we cannot sample from this population directly; we have to take plane sections. It is impossible to estimate Nv, the number of points or objects per unit volume, from plane sections in the sense of Table 10.2. One indica- tion of this is the mismatch of dimensions or units. For example, Sv = S(mineral)/V(rock) is in units length2/length3 = lengthy; so are the other terms in the same row. Now NV is in units length-3, and so we would naively expect not to be able to estimate it from lower-dimensional sections. Notice that V, S. and ~ are "aggregate" quantities, defined as integrals over the object of interest, whereas N is an "individual" quantity with no such interpretation in general. Miles [60] gives an elegant sketch proof jus- tifying the estimation of aggregate quantities as a straightforward exchange of integration and expectation. The fundamental problem is that a plane section through a particle pop- ulation is a biased sample of the population. To see this, visualize the entire sampling material sliced thinly end-to-end by a series of parallel planes. Randomly choose one of the slices with equal probability. The chance that a given particle is represented on this slice depends on the number of slices through that particle, i.e., is proportional to the projected height of the particle in the direction normal to the section planes. Hence the sampling design has a bias in favor of larger particles. There are essentially three responses to this problem. We can attempt to numerically "correct" our data for the effect of the sampling bias; we can choose to measure different variables that are more "natural" in this sampling design; or we can change the sampling design so that it becomes unbiased. In the correction approach to estimating Nv, the two-dimensional quan- tity, we would naively think of using is QA, the number of observed particle profiles per unit area of section. This is indeed related to Nv through the DeHoff-Rhines equation QA = ~ Nv (e.g., [88, p. 142]), where H is the mean projected height or mean caliper diameter (i.e., the average over all particles Xi of the mean projected height

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187 H(Xi) defined in (10.12) below). Estimation of particle number is thus confounded by particle shape and size (or involves a nuisance parameter associated with shape and size). Even in the happy case where ah particles have the same known shape, the distribution of sizes is usually unknown, and it is hard to estimate H from plane sections. In the second approach, we measure sample quantities only when they are three-dimensionaBy meaningful. For example, if the objective is to study the proportion of "type X" cells in a given tissue, it is not useful to count cells appearing on the section plane, since there is no direct relation between cell sections and cells. Instead, one should measure the area fraction AA of type X cells on section, because this can be translated directly into an estimate of the volume fraction Vv of type X cells. 10.2.4 Unbiased} Counting and Sampling A better solution to the problems of sampling bias mentioned above is to avoid them altogether by devising another, unbiased, sampling method. One example is disector sampling [79,28,273. A disector is a pair of parallel plane sections a fixed distance apart; often these are two consecutive slices through the material. We count a particle only if it appears on one section and not on the other. This gives each particle an equal probabilit.y of being sampled. The only assumptions needed are (1) that no particle is small enough to fall between two section planes at this distance and (2) that the experimenter can establish the identity of each particle, i.e., can tell whenever the same particle has been sectioned on two different planes. Sampling bias is present even in two dimensions. Figure 10.1a shows a sketch of a microscope field-of-view with cell profiles visible. The object is to determine NA, the number of profiles per unit area. A frame F of known area has been superimposed on the image. Naively one would just count all the objects that lie in or on the frame F and divide by the area A(F). The features so counted are shaded in Figure 10.1a. This counting rule, dubbed plus-sampling by Miles [59], clearly produces a biased sample of profiles. If we imagine the field-of-view to be placed at random on the microscope slide, the larger profiles have a greater probability of being sampled. Hence the plus-sampled estimate of NA is biased: the expected number of profiles counted is greater than NA x A(F). An alternative is minus-sampling: count only those profiles that are completely inside the frame F (~59], illustrated in Figure 10.1b). As the name suggests, this counting rule is negatively biased. Smaller profiles have a

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188 ~~'~~~~~~'''~0 0 ~ O ~00 ~ ~ Oo ~ Mono ( 'hO~o~( (a) ~ (b) (A f FIGURE 10.1: Two biased counting rules for planar profiles: (a) plus- sampling, (b) minus-sampling. greater probability of being sampled and counted. Profiles that are actually larger than F Will never be counted. A better suggestion is to count only fractionally the profiles that hit the boundary of the frame. Count profile Xi with weight A(Xi n F)/A(Xi), i.e., the weight is the fraction of area of that profile that is within the window. Using a mean-content formula for windows (610.3.3), we can verify that the integral of this weight over all translations of F is A(F), SO that 1 is an unbiased estimator of NA. An alternative which does not require area calculations is the associated point method [594. Suppose that for any profile X, a unique point v(,Y) is specified; for example, the centroid of X or the bottom left corner. It is not necessary that v(X) be inside X; we assume only that v(X) is equivarian under translations, v(X + t) = v(X) + t for all vector translations t (if X is shifted then the associated point shifts by the same amount). Then an unbiased estimate of NA is to count the number of profiles whose associates] points fall inside F. and divide by A(F). See Figure 10.2a.

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189 (a) OF K:0~0 ~ (W ~ ~ ~ ~ ~ ~> FIGURE 10.2: Two unbiased counting rules for planar profiles: (a) associ- ated point rule, (b) tiling rule. An even easier alternative suggested by Gundersen [25] employs the spe- cial frame illustrated in Figure 10.2b. The solid line, around two sides of the frame and extending to infinity in two directions, is a "forbidden line"; any profile that touches it is not counted. Otherwise any particle that inter- sects the sampling frame, wholly or partially, possibly crossing the dotted boundary, is counted. The rationale for this rule is, briefly, that if the in- finite two-dimensional plane were tiled with copies of this sampling frame (like stacked chairs), then any profile would be counted by exactly one of the frames. Plate 10.2 (preceding page 71) shows the unbiased estimation of Nv for nuclei in human renal giomeruTus using a combination of Gundersen's tiling rule and the disector. Two optical section planes (i.e., different positions of the microscope focal plane) with a separation of 4 ,um are shown. To the left is the top (Iook-up) plane; to the right is the bottom (measuring) plane on which is superimposed a randomly translated tessellation of rectangular counting Frances. Nuclei seen clearly on the Took-up plane are not counted; on the measuring plane, three new nuclei have come into focus in the counting rectangle just below the center. The counting rectangles have real area 527,um2, and so our estimate of Nv is QA = 3/~4 x 527) = 0.001423,um-3, or roughly 1.4 x 106 nuclei per cubic millimeter of gIomeruTus.

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190 10.2.5 Spatial Interpretation and Inverse Problems Its founders envisaged stereology as the spatial interpretation of sections, meaning not only quantitative estimation but also more qualitative reason- ing about three-dimensional geometry, including shape and topology. But spatial reasoning is confused by sampling effects. A single three-climensional object may appear on section as several unconnected objects. A section of a three-dimensional object has smaller diameter than the object itself; while the distance between two objects, or two surfaces (e.g., the thickness of a biological membrane) appears greater on section than in three dimensions. A given three-dimensional object may look very different on different section planes; different three-dimensional objects may fortuitously have identical plane sections. As we have seen, plane sections and rectangular sampling windows gen- erate biased samples of a particle population, since larger particles have a greater probability of being "caught." Other more subtle biases are caused by selecting a particular orientation for the section plane (for example, al- ways slicing muscle tissue transverse to the muscle fibres) or selecting sec- tions where a particular feature is visible. "Real" and "ideal" geometry also differ. Since physical slices of biological tissue have nonzero thickness, the microscope image is actually a projection through a translucent stab of material onto the viewing plane. This is the Holmes effect: images of sectioned objects are larger than they would be for an ideally thin plane section, and some objects may be obscured by others. The traditional response was "correction" based on an i(leal model, for example, assuming the particles are perfect spheres. WickseD [94,95] showed that, for a population of spheres, both Nv and the size distribution of the spheres can be determined from sections: if F is the distribution function of sphere radii and G the distribution of circle radii observed on section, then (under suitable sampling conditions [39,70,804) G has probability density roo g(s) = ~ ,,/ (r2 _ 52)-~/2 dF(r) . This is an integral equation of Abe! type. It is invertible: 2 o 1F(r) = - ,u J (t2 _ r2)-1/29(t) aft, so that F can be uniquely recovered from G. Implicitly this includes the estimation of mean sphere radius ~ so that Nv can also be determined.

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191 Similar equations have been encountered in the estimation of the thickness distribution of a biological membrane [42] and the orientation distribution of a curved surface [16~. This is a typical inverse problem, in which an unknown distribution or function is related to an observable function by an integral equation or other operator. The difficulty here is that the inversion of the equation is numer- ically unstable. For example, the circle distribution G must always have a density. Thus, if we apply a naive inversion procedure to the empirical distribution of circle radii obtained from observations of n circles, the in- verted F is not a distribution function [86~. Again, substituting ~ = 0 in the inversion formula shows that ,~ is proportional to the harmonic mean of G; the estimate offs wiD have poor sampling properties. Part of the trouble is that we are attempting to estimate a whole func- tion F nonparametricaDy without constraints. An alternative is to moclel F parametrically and estimate the parameters from observations of G. Nichol- son [65,66,67] and Watson [85] also showed that some linear functionals of F. such as its moments, can be estimated reliably from samples of G. More sophisticated approaches to inverse problems are mentioned in chapter 2 of this report. In the WickseB context, statisticians have recently proposed kerned smoothing methods [81,14,32,37,83] and iterative methods such as the EM algorithm combined with smoothing t784. Apart from the considerable numerical hitches, some practical objections to the Wicksell approach are that the geometrical mode! is unrealistic and untestable (cells are not perfect spheres); extra factors such as the Holmes effect wiD distort the kernel frisk); the amount of data collected in stere- ological experiments will rarely be sufficient to form a stable estimate of F. By the 1970s there had been many dubious or even erroneous attempts to avoid section effects, and theoretical stereologists evolved the narrower "party line" that it is only possible to reliably estimate certain aggregate three-dimensional quantities such as volume and surface area. More recently, additions to the list of fundamental formulas (Table 10.2) have made it possible to estimate parameters such as the mean squared particle volume, without any assumptions about particle shape. The list of parameters that can be reliably estimated without shape assumptions now includes some quantities related to curvature, orientation, and "shape."

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206 der this sampling regime, the particles have been selected with probabilities proportional to their individual volumes, Pi=PIXiselected)= ~ ~i) I. The cubed intercept lengths estimate the individual volumes; so the mean cubed intercept length is an estimate of the volume-weighted mean particle volume. vv = ~piV(Xi) = Ei V(rX)y i.e., this is the ratio of mean square volume to mean volume. The mean volume can be estimated separately from estimates of total volume and total number; thus we have reliable (approximately unbiased) estimates of the first two moments of particle volume. Methods exist for some higher moments. In some applications, particularly in pathology, the mean square volume (or variance of volume, etc.) has proved very useful in detecting differences between particle populations. Another application of (10.16) is useful in studying materials that do not consist of separate particles. Let Y be any set in three dimensions. For example, Y might be the union of all the cells in a tissue, or the empty space in a porous material. At any point x, define the star set S(x,Y) to be the set of all points y such that the line segment from x to y lies wholly inside Y. See Figure 10.4. If x ~ Y. then S(x,Y) is empty; otherwise, S(x, Y) is a "star-shaped" set, and if Y is convex, then S(x,Y) = Y. Consider the mean star volume, i.e., the mean of V(S(x,Y)) over all points x. This can be estimated on plane sections by the mean cubed length of an intercept through a point in the section. The star volume gives us an interesting measure of the average "local size" of holes in a porous material. Variations on the star volume, involving other moments of intercept length, have recently been considered as indicators of "shape" t294. Covariance, and other second-order parameters, can be estimated with- out bias. This is easiest to describe when X is a stationary random set in R3. The (noncentered) spatial covariance of X at lag h ~ R3 is C(h) = E lx(O) 1X(h), where lx is the indicator function of the set X. In other words, this is the expected volume fraction of points x in space where both x and x + h simultaneously lie inside X. If we are willing to assume that X is isotropic, then C(h) depends only on the length the and not on direction, and we

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207 \ - FIGURE 10.4: The star volume. can estimate C(h) as a function of the from the sample covariance of plane sections of X. This has been applied to extract detailed information about a material [33,774. Second-order statistics have also been used to define indices of mineral liberation t19,184. Non-uniform sampling designs are a very important development. As re- marked in 10.3.3, the general formulas for estimating quantities other than volume require random section planes with (roughly speaking) uniform dis- tribution over all possible orientations and all possible positions. However, many experimenters cannot adhere to this requirement. For example, about a third of all stereological applications require that the section plane be cut in a particular direction, either for physical reasons, or because the structure of interest can only be identified when cut this way. A common case is "vertical" sectioning, where the section plane must be aligned with a specified axis, in other words, normal to some well-defined plane we can call the "horizontal." Thus, there is only one degree of rota- tional freedom for plane orientation and one degree of translational freedom. An unbiased estimate of surface area from vertical sections has recently been found [5] that uses a test grid consisting of cycloid arcs. Sampling designs that are non-uniform in position and orientation have recently been studied [41,90,91,92,934. Mattfel~t and Mall [56,55] proposed samples involving three mutually orthogonal section planes.

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208 10.4.4 Research Frontiers Here we speculate about future advances (other than those already in progress and covered above). Three Dimensions New imaging modalities (such as confocal optical microscopy, infrared Fourier transform imaging) have been developed that can "see" directly into three- dimensional structures such as biological soft tissue and solid bone. Three- climensional images can also be reconstructed computationally from serial optical sections or tomographic data. Rather than making stereology redun- dant, this technology has releaser! a flood of interesting new problems. Stere- ological sampling techniques are needecl, e.g., for counting three-dimensional particles [38], and the methods of two-dimensional spatial statistics (see chapters 4 and 7) need to be aciapted and refined for three dimensions :6,492. Structured Models One reason for the overwhelmingly "nonparametric" character of stereology is that explicit stochastic process moclels have not succeecled in reproducing the very high degree of organization seen in real (especially biological) mi- croscopic structures. This may change in the next five years. Much recent activity in stochastic geometry tS0] is focusing on models where the real- izations have a prescribed, ordered appearance such as random tessellations t63], random dense packings, and random fibre processes. Markov Models Particularly promising is the development of several kinds of Markov models for spatial processes t1,7,10,73,71,723. These are one step more complex than completely random Poisson processes, in that a stochastic interaction is al- Towed between "neighbouring" elements of the process, for example, pairwise interactions between the points in a point process. Markov point processes and random sets can easily be simulated using Monte CarIo methods, and they are convenient for likelihood-based inference [684. Bootstrap Methods Bootstrap resampling methods were introduced to stereology by Hall t31,33] in connection with the point-counting estimator of area fraction AA. The basic idea was to break the sampling region into strips or pieces that are

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209 sufficiently separated for any dependence to be ignored, and to resample these pieces as if they were i.i.d. observations. It seems likely that such methods will prove a useful alternative to parametric modeling, as a way of getting information about variances and confidence levels. The difficulty is in finding acceptable ways of bootstrapping a spatial process with all its inherent spatial dependence. Bibliography [1] Arak, R. J., and D. Surgailis, Markov random fields with polygonal realizations, Probability Theory and Relatecl Fields 80 (1989), 543-579. t2] Ayala, G., Inferencia en Alodelos Booleanos, Ph.D. dissertation, Uni- versity of Valencia, Spain, 1988. [3] Baddeley, A. J., Stochastic geometry: an introduction and reading-list, Int. Stat. Rev. 50 (1982), 179-183. [4] Baddeley, A. J., and P. Averback '7. Micros. 131 (1983), 323-340. Stereology of tubular structures, [5] Baddeley, A. ]., H. J. G. Gundersen, and L. M. Cruz-Orive, Estimation of surface area from vertical sections, IT Micros. 142 (1986), 259-276. [6; Baddeley, A. J., C. V. Howard, A. Boyde, and S. Reid, Three- dimensional analysis of the spatial distribution of particles using the tandem-scanning reflected light microscope, Acta Stereologaca 6 (sup- plement II, 1987), 87-100. [7] Baddeley, A. J., and J. M01ler, Nearest-neighbour Markov point pro- cesses and random sets, Int. Stat. Rev. 57 (1989), 89-121.

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210 [8] Leclerc, G. L., Comte de Buffon, Essai d'arithmetique morale, in Supplement a i'Histoire Naturelle, vol. 4, 1777. [9] Cahn, J. W., The significance of average mean curvature and its de- termination by quantitative metallography, Trans. Amer. Inst. Min.. Mett., Pet. Eng. 239 (1976), 610. [10] Clifford, P., Markov random fields in statistics, in John Hammers1ley Festschrift, to appear, 1990. t11] Cochran, W. G., Sampling Techniques, 3rd edition, John Wiley anti Sons, New York, 1977. [12] Coleman, R., An Introduction to Mathematical Stereology, Memoirs no. 3, Department of Theoretical Statistics, University of Aarhus, Den- mark, 1979. [13] Cruz-Orive, L. M., Best linear unbiased estimators for stereology, Bio- metrics 36 (1980), 595-605. t14] Cruz-Orive, I.. M., Stereology: recent solutions to owl problems and a glimpse into the future, Acta Stereo70gica 6/ (1987), 3-18. [15] Cruz-Orive, L. M., On the precision of systematic sampling: a review of Matheron's transitive methods, '7. Micros. 153 (1989), 315-333. [16] Cruz-Orive, L. M., H.. Hoppeler, O. Mathieu, and E. R. Weibel, Stere- ological analysis of anisotropic structures using directional statistics, Appl. Stat. 34 (1985), 14-32. [17] Cruz-Orive, L. M., and E. R. Weibel, Sampling designs for stereology, '7. Micros. 122 (1981), 235-257. [18] Davy, P. J., Liberation of points, fibres and sheets, pp. 69-78 in Proceed- ings of the Oberwolfach Conference on Stochastic Geometry, Teubner, Leipzig, 1984. [19] Davy, P. J., Probability models for liberation, J. Appl. Proh. 21 (1984), 260-269. [20] Davy, P. J., and R. E. Miles, Sampling theory for opaque spatial spec- imens, J. R. Stat. Soc., B 39 (1977), 56-65. [21] DeHoff, R. T., The quantitative estimation of mean surface curvature, Trans. Amer. Inst. Min., Mett., Pet Eng. 239 (1967), 617.

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