Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
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
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
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.
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
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).
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
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
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.
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.
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.
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."
192 10.2.6 Stochastic Moclels Stereological inference and spatial interpretation are Circuit because we simultaneously have not enough data (important three-dimensional infor- mation is lacking) and too much data (the tw>dimensional images are not analyzed closely). Stochastic models can bridge this gap. Explicit Moclels At one extreme, we could build a probability moclel for the entire spatial structure X using random set models from stochastic geometry t34,4S,53, 77,804. An explicit, parametric model would contain information about the sizes, positions, shapes, relative arrangement, and topological relationships of components in X, which conic! be estimated by comparatively familiar statistical methods. Explicit models in stochastic geometry are mostly ana- Togues of point processes, the Poisson, Cox, cluster, and Markovian cate- gories described in chapter 7. Some statistical theory is available for them t2,33,69,71,77], and they have proved to be excellent descriptions of some simple structures such as rock fractures and crystalline materials [774; but realistic models for the highly organized structures of biology and ecology still etude us. Stationary Models "Nonparametric spatial modeling" is a less demanding approach where the random spatial process X is not explicitly described, but is assumed to be stationary (certain (distributions or moments are invariant under translations and/or rotations). Then we can nonparametrica~y estimate the moments or distributions associated with the process [80, chap. 43. All the standard stereological results can be rederived in this context (see t57,5S,644) since in fact it is a reformulation of the same sampling problem. The reformulation emphasizes how little need be assumed about the spatial structure X, and suggests new estimators. For example, the locational interaction (such as clumping or dispersion) between parts of a spatial structure can be described by the second-order moment characteristics of the process, which can be estimated nonparametrically from sample data. The I: function for point processes [70i, described in chapter 7, is one instance.
193 Semiparametric Models Commonly, only a part of a spatial structure X is of interest. If only that part is modeled, we have a semiparametric statistical model. For example, the thickness of a curved tube could be modeDed by a parametric family of distributions for the radius, without specifying shape or location except to assume that the process is stationary [44. The distribution of directions in a structure (e.g., surface normal vectors, curve tangent vectors) could be modeled by a parametric family of distributions on the unit sphere [164. In a material consisting of several phases or compartments, one can test whether the arrangement of phases is "random" or whether some phases tend to be associated, by applying standard discrete data models t504. Data Moclels At the other extreme would be a statistical mode! for the stereological data obtained from a series of samples Ti. For example, Cruz [13] proposed a proportional linear regression model for, say, A(Y n Ti) against A(X n Ti). This model has been criticized t43], and justifications must remain largely empirical, because it is difficult to derive any distributional theory from probabilistic models of the structure or the sampling design. 10.3 Statistical Theory Stereological methods can be applied with minimal knowledge of the three- dimensional structure under study. However, the sampling rules must be strictly followed; the experimental protocol must generate a random plane or probe with the correct distribution required by stereological theory. In this section, we describe that theory, and show how simple design mistakes can lead to catastrophic errors. 10.3.] What To Estimate It was believed for many years that the normal human brain, alone among all organs, loses cells without replacing them. This was established repeat- edly from estimates of Nv (ceH number per unit volume) at different ages. However, the quantity of real interest is the total cell number N. not Nv. In 1985, Haug [35,36] pointed out that, since younger tissues shrink more during fixation (chemical treatment prior to embedding and sectioning), the
194 total brain volume after fixation was effectively an increasing function of age, and this could account for the decrease in Nv estimates. The situation is still unresolved because of other uncontrolled variables; but it may be that the wrong scientific question was pursued for 20 years. This emphasizes the distinction between a total quantity ~ = p(Y) and a relative quantity ~v= V(X)' where X is the containing set and Y is the "feature" of interest (Y C X). Estimation of absolute and relative quantities is also different. We can convert estimates of ~ to TV and vice versa, given an estimate of V(X); but statistical properties of the estimators are not preserved. For example' the ,. ~ , expectation of a ratio of random variables is not generally equal to the ratio of their expectations. Sampling designs and estimators that are unbiased or optimal for estimating .Bv may not be appropriate for JIB and vice versa. 10.3.2 Inference Statistical inference is called design-based if it relies on the randomness in the sampling design. Expectations are averages over Al possible outcomes of the sampling. In design-based stereology it is assumed that the geometrical object X is fixed and the sampling probe T is random. Meanwhile, inference is called model-based if it imagines the sampling population was generated by a stochastic model. Expectations are averages over aD hypothetical real- izations of this model. in model-based stereology, it is assumed that X is (a bounded sample from) a realization of a random process, and the sampling probe T is arbitrary, say fixed. This is mainly an issue of correctly specifying the population to which we wish to extrapolate statistical inferences. The design-based approach corre- sponds to finite population inference for survey samples [11] or randomized design inference, while the model approach corresponds to superpopulation inference. Miles [60] distinguishes three kinds of inference in stereology: Restricted case: The specimen X is a nonrandom, bounded set that is the sole object of interest. For instance, a whole organ from an experimental animal could be available for study. Typically we want to estimate the total volu~ne, surface area, etc., of the organ.
195 Extender} case: The specimen X available for examination is but a por- tion sampled from a much larger object W. For instance, a rock sample is typically taken from a large outcrop of rock, and we wish to make inferences about the latter. Either total gold volume or relative gold volume fraction might be of interest. Random case: A stochastic process really exists that generates the in- ternal structure of the sample. That is, the specimen X is a fixed set, but the feature Y inside X is generated as Y = X n z where Z is a "random set" or "spatial stochastic process." For instance, a metallurgist will regard a small piece of steed cut from a bar, formed at a known temperature, and so on, as a sample from the infinite hy- pothetical reservoir of steel that could be formed under those same conditions. Quantities like total volume are meaningless here; we are mainly interested in fractions per unit volume of steel. In the restricted case, we are totally dependent on the randomness of the sampling probe T to guarantee validity of the method; but apart from this we do not need to make any unverifiable assumptions. In the extended case, it must typically be assumed that X was sampled "randomly" from W. For some purposes, it is not valid to sample a rock outcrop by breaking off a piece with a hammer, since the breakage surfaces will usually depend on the internal rock structure. In the random case Y must be independent of X; that is, the internal structure must not depend on the external boundaries of the specimen. This would be inappropriate for objects such as biological organs, which have many levels of internal organization. 10.3.3 Geometrical Identities Unbiased estimation of properties of a set X from observations of the in- tersection X n T is possible thanks to the mean-content formulas or section formulas of integral geometry [76,89], which have the general form |-- (~(XnT),dp(T)=C~(X)~ (10.3) positions of T where cY, ,5 are geometrical quantities such as those listed in Table 10.1, and c = cap is a constant. Here If is a so-called "invariant measure" on the space of all possible probes T; basically, this is an appropriate generalization of
196 Lebesgue measure, and so the integral represents "uniform integration" over positions of T. A simple example is the statement that the volume of a three-dimensional object is the integral of the areas of its plane slices: roo / A(X n Th) dh = V(X), J_OO (10.4) where Th is the plane {(x, y, z): x = h). This is known as Cavalieri's prin- ciple. In simple terms, the volume of an arbitrarily shaped potato can be determined by slicing the potato into infinitely thin parallel slices ant! sum- ming the areas of the slices. The slicing direction is fixed and arbitrary; we could also average over ah orientations, giving //~ A(X n TW,h) dh ~ = 2~V(X), (10.5) where TW,h denotes the plane with direction given by its unit normal vector co and displacement h from the origin. This averaged version is no longer practicable for potatoes, since after slicing end-to-end in direction Hi, we have to reassemble the object and slice it end-to-end from another angle ~2, and so on. The basic mean content formulas in three dimensions are summarized in Table 10.3. In general, the formulas involving plane sections or line probes require us to average over all orientations. For example, the surface area S(Y) of a curved surface Y C R3 can be determined from the lengths of plane sections, I/ ICY n TW,h) Ah ~ = 2 S(Y), (10.6) but in this case there is no analogue of (10.6) for planes with fixed orienta- tion. The surface area of a potato cannot be determined from the boundary lengths of parallel slices, unless we are permitted to reassemble and resTice the object many times. A better alternative for surface area is to use the mean content result in which T is a one-dimensional line. Thus we would repeatedly impale the potato on an array of linear spikes, changing the potato's orientation each time, and count the total number of points where the spikes penetrated the surface. Other sampling probes can also be used. Instead of infinite two-climension- al planes, we can take bounded sampling windows within a plane; the results here are similar except that the right-hand sides also involve the area of the
197 TABLE 10.3: Typical Mean Content Results (3 dimensions) Object X Probe T Resulting X n T Desired ,0 Required Solid domain plane plane domain~s) A linets) linear domain~s) V ~ pointts) pointts) V P Surface plane plane carvers)- line point~s) S ~ Space curve plane points Q Surface plane plane curve~s) M C sampling window. Instead of a single plane Th in (10.4) we may take a stack of equally-spaced parallel planes Gh = { ,Th2s'Ths'Th,Th+s,Th+2s' ); summing the contributions A(X n Th+ks) in (10.4) gives as J A(X n Oh) dh = V(X). o (10.7) Note that the range of integration is now the bounded interval t0, s) because the stack of planes is uniquely specified by its "starting position" h ~ A, s). To take stock of these results we note that 1. They do not depend on the "shape" or position of the objects X, and are true under very minimal regularity conditions; 2. They are valid only when integration is performed "uniformly" over ah positions of T (in most cases this requires averaging over orientations); and 3. They are statements about integrals or mean values only. For some time, stereologists thought that opportunities for finding new mean value formulas were severely restricted by (c). This turned out to be pes- simistic, because many properties of a geometrical object can be expressed as integrals. For example, some powers of volume V(X)m can be stereolog- ically determined. Again, the orientation distribution of a curved surface Y in R3 is the probability distribution of the unit normal vector at a uniformly distributed random point on Y. This is a distribution Q on the unit sphere
198 S2 defined by Q(U) = A(YU)/A(Y) for U c S2, where Yu is the subset of Y consisting of all points y ~ Y. where the unit normal w(y) lies in U. The observed orientation distribution of a two-dimensional plane section Y n T is related to Q by an integral equation reminiscent of WickseD's equation for particle size [164. 10.3.4 Design-Basec! Estimation The design-based approach is analogous to sampling design for finite pop- ulations [11] but has interesting geometrical complications. The set X is fixed (Miles's restricted or extended models); the probe T is generated by a random sampling design. For example, T could be a single random plane (the analogue of simple random sampling) or a stack of parallel planes (the analogue of systematic sampling). The choice whether to estimate total or relative quantities (V(X) or Vv) affects the choice of sampling design. Thus we need to convert mean content formulas (10.3) into results of the form Eol(Y n T) = CT,B(Y), (10.8) MAY n T) E coax n T) Ecl!(Y n T) Ec~'(X n T) IVY) ~'(X) p(Y) 4'(X) (10.9) (10.10) where Y C X are fixed sets, T is a random probe, and E denotes expectation with respect to the distribution of T. In (10.8) CT iS a constant associated with this distribution; in the other versions c = c~,B/c~3' is a "geometrical constant." Usually or' = A and 6' = V so that in (10.9 and 10.10) we are estimating,l~v from ~A- The stereological equivalent of a uniform random sample is a so-called isotropic uniformly random (lUR,) probe. Suppose we wish to generate a random probe T intersecting a set X C Rn. The probe is said to be {UR if it has constant probability density with respect to the invariant measure ,~b that features in (10.3~: dP(T) = ~ dy(T) if T n X ~ ~ (10.11) = 0 ifTnX=~.
199 Here, H(X) is the appropriate normalizing constant H(X)= J dot, TnX/0 ,, i.e., the total ,u-measure of all positions of T that intersect X. (10.12) For example, if the "probe" T is just a single point, the invariant measure ,u is Lebesgue (volume) measure, so that H(X) = V(X), and an TUR point probe hitting X is just a random point uniformly distributed in X. As a less trivial example, a straight line T in two dimensions is uniquely specified by its direction ~ and its distance from the origin: T(8,p) = ~,fx,y): jocose + using = p), where ~ ~ [O,~r), p ~ R. The invariant measure for lines turns out to be t4S,76] dR = dp db , i.e., Lebesgue measure on the (8,p) coordinate space. Thus, an TUR line probe T hitting X is a line with random coordinates ~ and p, such that the pair (8,p) is uniformly distributed over the permissible range {~e, p) : T(8, p) n X 7{ 0) . In the special case where X is a disc of rallies r centered at the origin, an {UR line through X is generated by making ~ and p indepenclent and uniformly distributed over tO,~r) en c! [-r,r], respectively; hence the term TUR. However, note that for a general set X the coordinates of an {UR line are not independent and their marginal distributions are not uniform. A practical method of generating {UR lines through an arbitrary set X is to enclose X by a larger disc D ~ X, generate a sequence of JUR lines intersecting D, and take the first line that happens to intersect X. This is just an application of the rejection method of Monte Cario simulation. The probability that an JUR line through X wiB hit Y C X is H(Y)/H(X) with H as defined in (10.12~. In other words, the probability that an {UR line intersects a particular target is related to the mean projected height of the target. This does not depend on the position of Y within X; so in a sense the {UR fine is a uniform sample through X. Returning to the estimation problem, clearly we can derive (10.~) from (10.3) by taking T to be an lUR probe hitting X, so that Eol(Y n T) = Positions of T Ol(Y n T) dP(<T)
200 = H(X' | ~(Y n T) d,U(T) H~X'7B( ); so we have an unbiased estimator of 3(Y), provided we can determine the normalizing constant H(X). However, a similar argument will not work for formulas (10.9) such as Delesse's principle. The problem is that the expectation of a ratio of random variables in general does not equal the ratio of their expectations. Histor- icaBy there were many incorrect derivations of Delesse's principle; but the result is just not true for JUR planes. Miles and Davy [20,61] showed that a solution is to take T with the weighted distribution with probability density proportional to c~'(X n T), dP(T) = ~ ~ EXIT) (TV), where a' must be nonnegative (e.g., A or ]; but not C). The normalizing constant is G(X) = J '(X n T) dR(T) = Cow, ;'(X) - Then. using Ew to denote averages with respect to this weighted distribu- tion, we have Or(YnT) Ew ct'(X n T) ~ 'AX n Ty ( ) /a'fX n T) GrXy (At) G(X)-i / (My n T) ~Il(T) Cry' p(Y) cCy',(~' ,B'(X) This holds provided or'(X n T) > 0 whenever cr(Y n T) ~ O. Note that the proportionality constant c is now a geometrical constant not dependent on T. Another, closely related, solution is to estimate the numerator and denominator of (10.10) separately on a large number of replicated samples: in other words, when replication is present, take the ratio of means, not the mean of the ratios.
201 The problem encountered here was that plane sections and other stere- ological analogues of simple random sampling actually do not yield fixed sample size. The sample mean is biased when the sample size is random. We must instead use samples with probability proportional to size, or take replicated estimates and numerically weight them in proportion to size. Most stereological sampling designs do not have fixed sample size. Dif- ferent plane sections of a bounded three-dimensional object have different size and shape. Thus, simple random sampling does not generalize easily to most stereological situations. There are some exceptions: a sampling window is a fixed-size sample, if the object of interest always fibs the entire window. Of course, systematic sampling does generalize well to stereology, as we have seen. Stereologica] estimates based on grids of points, lines, and the like, are essentially systematic sampling estimates. Cavalieri's principle for a stack of planes (10.7) is just an application of (10.2) to the function f (h) = A(X n Th) appearing in the original CavaTieri formula (10.4~. The parameter space describing ah positions of a grid or systematic sample is totally bounded, and the invariant measure ,u can be integrated over the entire space. In (10.7) the position of a plane grid was specified by a value h ~ [0, s). Thus an lUR grid is defined to have uniform probability density with respect to ,u over the entire space, dP(T) = H dy(T), (10.13) where the normalizing constant is now the total ,u measure of all positions of T. E = ~ dy(T). Typically, H depencls only on the grid spacing. Thus, estimation of a popu- lation total according to (lO.S) is relatively easy when T is a systematic grid. An unbiased estimate of the volume of a potato can be obtained by cutting it into thick slices by paraDe! planes at constant separation 1, summing the areas of the slices, and multiplying by cl. 10.3.5 Model-Based Estimation In model-based stereology we convert (10.3) into E ANY n T)~ = c~'g~y'¢'E ~3(Y)' , (10.14)
202 X. The probe l where Y is a random set Y = Z n x inside a fixed three-dimensional domain ~ is now arbitrary (say, fixed). This time E is present on both sides and denotes expectation with respect to the random structure Y. Note the denominators are constant. The nonparametric modeling approach described in §10.1.5 is simply to assume that the random process Z is statistically stationary and derive (10.14) by studying moments associated with Z. For example, here is a sketch proof of the model-based Delesse formula, A(Y n T) E V(Y) A(X n T) V(X) (10.15) Suppose the random process Z is such that for any x ~ R3 the indicator variable ~ ~ { 0 if not is a weD-defined random variable. :Let pax) = El(X) = P{x ~ Z). Then EV(Y) = EJ-XI(x)dx = ~/XEl~x)dx = J^xp~x)dx by exchanging integration and expectation. Assume Z is prst-order station- ary in the sense that pox) = p does not depend on x. Then this integral iS EV(Y) = p V(X) By a similar argument EA(Y n T) = p A(X n T), so that both sides of (10.15) are equal to p, and the result is proved. This example needed only a simple exchange of integration and expec- tation. For the other stereological formulas, we need the integral geometric results (10.3), and first-orcler stationarity assumes (roughly) that certain first moments associated with Z are invariant under translation and/or ro- tation. The formal apparatus is the moment theory of stationary random measures [57,5S,804. Other, higher-order expectations can be calculated similarly. For exam- ple, the variance of AA can be expressed in terms of the secon(l-order charac- teristics of the process. We now need to assume Z is seconcl-order stationary,
203 which in this case means that El~x)~(y) = Prig ~ Z and y ~ Z) = Rex, y) depends only on yx. The resulting formula gives the variance as an integral in terms of r: this is equivalent to the basic variance result of geostatistics (see chapter 5~. · c~ ~ w ~ ~ Characteristics of "infinite order" can be considered exactly as in the design-based case, for example, the orientation distribution of a curved sur- face. 10.4 Recent Research ant! New Directions 10.4.1 Variance of Systematic Sampling Systematic sampling usually leads to more efficient estimation than simple random sampling with comparable sample size. However, there are fewer general results about the variance under systematic sampling because this depends on the "structure" of the population t11~. At worst, there could be a periodicity in the population that matches the periodicity of the sys- tematic sample, and the variance would be elevated. The classic example is an army population where every tenth serial number is allocated to a sergeant. In stereology, such cases do arise when a biological structure such as a corrugated sheet is sampled by a test grid with the same spacing. The estimator of the area of a plane set based on a point grid has re- cently been studied extensively [15,29,45,51,54] using earlier results about the systematic sampling estimator (10.2) of an integral [524. The variance of (10.2) is var(I) = ~ ~ 9(iA)J 9 ' j oo where 9 is the covariogram of f boo g(x) = J f(y)f(x + y) dy, oo see [52] and chapter 5. For a wide class of applications, ( ) 6g ( ) as ~ ~ 0. The point-counting estimator of area of a plane set has been found [46,47,15,29] to have variance var(A) ~ 0.0724 La3
204 as a ~ O. where ]; is the perimeter length of the set, this being a good approximation for a wide variety of shapes. 10.4.2 E`ractionator Sampling A simple yet extremely powerful application of systematic sampling is the fractionator technique t264. Suppose we wish to estimate the total volume or number of cells in a large animal. Effectively, we are in the "extended case" where it is not feasible to study more than a tiny sample from the object. Worse, it would seem that we have to generate a uniformly distributes! random sample of this complex object in order to get valid estimates. On the other hand, it is easy to generate a systematic sample of an animal. We start by dismembering the animal in any fashion we choose and arranging the pieces in arbitrary order (e.g., in ascending order of size; or at whim). Then we take a systematic sample of this finite population (inverse sampling fraction kit ~ and throw away the remaining material. The retained sample is then cut into smaller pieces and again arranged in arbitrary order; a systematic subsample of this material (inverse fraction k2) is taken. The process is repeated until we have a subsample that is small enough to analyze microsconicaliv. Then we apply a design-based method to estimate the total amount of material in this ultimate subsample. Finally the total for the entire animal is estimated by the subsample total estimate times the product of the successive inverse sampling fractions kit · kn. Clearly this estimator is unbiased. Sampling fractions as Tow as 10-9 are routinely used in brain tissue, meaning that only ~ 100 cells are actually counted. At the lowest level of the experiment we still have the problem of esti- mating the total number (say) of cells in the sample. But here we can often employ a modification of the disector method. If the last stage of subsam- pling is carried out by slicing the material into sections an(1 systematically subsampling the sections, then we just apply the disector counting rule to each section, and sum the disector counts. This does not require knowledge of section thickness. Indeed the section planes can be separate(1 by different distances, anti even be nonparallel [26,2S,274. Little is known theoretically about the variance of fractionation sam- pling, although the estimator clearly has a martingale structure. Current practice is to form a jackknife estimate of variance, by initially dividing the specimen into two comparable halves, forming a fractionator estimate from each half, and estimating variance from the absolute difference of the two estimates.
205 ,{ , ~ ~ ~ ~ 1 ) +(_~/+ , / + ( ~ I ~ FIGURE 10.3: A point-sampled intercept through a plane section of a par- ticle. 10.4.3 New Estimation Formulas Perhaps the most exciting area of stereology is the discovery of new mean content results (10.3!. Some of the new quantities ,0? are associated with "shape," "size," orientation, curvature, or spatial arrangement. Other re- sults apply in sampling situations where it was previously thought impossible to estimate anything. Let x be a point inside a three-dimensional set X, which we assume convex (for convenience only). Let ((x,~) be the infinite ray (half-line) through x in direction I, where ~ is a unit vector. Then the mean cubed length of the intersection between this line and X is proportional to the volume of X: 4i ,/ L(Xnt~x, ~3 d&= 43 V(X); (10.16) this is an application of elementary calculus. Note that x is a fixed, arbitrary point. A similar but more complicated formula holds if and/or x is outside X. X is not convex This can be used t40] to estimate the mean squared volume of particles in a three-dimensional population. First take an area-weighted plane section of the sample material; superimpose a point grid on the section, and at every grid point which hits a particle profile, place a line in a random direction through the grid point and measure the cubed intercept (i.e., length of the intersection between the line and the particle profile). See Figure 10.3. Un-
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
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.
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
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.
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.
211 [22] Delesse, M. A., Procede mecanique pour determiner la composition des roches, C. R. Acad. Sci. Paris 25 (1847), 544. t23] Glagolev, A. A., On geometrical methods of quantitative mineralogic analysis of rocks, Trans. Inst. Econ. Min. (Moscow) 59 (1933), 1. [24] Gundersen, H. J., and R. 0sterby, Optimizing sampling efficiency of stereological studies in biology: or 'Do more less well!', ~7. Micros. 121 (1981), 65-74. [25] Gundersen, H. J. G., Estimators of the number of objects per area unbiased by edge effects, Micros. Acta 81 (1978), 107-117. [26] Gundersen, H. J. G., Stereology of arbitrary particles. A review of un- biased number and size estimators and the presentation of some new ones, in memory of William R. Thompson, ~1. Micros. 143 (1986), 3-45. [27] Gundersen, H. J. G., and others, The new stereological tools: disector, fractionator, nucleator and point sampled intercepts and their use in pathological research and diagnosis, Acta Pathol. Immun. Scand. 96 (1988), 857-881. [28] Gundersen, H. J. G., and others, Some new, simple and efficient stere- ologica] methods and their use in pathological research and diagnosis, Acta Pathol. Immun. Scand. 96 (1988), 379-394. t29] Gundersen, H. J. G., and E. B. Jensen, The efficiency of systematic sampling in stereology en cl its prediction, J. Micros. 147 (1987), 229- 263. [30] Hadwiger, H., Vorlesungen ueber Inhalt, Oberflaeche und Isoperimetrie, Springer-VerIag, Bertin, 1957. t31] HaH, P., Resampling a coverage pattern, Stoch. Processes Appit. 20 (1985), 231-246. t32] HaL, P., and R. L. Smith, The kernel method for unfolding sphere size distributions, '7. Comput. Phys. 74 ( 1988), 409-421. [33] HaD, Peter, An Introduction to the Theory of Coverage Processes, John Wiley anct Sons, New York, 1988. [34] Harding, E. F., and D. G. Kendall, eds., Stochastic Geometry: A Trib- ute to the Memory of Rol1/o Davidson, John Wiley and Sons, New York, 1974.
212 [35] Haug, H., NervenheilLunde 4 (1985), 103-109. [36] Haug, H., History of neuromorphometry, J. Neurosci. Methods 18 (1988), 1-17. [37] Hoogendoorn, A. W., Estimate the weight undersize distribution for the Wicksell problem, Statistica Neeriandlica, to appear, 1990. [38] Howard, C. V., S. Reid, A. J. Baddeley, and A. Boyde, Unbiased es- timation of particle density in the tandem-scanning reflected light mi- croscope, '7. Micros. 138 (1985), 203-212. [39] Jensen, E. B., A design-based proof of Wicksell's integral equation, AT. Micros. 136 (1984), 345-348. [40] Jensen, E. ]3., and H. J. G. Gundersen, The stereological estimation of moments of particle volume, J. Appl. Prob. 22 (1985), 82-98. [41] Jensen. E. B., and H. it. G. Gundersen, Fundamental stereological for- mulas based on isotropically orientated probes through fixed points with applications to particle analysis, J. Micros. 153 (1989), 249-267. t42] Jensen, E. B., H. J. G. Gundersen, and R. 0sterby, Determination of membrane thickness from orthogonal intercepts, A. Micros. 115 (1979), 19-33. t43] Jensen, E. B., and R. Sundberg, Statistical models for stereological inference about spatial structures; on the applicability of best linear unbiased estimators in stereology, Biometrics 42 (1986), 735-751. [44] Jensen, E. B., A. J. Baddeley, H. J. G. Gundersen, and R. Sunc~berg, Recent trends in stereology, Int. Stat. Rev. 53 (1985), 99-108. [45] Kellerer, A. M., Exact formulas for the precision of systematic sampling, J. Micros. 153 (1989), 285-300. [46] Kendall, D. G., On the number of lattice points inside a random ova], Q. J. Math. (Oxford!) 19 (1948), 1-26. t47] Kendall, D. G., and R. A. Rankin, On the number of points of a given lattice in a random hypersphere, Q. ]. Math. (Oxfor~j 4 (series 2, 1953), 178-189. [48] Kendall, M. G., and P. A. P. Moran, Geometrical Probability, Griffin's Statistical Monographs and Courses no. 10, Charles Griffin, London, 1963.
213 [49] Konig, D., N. Blackett, C. J. Clem, A. M. Downs, and J. P. Rigaut, Orientation distribution for particle aggregates in 3-D space based on point processes and laser scanning confocal microscopy, Acta Stereolog- ica 8/2 (1990), 213-218. [50; Kroustrup, J. P., H. J. G. Gundersen, and M. Vaeth, Stereological analysis of three-dimensional structure organization of surfaces in mul- tiphase specimens: statistical models and model-inferences, I. Micros. 149 (1988), 135-152. t51] Matern, B., Precision of area estimation: A numerical study, '7. Micros. 153 (1989), 269-284. t52] Matheron, G., Les Variables Regionalisees et leer Estimation, Masson, Paris, 1965. t53] Matheron, G., Random Sets and Integral Geometry, John Wiley and Sons, New York, 1975. [54] Matifel~t, T., The accuracy of one-dimensional systematic sampling, ]. Micros. 153 (1989), 301-313. [55] Mattfel~t, T., H.-J. Moblus, and G. Mall, Orthogonal triplet probes: an efficient method for unbiased estimation of length and surface of objects with unknown orientation in space, '7. Micros. 139 ~ 1985), 279-289. t56] Mattfel~t, T., and G. MaD, Estimation of length and surface of anisotropic capillaries, A. Micros. 135 (1984), 181-190. [57] Mecke, J., and D. Stoyan, Formulas for stationary planar fibre processes rGeneral theory, Math. Operationsforsch Stat., Ser. Stat. 1 2 ~ 19 80), 267-279. [58] Mecke, J., and D. Stoyan, Stereological problems for spherical particles, Math. Nach. 96 (1980), 311-317. [59] Miles, R. E., On the elimination of edge-effects in planar sampling, pp. 228-247 in Stochastic Geometry: A Tribute to the Memory of Rollo Davidson, E. F. Harding and D. G. Kendall, eds., John Wiley en c! Sons New York, 1974. t60] Miles, R. E., The importance of proper model specification in stereol- ogy, pp. 115-136 in Geometrical Probability and Biological Structures: Bu~on's 200th Anniversary, R. E. Miles and J. Serra, eds., Lecture Notes in Biomathematics, No. 23, Springer-Ver]Lag, New York, 1978.
214 [61] Miles, R. E., and P. J. Davy, Precise and general conditions for the validity of a comprehensive set of stereological fundamental formulas, ~1. Micros. 107 (1976), 211-226. [62] Miles, R. E., and P. J. Davy, On the choice of quadrats in stereology, ]. Micros. 110 (1977), 27-44. [63] Mailer, J., Random tessellations in R3, Adv. Appl. Prob. 21 (1989), 37-73. t64] Nagel, W., Dunne Schnitte von stationare raumlicher1 Faserprozessen, Alathematisch Operationsforsch Stat., Ser. Stat. 14 (1983), 569-576. [65] Nicholson, W. L., Estimation of linear properties of size distributions, Biometrika, 57 ~ 1970), 273-297. [66] Nicholson, W. L., Estimation of linear functionals by maximum likeli- hood, IT. Micros. 10 7 ~1976), 323-336. [67] Nicholson, W. L., Application of statistical methods in quantitative microscopy, J. Micros. 113 (1978), 223-239. [68] Ogata, Y., and M. Tanemura, Likelihood analysis of spatial point pat- terns, J. R. Stat. Soc., B 46 (1984), 496-518. [69] Preteux, F., and M. Schmitt, Boolean texture analysis and synthesis, pp. 379-400 in Image Analysis and Mathematical Morphology. Volume If: Theoretical Advances, J. Serra, ea., John Wiley and Sons, New York, 1988. [70; Ripley, B. D., SpatialStatistics, John Wiley and Sons, New York, 1981. [71] Ripley, B. D., Statistical Inference for Spatial Processes, Cambridge University Press, Cambridge, 1988. [72] Ripley,B. D., Gibbsianinteractionmodels,pp. 1-19in SpatialStatis- tacs: Past, Present and Future, D. A. Griffiths, ea., Image, New York, 1989. [73] Ripley, B. D., and F. P. Kelly, Markov point processes, ]. London Math. Soc. 15 (1977), 188-192. [74] Rosiwal, A., Uber geometrische Gesteinsanalysen, Verb. K. A. Geol. Reichsanst. Wien, 1898, 143.
215 [75] S ant alo, L. A., Introduction to Integral Geometry, Actualites Scien- tifiques et Industrielles, no. 119S, Hermann, Paris, 1952. t76] Santalo, I`. A., Integral Geometry and Geometric Probability, Ency- clopedia of Mathematics and Its Applications, vol. 1, Addison-WesTey, 1976. t77] Serra, J., Image Analysis and Mathematical Morphology, Academic Press, New York, 1982. t78] Silverman, B. W., M. C. Jones, J. D. Wilson, and D. W. Nychka, A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography, .7. R. Stat. Soc., B 52 (1990~. [79] Sterio, D. C., The unbiased estimation of number and size of arbitrary particles using the disector, '7. Micros. 134 (1984), 127-136. t80] Stoyan, D., W. S. Kendall, and J. Mecke, Stochastic Geometry and its Applications, John Wiley and Sons, Chichester, 1987. t81] Taylor, C. C., A new method for unfolding sphere size distributions, '7. Micros. 132 (1983), 57-66. [82] Tomkeieff, S. T., Linear intercepts, areas and volumes, Nature 155 (1945), 24. t83] van Es, B., and A. Hoogendoorn, Kernel estimation in WickseD's cor- puscle problem, Biometrika 77 (1990), 139-145. :84] Van Blaschke, W., Voriesungen uber Integraigeometrie, Chelsea, New York, 1949. [85] Watson, G. S., Estimating functionals of particle size distributions, Biometrika 58 (1971), 483-490. [86] Watson, G. S., Characteristic statistical problems of stochastic ge- ometry, in Geometrical Probability and Biological Structures: Buffon's 200th Anniversary, Lecture Notes in Biomathematics, no. 23, Springer- VerIag, New York, 1978. [87] Weibel, E. R., Stereological Methods, 1. Practical Methods for Biological Morphometry, Academic Press, London, 1979. [88] Weibel, E. R., Stereological Methods, 2. Theoretical Foundations, Aca- demic Press, London, 1980.
216 [89] Weil, W., Stereology: a survey for geometers, pp. 360-412 in Convexity and its Applications, P. M. Gruber and J. M. Wills, eds., Birkhauser, Stuttgart, 1983. [90] Weil, W., Point processes of cylinders, particles and flats, Acta Appli- candae Mathematicae 9 ~ 1984), 103-136. [91] Weil, W., Expectation formulas and isoperimetric properties of Boolean models, A. Micros. 151 (1988), 235-245. [92] Weil, W., Translative integral geometry, pp. 75-86 in Geobild 89, A. Hubler et al., eds., Akademie Verlag, Berlin, 1989. t93] Weil, W., Iterations of translative integral formulas and nonisotropic Poisson processes of particles, Mathematische Zeitschrift, to appear 1990. [94] Wicksell, S. D., The corpuscle problem, I, Biometrika 17 (1925), 84-89. [95] Wicksell, S. D., The corpuscle problem, II, Biometrika 18 (1926), 152- 172. Bibliographic Notes Introductory references to stereology are [3,12,44] for statisticians, tS7,8S, 2S,27,14] on applied stereology especially in biology, tS0,48] on probabilistic modeling, and [89] from the viewpoint of convex geometry. Many research papers on stereology appear in the Journal of Microscopy and Acla Stereo- logica, which are the official journals of the International Society for Stere- ology; and in Biometrika, Journal of Applied Probability, and Advances in Applier! Probability.