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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 twodimensional plane section through the material.
It now embraces all geometrical sampling operations, such as clipping a two
dimensional image inside a window, taking onedimensional 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 threedimensional
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
threeclimensional 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 LowerDimensional 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 twodimensional 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 threedimensional 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 threedimensional 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 tissueairway 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 twodimensional
systematic sample of the continuous twodimensional 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 n3/2 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 length3, and so we would
naively expect not to be able to estimate it from lowerdimensional 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 endtoend 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 twodimensional 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
DeHoffRhines 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 threedimensionaBy 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 fieldofview 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 plussampling by Miles [59], clearly produces
a biased sample of profiles. If we imagine the fieldofview to be placed at
random on the microscope slide, the larger profiles have a greater probability
of being sampled. Hence the plussampled estimate of NA is biased: the
expected number of profiles counted is greater than NA x A(F).
An alternative is minussampling: 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) minussampling.
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 meancontent 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 twodimensional 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 (Iookup) 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 Tookup 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,um3,
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 threedimensional geometry, including shape and topology. But
spatial reasoning is confused by sampling effects. A single threeclimensional
object may appear on section as several unconnected objects. A section of a
threedimensional 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 threedimensional object may look very different on different section
planes; different threedimensional 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
1—F(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
threedimensional 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 volumeweighted 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 "starshaped" 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 secondorder 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. Secondorder statistics have also been used to define
indices of mineral liberation t19,184.
Nonuniform 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 welldefined
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 nonuniform 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 threedimensional
particles [38], and the methods of twodimensional 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 likelihoodbased inference [684.
Bootstrap Methods
Bootstrap resampling methods were introduced to stereology by Hall t31,33]
in connection with the pointcounting 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.
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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.