Chapter 12
Unfolding Size Distributions
In the preceding chapter, methods were discussed for predicting the
distribution of intercepts that a random probe would make with objects in a threedimensional space. This kind of modeling can be done using either line or plane
probes. With line probes the length of the intersection can be measured, and with
plane probes the area as well as the shape of the intersection can in principle be
determined. This approach requires making several assumptions: the size and shape
of the objects being sampled must be known, and the probes must be isotropic,
uniform and random with respect to the objects.
In actual practice, the experiment is carried out in the opposite direction. A
real specimen is probed with lines or planes, and the data on intercept lengths, areas
and perhaps shape information are recorded. Then the goal is to calculate the
(unknown) sizes of the objects that must have been present in order to produce those
results. Note that it is not possible to determine the size of any single object this
way, from the intersection that the probe makes with it. It is only in the statistical
aggregate that a distribution of objects can be inferred.
And even then it is far from easy to do so. There are several aspects
to the problem, which will be discussed below. First, mathematically the inverse
problem is ill-conditioned. The small statistical variation in measured data
(which arises because a limited number of measurements are made, even if they are
an unbiased sample of the specimen) grows to a much larger variation in the
calculated answer because of the ways that these variations propagate through the
calculation.
Second, a critical assumption must be made for the method to be applied at
all: we must assume that we know what the shape of the objects is, and usually
that they are either all of the same shape or have a very simple distribution of
shapes. The most popular shape assumption is that of a sphere, because its symmetry makes the mathematics easy and also relaxes the requirements for isotropic
sampling.
But in fact not very many structures are actually spheres, and even a small
variation from the assumed shape can introduce quite a large bias in the calculated
results. To make matters worse, many real specimens contain objects that have a
wide variety of shapes, and the shape variation is often a function of size. This presents major problems for unfolding techniques.
In spite of these difficulties, the unfolding approach was the mainstay of classical stereology for decades and is still used in spite of its limitations in many situations. Certainly there are specimens for which a reasonable shape assumption can
be made—spheres for bubbles, cubes for some crystalline materials (e.g., tungsten
carbide), cylinders for fiber composites, and so on. And the use of computer-based
297
298
Chapter 12
measurement systems makes it possible to collect enough data that the statistical
variations are small, limiting the extent of the errors introduced by the inverse solution method.
Linear Intercepts in Spheres
The preceding chapter developed the length distribution of lines intersecting
spheres. The frequency distribution for a sphere of diameter D is simply a straight
line, and when measured intercept lengths are plotted in a conventional histogram
form we would expect to see a result as shown in Figure 12.1. The presentation of
data in histogram form is the most common way to deal with measurement data of
all types. The bin width d is typically chosen to permit each bin to accumulate
enough counts for good statistical precision, and enough bins to show the shape of
the distribution. In most cases from 10 to 15 bins are linearly spaced in size up to
the largest value obtained in measurement.
One consequence of the number of bins used is the ability to measure
spheres (or other objects) whose sizes vary over that range of dimensions. Ten or
15 bins allows determining spheres in that number of sizes. If the size range of the
spheres is much larger, say 50 : 1, and information on the smallest is actually needed,
then a histogram with at least that many bins is required. This greatly increases the
Figure 12.1. Frequency distribution of linear intercepts in a sphere of diameter D, shown
as a continuous plot and as a discrete histogram. (For color representation see the
attached CD-ROM.)
Unfolding Size Distributions
299
Figure 12.2. Frequency distribution of linear intercepts from a mixture of sphere sizes,
shown as a continuous plot and a discrete histogram. (For color representation see the
attached CD-ROM.)
number of measurements that must be made to obtain reasonable counting
precision.
When a mixture of sphere sizes is present, the measured distribution represents a summation of the contribution of the different sizes each in proportion to
the abundance of the corresponding size. As shown in Figure 12.2, the result is that
the histogram bins corresponding to smaller intercept lengths contain counts from
intersections with many different size spheres, while the largest bin contains counts
only from the largest spheres.
This offers a straightforward graphical way to unfold the data (Lord & Willis,
1951). As shown in Figure 12.3, knowing that the largest bin contains counts only
from the largest spheres and that those spheres should also generate a proportionate number of shorter intercepts allows subtracting the expected number of shorter
intercepts from each of the other (smaller) bins. This process can then be repeated
for the next bin, using the number of counts remaining, and so on.
Notice that the number of counts in the smaller size bins (and hence the estimated number of smaller size spheres) is obtained by successive subtractions. Subtraction is a particularly poor thing to do with counting data, since the difference
between two numbers has a standard deviation that is the sum of the deviations of
the two numbers whose difference was taken. This means that in effect the
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Chapter 12
Figure 12.3. Unfolding the linear intercept distribution from a mixture of sphere sizes.
Lines project the histogram bin heights onto a vertical axis and the increments given
the relative proportion of each sphere size. (For color representation see the attached
CD-ROM.)
uncertainty of the estimated number of smaller size spheres grows rapidly and
depends strongly on the number of larger size spheres and the precision with which
they are determined.
Hence the need to obtain large numbers of interception counts, since the
standard deviation of any counting data is simply the square root of the number
counted. In other words if you count one hundred events the one-sigma uncertainty
–—
is ÷100 = 10 out of 100 or 10%, and to reduce this to 1% precision would require
counting 10,000 events.
Plane Intersections
Linear intercepts were primarily used when manual measurements were
required to obtain data, because they could be performed by drawing (random)
lines on images of sections and measuring the length of the intersections. With
modern computer-based instruments, it is usually easier to measure the intersection
areas made by the sampling plane with the objects. It is very difficult to use plane
probes that must be isotropic, uniform and random (IUR) in a real object because
once a single plane section has been taken, other planes that might intersect that
one cannot be generated. However, for specimens which are themselves IUR any
plane is as good as another, and examination of plane sections is a very common
approach.
Every section through a sphere is a circle as shown schematically in
Figure 12.4. Figure 12.5 shows an image of pores in an enamel coating on steel. The
pores may be expected to be spherical as they result from the evolution of gas
bubbles in the firing of the enamel, and all of the intersections are observed to be
circular. The distribution of the sizes of circles can be calculated analytically. The
probability of obtaining a circle of radius d ± dd (corresponding to counts that will
fall into one bin of a histogram) from a sphere of diameter D is equal to the
vertical thickness of a slice of the sphere with that diameter. The shape of the
distribution is
Unfolding Size Distributions
301
a
b
c
d
e
Figure 12.4. Different size circles are produced by sectioning of a sphere. (For color
representation see the attached CD-ROM.)
P(dcircle) = d/(D · (D2 - d 2)1/2)
(12.1)
as shown in Figure 12.6.
As for the linear intercept case, if there are several different sizes of spheres
present in the sample a particular size of intercept circle could result from the intersection of the plane with several different sphere sizes. The resulting measurement
302
Chapter 12
Figure 12.5. Example of sectioning of spheres: bubbles in a fired enamel coating.
histogram would show the superposition of data from the different sizes in proportion to their relative abundance and to their size (it is more likely for the plane
to strike a large sphere than a small one).
This distribution might be unfolded sequentially and graphically as shown
before for the linear intercepts, since the largest circles could only come from the
largest spheres. Then as before, a corresponding number of intercepts would be subtracted from each smaller bin and the process repeated until all sizes are accounted
for. This is not a very practical approach because of the tendency for statistical
Figure 12.6. The distribution of planar intercepts through a sphere is proportional to
the vertical thickness dz of a slice of the sphere covering a range of circle sizes dr.
Unfolding Size Distributions
303
variation to concentrate in the smaller size ranges, and instead a simultaneous solution is generally used.
For the case of spheres, this method was first described by Saltykov (1967)
and has been subsequently refined by many others, with an excellent summary paper
by Cruz-Orive (1976) that includes tables of coefficients suitable for general use. The
method is simply to solve a set of simultaneous equations in which the independent
variables are the number of circles of various sizes (measured on the images and
recorded as a histogram) using the coefficient matrix, to obtain the numbers of
spheres of each size present in the material.
The matrix is determined by calculating as discussed above, from geometric
probability, the frequency distribution for the number of circles in each size class
due to three dimensional objects of each size. This can be written as
NAi = a¢ij · NVj
(12.2)
where the subscript i ranges over all size classes of circles in the measured histogram
and the subscript k covers the corresponding sizes of spheres in the (unknown) distribution of three-dimensional objects. The equations are more readily solved by
using the inverse matrix a, so that
NVj = (1/d)a ij · NAi
(12.3)
where d is the size of the bin used in the histogram. A typical a matrix of coefficients is shown in Table 12.1.
As an example of the use of this method, Figure 12.7 shows images
of nodular graphite in a cast iron. The sections are all at least approximately
circular and we will make the (typical) assumption that the nodules are spherical.
Figure 12.8a shows a distribution of circle sizes obtained from many fields of
view covering the cross section of the enamel. Converting these NA data using
the coefficients in Table 12.1 produces the plot of sphere sizes shown in Figure
12.8b.
Other Shapes
The “sphere unfolding” method is easily programmed into a spreadsheet, and
is often misused when the measured objects are not truly spheres. Of course, similar
matrices for a’ can be computed for other shapes of objects, and inverted to produce
a matrices. In fact, this has been done for an extensive set of convex shapes including polyhedra, cylinders, etc. (see for example Wasén & Warren, 1990). It is well to
remember, however, that the results are only as useful as the assumption that the
three-dimensional shape of the objects is known and is the same for all objects
present.
Another shape model that has been applied to the “nearly spherical” shapes
that arise in biological materials in particular (where membranes tend to produce
smooth boundaries as opposed to polyhedra) is that of ellipsoids. These may either
be prolate (generated by revolving an ellipse around its major axis) or oblate (generated by revolving the ellipse around its minor axis), and can be used to approximate a variety of convex shapes.
NV(1)
NV(2)
NV(3)
NV(4)
NV(5)
NV(6)
NV(7)
NV(8)
NV(9)
NV10)
NV(11)
NV(12)
NV(13)
NV(14)
NV(15)
NV(1)
NV(2)
NV(3)
NV(4)
NV(5)
NV(6)
NV(7)
NV(8)
NV(9)
NV10)
NV(11)
NV(12)
NV(13)
NV(14)
NV(15)
Alpha matrix for sphere unfolding (Cruz-Orive)
NA(2)
NA(3)
NA(4)
NA(5)
NA(1)
0.26491 -0.19269 0.01015 -0.01636 -0.00538
0.27472 -0.19973 0.01067 -0.01691
0.28571 -0.20761 0.01128
0.29814 -0.21649
0.31235
Alpha matrix for sphere unfolding (Saltykov)
NA(1)
NA(2)
NA(3)
NA(4)
NA(5)
0.1857 -0.0750 -0.0261 -0.0132 -0.0080
0.1925 -0.0776 -0.0270 -0.0136
0.2000 -0.0804 -0.0280
0.2085 -0.0836
0.2182
NA(6)
-0.00481
-0.00549
-0.01751
0.01200
-0.22663
0.32880
NA(6)
-0.0054
-0.0083
-0.0140
-0.0290
-0.0872
0.2294
NA(7)
-0.00327
-0.00491
-0.00560
-0.01818
0.01287
-0.23834
0.34816
NA(7)
-0.0039
-0.0055
-0.0085
-0.0146
-0.0301
-0.0913
0.2425
NA(8)
-0.00250
-0.00330
-0.00501
-0.00571
-0.01893
0.01393
-0.25208
0.37139
NA(8)
-0.0028
-0.0039
-0.0056
-0.0088
-0.0151
-0.0319
-0.0961
0.2582
NA(9)
-0.00189
-0.00250
-0.00332
-0.00509
-0.00579
-0.01977
0.01527
-0.26850
0.40000
NA(9)
-0.0022
-0.0029
-0.0040
-0.0057
-0.0090
-0.0155
-0.0329
-0.1016
0.2773
NA(10)
-0.00145
-0.00186
-0.00248
-0.00332
-0.00516
-0.00584
-0.02071
0.01704
-0.28863
0.43644
NA(10)
-0.0016
-0.0022
-0.0028
-0.0041
-0.0058
-0.0091
-0.0163
-0.0346
-0.1081
0.3015
NA(11)
-0.00109
-0.00139
-0.00180
-0.00242
-0.00327
-0.00518
-0.00582
-0.02176
0.01947
-0.31409
0.48507
NA(11)
-0.0013
-0.0016
-0.0021
-0.0028
-0.0040
-0.0059
-0.0094
-0.0168
-0.0366
-0.1161
0.3333
NA(12)
-0.00080
-0.00101
-0.00129
-0.00169
-0.00230
-0.00315
-0.00512
-0.00565
-0.02293
0.02308
-0.34778
0.55470
NA(12)
-0.0009
-0.0012
-0.0016
-0.002
-0.0027
-0.0038
-0.0058
-0.0095
-0.0174
-0.0386
-0.1260
0.3779
Table 12.1. Alpha matrices to unfold sphere size distribution from measured circle sizes
NA(13)
-0.00055
-0.00069
-0.00087
-0.00113
-0.00150
-0.00208
-0.00288
-0.00488
-0.00516
-0.02416
0.02903
-0.39550
0.66667
NA(13)
-0.0007
-0.0007
-0.0010
-0.0013
-0.0018
-0.0026
-0.0037
-0.0057
-0.0093
-0.0178
-0.0408
-0.1382
0.4472
NA(14)
-0.00033
-0.00040
-0.00051
-0.00066
-0.00087
-0.00117
-0.00167
-0.00234
-0.00427
-0.00393
-0.02528
0.04087
-0.47183
0.89443
NA(14)
-0.0004
-0.0006
-0.0006
-0.0009
-0.0010
-0.0015
-0.0021
-0.0031
-0.0051
-0.0087
-0.0171
-0.0420
-0.1529
0.5774
NA(15)
-0.00013
-0.00016
-0.00020
-0.00026
-0.00034
-0.00045
-0.00062
-0.00094
-0.00126
-0.00298
-0.00048
-0.02799
0.08217
-0.68328
2.00000
NA(15)
-0.0001
-0.0002
-0.0003
-0.0004
-0.0005
-0.0006
-0.0009
-0.0013
-0.0020
-0.0033
-0.0061
-0.0130
-0.0360
-0.1547
1.0000
Unfolding Size Distributions
305
a
b
Figure 12.7. Images of graphite nodules in cast iron.
a
b
Figure 12.8. a) The distribution of measured circle sizes from the cast iron in Figure
12.7, and a) the calculated distribution of sphere sizes that generated it. (For color representation see the attached CD-ROM.)
306
Chapter 12
All of the sections made by a plane probe passing through an ellipsoid
produce elliptical profiles (Figure 12.9). These can be characterized by to dimensions, for instance the lengths of the major and minor axes. DeHoff (1962) presented a method for unfolding the size distribution of the ellipsoids, which are
assumed to have a constant shape (axial ratio q) but varying size (major axis length),
from the size distribution of the ellipses. This is a modification of the sphere method
in which
NVj = (K (q ) d ) ◊ Â a ij N Ai
(12.4)
d is the size increment in the histogram, and the additional term K(q)
is a shape dependent factor that depends on the axial ratio of the ellipsoid q and
whether the ellipsoids are prolate or oblate; this factor shown in Figure 12.10 for
both cases.
Note that it is necessary to decide independently whether the generating
objects are oblate or prolate, since either can produce the same elliptical profiles on
the plane section image. If this is not known a priori, it can sometimes be determined by examining the sections themselves. If the generating particles are prolate,
then the diameters of the most equiaxed sections will be close in size to the width
of those with the highest aspect ratio, while if the particles are oblate then the most
a
b
c
d
Figure 12.9. Ellipses formed by planar sections with prolate (a..d) and oblate (e..g)
ellipsoids. (For color representation see the attached CD-ROM.)
Unfolding Size Distributions
307
e
f
g
Figure 12.9. Continued
a
b
Figure 12.10. K shape factors for prolate (a) and oblate (b) ellipsoids as a function of
the axial ratio q of the generating ellipse.
308
Chapter 12
Figure 12.11. Elliptical profiles from prolate and oblate ellipsoids. (For color representation see the attached CD-ROM.)
equiaxed sections will be close to the length of those with the highest aspect ratio
(Figure 12.11). In either case, the axial ratio q is taken as the aspect ratio of the
most elongated profiles and used to obtain k from the graph.
It is in principle possible to extend the unfolding method of analysis to
convex features that do not all have the same shape. The profiles observed in the
plane section can be measured to determine both a size and shape (e.g. an area and
eccentricity) and a two-dimensional histogram of counts constructed (Figure 12.12).
From these data, calculation of both the size and shape of the generating ellipsoids
or other objects should be possible. The calculation would be of the form
NVij = Â a ijkl ◊ N Akl
(12.5)
where the subscripts i, j cover the size and shape of the three-dimensional objects
and k, l cover the size and shape of the intersection profiles. Calculating the fourdimensional a’ matrix and inverting it to obtain a is a straightforward extension of
the method shown before, although selecting the correct three-dimensional shape
and its variation is not trivial. The major difficulty with this approach is a statistical one. The number of counts recorded in many of the bins in the two-dimensional
NA histogram will be sparse, especially for the more extreme sizes and shapes. These
are important to the solution of the equations and propagate a substantial error
into the final result.
Figure 12.12. An example of a two-dimensional histogram of size and shape of intersections made by a plane probe. (For color representation see the attached CD-ROM.)
Unfolding Size Distributions
309
Shape information from the profiles can be useful even when a single shape
of generating object is present. Ohser & Nippe (1997) have shown that measuring
the shapes of intersections with cubic particles (which are polygons with from 3 to
6 sides) can significantly improve the determination of the size distribution of the
generating cubes.
Simpler Methods
The unfolding of linear intercepts is one of the simplest methods available,
in use long before modern computers made the solution of more complicated
sets of equations easy. It can be performed using other distributions than the
linear one that applies to spheres, of course. If the actual shape of the objects present
is known, the exact distribution could be generated as discussed in the preceding
chapter and used for the unfolding. A relatively compact and general approach
has been proposed for the case of other convex shapes by several investigators
(Weibel & Gomez, 1962; DeHoff, 1964). Using the usual nomenclature of NV
(number per unit volume), NA (number per unit area), VV (volume fraction), PL
(intersections per length of line) and PP (fraction of points counted), the relationships proposed are
NV = (K b) ◊ N A
32
VV
12
(12.6)
and
NV = 2g ◊ NA PL PP
(12.7)
Note than both procedures require determining the total volume fraction of
the particles (VV or PP) as well as the number of objects per unit area of section
plane, and one requires determining the mean linear intercept (the inverse of PL).
The parameter K takes into account the variation of the size of features from their
mean, and is often assumed to have a value between 1.0 and 1.1 for many real structures that constitute a single population (e.g., cell organelles). Figure 12.13 shows
K as a function of the coefficient of variation of the sizes of the objects.
Figure 12.13. The shape distribution parameter K as a function of the relative standard
deviation of the mean diameter.
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Chapter 12
Table 12.2. Shape coefficients for particle counting
Shape
Sphere
Prolate Ellipsoid 2 : 1
Oblate Ellipsoid 2 : 1
Cube
Octahedron
Icosahedron
Dodekahedron
Tetrakaidekahedron
g
6.0
7.048
7.094
9.0
8.613
6.912
6.221
7.115
b
1.38
1.58
1.55
1.84
1.86
1.55
1.55
1.55
Notice that the size distribution of the intercept lengths is not used here.
Instead, the fact that the mean value of the intercept length of a line probe through
a sphere is just (p/4) times the sphere diameter is used (and adjusted for other shape
objects). b and g are shape factors that depend upon an assumed shape for the threedimensional objects. They vary only slightly over a wide range of convex shapes, as
indicated in Table 12.2 and Figure 12.14. This method represents a very low
cost way to estimate the number of objects present in three dimensions based on
straightforward measurements taken on plane sections. However, it is presently
giving way to more exact and unbiased techniques described under the topic of the
“new” stereology.
Lamellae
Linear probes are also useful for measurement of other types of structures.
One of the advantages of linear probes is the ease with which they can be produced
with uniform, random and isotropic orientations (which as noted above is very difficult for plane probes). Drawing random lines onto a plane section can be used to
measure intercept lengths, either manually or using a computer.
Lamellar structures occur in many situations, including eutectic or other
layered structures in materials, sedimentary layers in rocks, membranes in tissue,
Figure 12.14. The shape coefficient b for ellipsoids and cylinders of varying axial
ratio.
Unfolding Size Distributions
311
Figure 12.15. A plane section through a layered structure produces an apparent layer
thickness that is generally larger than the true perpendicular dimension. (For color representation see the attached CD-ROM.)
and so on. Because these structures will not in most cases lie in a single known orientation perpendicular to the section plane, the apparent spacing of the layers will
be larger than the true perpendicular spacing (Figure 12.15). There is no guarantee
that on any particular section plane the true spacing will be revealed.
Random (IUR) linear intercepts through a layer of true thickness t produce
a distribution of intercept lengths as shown in Figure 12.16. If the data are replotted as a histogram of 1/l, the inverse of the measured intercept length, a much
simpler distribution is obtained (Gundersen, 1978). The mean value of this
Figure 12.16. Plots of the intercept length distribution for linear intercepts through a
layer. (For color representation see the attached CD-ROM.)
312
Chapter 12
triangular distribution is just (2/3) (1/t) so the true layer thickness can be
calculated as
t = 3 2 (1 lm )
(12.8)
where lm is the mean measured intercept length. Furthermore, if the layers are not
all the same thickness but have a range of thickness values, the distribution can be
unfolded using the same graphical technique as shown above for linear intercepts
through spheres.