Chapter 4
Classical Stereological Measures
This chapter reviews the most commonly used stereological measurements.
More detailed discussions and derivations may be found in the collection of traditional texts in the field (Saltykov, 1958; DeHoff and Rhines, 1968; Underwood,
1970; Weibel, 1978; Kurzydlowski and Ralph, 1995; Howard and Reed, 1998). Each
section of the chapter focuses on a manual stereological measurement. In each case,
the procedure is illustrated with a microstructure and a superimposed grid. Each
figure lists:
1. The probe population that is required in the measurement;
2. The specific probe that is used in the illustration;
3. The event that results when this probe interacts with the microstructure;
4. The measurement to be made and its specific result;
5. The stereological relationship that connects the measurement to the geometric property of the structure;
6. The calculation of the appropriately normalized version of the measured
result;
7. The calculation of the geometric property of the microstructure that is estimated from this measurement.
A discussion of the corresponding procedure used in computer-based image
analysis follows the descriptions of the manual measurements.
The chapter begins with the measurement of the area of features in a two
dimensional structure since this is easily visualized. Measurement of volume fraction, surface area density and line length are then reviewed.
Two-Dimensional Structures; Area Fraction from the Point Count
Figure 4.1 shows a two dimensional structure consisting of two feature sets.
Label the white background a and the colored regions b. The fraction of the area
of the structure occupied by the b phase is AAb. This area fraction is measured using
point probes. The population of points in this two dimensional world is sampled by
superimposing a grid of points on the structure. An outline of the measurement
process is given in the caption of Figure 4.1.
The square 5 ¥ 5 grid in Figure 4.1 constitutes a sample of 25 points (the
intersections in the grid) out of this population of all of the points that could be
identified in the area of the specimen. The event of interest that results from the
45
46
Chapter 4
Figure 4.1. Measurement of the area fraction of the dark phase AA. (For color representation see the attached CD-ROM.)
Probe population:
Points in two dimensional space
This sample:
25 points in the grid
Event:
Points lie in the phase
Measurement:
Count the points in the phase
This count:
8 points in the phase
Relationship:
·PPÒ = AA
Normalized count:
PP = 8/25 = 0.32
Geometric property:
AA = 0.32
interaction of the probe sample with the structure is, “the point hits the b phase”.
The actual measurement is simply a count of these events, i.e., the number of points
in the 5 ¥ 5 grid in Figure 4.1 that lie within the b areas. In Figure 4.1, this count
gives 8 points in the b phase; that count is the result for this placement of the probe
sample in this field of the microstructure. This point count is related to the area
fraction of the b phase in this two dimensional structure by the fundamental
stereological relationship,
PP
b
= AAb
(4.1)
The left side of this equation, ·PPÒ, is the “expected value” of the fraction
of points that hit the b phase in a sample set of grid placements; the right hand side
is the area fraction of b in the microstructure.
Classical Stereological Measures
47
In the example of Figure 4.1, the point fraction is 8/25 = 0.32. In practice
the grid will be placed on a number of fields, each producing a particular point
count for beta hits. This distribution of number of hits has a mean value P , and
a corresponding sample mean value for the point fraction, P P = P /PT, where PT is
the total number of points in the grid, 25 in this example. Following usual statistical practice, this sample mean value, P P, is use to estimate the expected value of
the point fraction ·PPÒ in the b phase for the population of points in the structure,
and, through equation (3.1), the area fraction of the b phase. The area fraction of
the a phase in this structure may be evaluated by counting hits in a, or simply by
subtracting AAb from 1.
Figure 4.2 shows a three phase structure composed of a, b and g areas.
In this example, a 5 ¥ 5 grid is placed on the structure. The procedure described
in the preceding section may be applied separately to hits in a, hits in b and hits in
the g phase. This placement of the grid gives 10 hits on the a phase, 8 on the b
particles and 7 hits on g. The corresponding point fractions of the three phases are
obtained by dividing by PT = 25. Estimates for the three area fractions for this single
Figure 4.2. Measurement of the area fractions in a three phase structure AAa, AAb, AAg.
(For color representation see the attached CD-ROM.)
Probe population:
Points in two dimensional space
This sample:
25 points in the grid
Event:
Points lie in each phase, a, b, g
Measurement:
Count the points in each phase
This count:
10 points in a, 8 points in b, 7 points in g
Relationship:
·PPÒ = AA
Normalized count:
PPa = 10/25 = 0.40; PPb = 8/20.32; PPg = 7/25 = 0.28
Geometric property:
AAa = 0.40; AAb = 0.32; AAg = 0.28
48
Chapter 4
placement of the grid are AAa = 0.40; AAb = 0.32; AAg = 0.28. Statistically valid
results will require replication of these counts on a number of appropriately chosen
fields.
Area fractions of constituents can be measured directly in computer assisted
image analysis. This is a straightforward application of the point count applied with
a very high density of points. In this case, each pixel in the image is a point probe.
The phase of interest in the count is segmented from the rest of the image on the
basis of its grey shade range or its color, perhaps along with some other geometric
characteristics of the phase, such as size or shape. Numerical gray shades or color
values for those pixels that satisfy the conditions set up to identify the phase are set
to black, and the remaining pixels are set to white. The “detected image” is thus a
“binary image”. The computer then simply counts black pixels in this binary image.
The point fraction is the ratio of this count to the total number of pixels in the
image. These counts can be made essentially in real time.
It might seem that the pixel count in an image analysis system would provide
a much greater precision in the estimate of AA than a manual count of a comparatively very limited number of points. However, analysis has shown that the manual
count may give about the same precision, as reflected in the standard deviation of
the counts for a collection of fields. This is partly because much of the variation in
AA derives from differences from one field to another. Also, there is a great deal of
redundant information in the pixel count, since many pixels lie in any given individual feature.
The point count method is most efficient when the grid spacing is such that
adjacent points rarely fall within the same feature, cell or region in the image (they
are then said to be independent samples of the structure). An advantage of this
method is that when the hits produced by the point grid are all independent, the
number of hits can be used directly to estimate the measurement precision, as discussed in Chapter 8.
The most difficult step in image analysis is detection or segmentation, i.e.,
making the computer “see” the phase to be analyzed as the human operator sees it.
There are always pixels included in the discriminated phase that the operator can
“see” are not part of the phase, and pixels not included that “should be”. Detection difficulties increase with the complexity of the microstructure. Such problems
frequently can be minimized with more careful sample preparation and additional
image processing and more steps in the analysis. In any experimental situation it is
necessary to balance the effort required to develop an acceptable sample preparation strategy and discrimination algorithm against the inconvenience of the manual
measurement which incorporates the most sophisticated of discrimination systems,
the experienced human operator with independent knowledge about the sample and
its preparation.
In these procedures for estimating area fraction of a phase it is not necessary to know the actual dimensions of the grid, since the measurement involves only
ratios of counts of points. The scale of the grid relative to the microstructure, and
the number of points it contains, do not influence the expected value relationship,
equation (4.1). These choices do influence the precision of the estimate obtained,
through the spread in the number of counts made from one grid placement to the
Classical Stereological Measures
49
next, but not the estimated expected value. For example, as a limiting case, if the
grid were very small in comparison with the features in the structure then most of
the time the entire grid would either lie within a given phase, or be outside of it.
For a 5 ¥ 5 grid, counts for most placements of the grid would give a count of 25
or zero. The mean number of counts would still be the area fraction for a large
number of observations, but the standard deviation of these counts would be comparatively large. A correspondingly large number of fields would have to be viewed
to obtain a useful confidence interval on the estimate of the mean.
The point count can also be used to estimate the total area of a single feature
in a two dimensional structure. The solution to this problem is an example of a sampling strategy that is pervasive in the design of stereological experiments called the
“systematic random sample”. Figure 4.3 shows a single particle of the b phase.
Figure 4.3. Area of a feature in two dimensions. (For color representation see the
attached CD-ROM.)
Probe population:
Points in two dimensional space
This sample:
6 ¥ 8 = 48 points in the grid
Calibration
l0 = 5.7 mm
Event:
Points lie in the feature
Measurement:
Count the points in the feature
This count:
20 points in the feature
Relationship:
A = l02·P Ò
Geometric property:
AA = (5.7 mm) · 20 = 650 mm2
50
Chapter 4
Design a grid of points with outside dimensions large enough to include the whole
particle. In order to estimate the area of the feature with a point count it is necessary to calibrate the dimensions of the grid at the actual magnification of the image.
A stage micrometer may be used to measure the overall dimensions of the grid and
compute the spacing between grid points. Let the spacing thus measured be l0, and
the total number of points in the grid be PT. For the 6 ¥ 8 point grid in Figure 4.3,
there are 48 grid points. Calibration of the magnification gives the grid spacing as
l0 = 5.7 mm. The population of points that the grid samples is contained in the area
AT given by PT · l02 (= 48 · (5.7)2 = 1560 mm2 in Figure 4.3).
Focus on the square box in the upper left corner of the area in Figure 4.3.
A specific placement of the grid may be specified by locating the upper left grid
point Q at any point (x, y) within this box. A given choice of (x, y) specifies the
location of the remaining 48 probe points in the grid. Imagine that the point (x, y)
is moved to survey all of the points in the corner box. Then the remaining grid
points systematically sample all of the points in the area AT, i.e., the population of
points of interest.
The number of points P that lie within the feature is counted for a given
placement of the grid (20 points hit the particle in Figure 4.3). The point fraction
PP for that placement of the grid may be computed as P/PT. If the point Q at (x,
y) that locates the grid is chosen uniformly from the points in the corner box then
the point fraction for this sample value is an unbiased estimate of the expected value
for the population ·PPÒ and equation (4.1) applies. Statistically speaking, the point
fraction in this experiment is an unbiased estimate of the fraction of the area AT
occupied by the b particle. The expected value relation is
PP = AA =
A
A
=
AT PT l02
(4.2)
The number of points PT in the grid in this experiment is fixed. The number
of points that hit the area of the figure, P, varies from trial to trial. Thus
PP =
P
PT
(4.3)
where ·PÒ is the expected value of the number of points observed within the feature
on each trial. Insert this result into equation (4.2) and solve for A
A = (PT l02 ) PP = (PT l02 )
P
= l02 P
PT
(4.4)
This simple result is the statistical equivalent of tracing the figure on graph
paper and counting the number of squares that lie in the figure. The area of the b
feature is seen to be the area associated with a given point (l02) times the number of
points that hit the figure. This is approximately true for any single placement of the
grid. The argument that leads to this perhaps obvious result demonstrates that the
area is exactly equal to the area of a grid square times the expected value of the
number of hits on the feature. This transforms the simple geometric approximation
into a statistical result, with the attendant potential for replication of the
Classical Stereological Measures
51
experiment, evaluation of standard deviation and estimate of the precision of the
result. For the b feature in Figure 4.3, the 20 hits for the placement of the grid shown
estimates the area of the feature to be 650 mm2.
Volume Fraction from the Point Count
The most commonly measured property of three dimensional feature sets
is their volume, usually reported as the volume fraction, VV, of the phase. This
property may be estimated using either plane, line or point probes; the simplest
and most commonly used measurement relies on point probes. The population of
points to be sampled by these probes is the set of points contained within the volume
of the specimen in three dimensional space. Point probes are normally generated
by first sectioning the sample with a plane, and generating a grid of points on the
plane section. As in the two dimensional structure described in the last section, the
event of interest is “point hits the b phase” where “b phase” is taken to mean
the set of features at the focus of the analysis. These points are simply counted.
The stereological relation that connects this point count with the volume of the
structure is
PP
b
= VVb
(4.5)
Visualize a specimen composed of two phases a and b. This structure is
revealed by sectioning it with planes and examining fields on these sections, as shown
in Figure 4.4. The population of points in the three dimensional specimen is probed
by the 5 ¥ 5 grid of points superimposed on this structure. For this sample, 5 points
lie within the b phase. This count is replicated on a series of fields on the set of
sectioning planes. The mean P and standard deviation sP of these counts are
computed. The mean point fraction, P /PT, is used to estimate the expected value
for the population of points, ·PPÒ, and, through equation (4.5), the volume fraction occupied by the b phase. For the field shown in Figure 4.4, this estimate is 5/25
= 0.25.
In order to obtain an estimate of the volume fraction of b in the three dimensional structure represented by the plane section in Figure 4.4 it is necessary to
repeat this measurement on a number of fields that are chosen to represent the population of points in three dimensions and average the result. Table 4.1 provides an
example of results that might be derived from a set of observations of 20 such fields.
The mean number of counts on these fields is 7.60 and the standard deviation of
the set of 20 observations is found to be 1.5.
The standard deviation of the population mean in this experiment is given
sP
by s P =
where n = 20, the number of readings in this sample. P ± 2s P is the
n
95% confidence interval associated with this set of readings. The result in Table 4.1
may be interpreted to mean that the probability is 0.95 that the expected value
of P for the population of points lies within the interval 7.60 ± 0.68. Since each
field was sampled with 25 points, the normalized point count and its confidence
interval is obtained by dividing both numbers by 25. There is thus an 0.95 probability that the expected value of the point fraction lies in the interval 0.277 to 0.231.
52
Chapter 4
Figure 4.4. Measurement of the volume fraction VV of a phase in three dimensions.
(For color representation see the attached CD-ROM.)
Probe population:
Points in three dimensional space
This sample:
25 points in the grid
Event:
Points lie in the phase
Measurement:
Count the points in the phase
This count:
5 points in the phase
Relationship:
·PPÒ = VV
Normalized count:
PP = 5/25 = 0.25
Geometric property:
VV = 0.25
Since the expected value of the point fraction estimates the volume fraction by equation (4.5), this range is also the confidence interval for the estimate of the volume
fraction.
Figure 4.5 shows a structure consisting of three phases, a, b and g. In this
structure the small g particles lie within the b phase; there are no g particles in the
a matrix. This is a very common structure in life science applications where
an organelle (g ) is a part of the b cells. The a phase is not of interest in the
Table 4.1. Point counts from the structure in Figure 4.4
P
7.60
sP
1.51
sP
0.34
P ± 2sP
7.60 ± 0.68
PP ± 2s PP
0.304 ± 0.027
VV ± 2sVV
0.304 ± 0.027
Classical Stereological Measures
53
Figure 4.5. Small particles of the g phase (black) are located within b features (gray)
in this three phase structure. There are no g particles in the a phase (white). A 70 point
grid is used to estimate the fraction of the volume of b occupied by g by separately estimating the volume fractions of the b and g phases in the structure. (For color representation see the attached CD-ROM.)
Probe population:
Points in three dimensional space
This sample:
70 points in the grid
Event:
Points lie in a, b or g phase
Measurement:
Count the points in each phase
This count:
30 points in b, 8 points in g
Relationship:
·PPÒ = VV
Normalized count:
PPb = 30/70 = 0.43; PPg = 8/70 = 0.11
Geometric property:
VVb = 0.43; VVg = 0.11; VVb,g = 0.11/(0.43 + 0.11) = 0.20
experiment; the fraction of the volume of the b cells that are occupied by g organelles
is the object of this example. There are 70 points in the grid. For the field shown,
30 points hit the b phase and 8 hit g; the remaining 32 points are in a. These counts
are replicated on a number of fields. The mean number of hits in each phase, P b
and P a, are computed, along with their standard deviations. Corresponding point
fractions are obtained by dividing by PT = 70 for this case. The resulting point fractions are used to estimate their corresponding expected values and hence the volume
fractions, VV b, VVg, and, by difference, VVa by applying equation (4.5).
The volume fraction of the structure occupied by the b cells including the g
organelles contained within them is the sum VVb + VVg. To find the fraction of that
volume occupied by g organelles, take the ratio, VVg/(VV b + VVg ). For the single field
shown the estimates are: VVg = 8/70 = 0.11; VV b = 30/70 = 0.43 and VVa 0.46. The
fraction of the volume of the b regions occupied by g is 0.11/(0.43 + 0.11) = 0.20.
In this analysis it is important to obtain valid estimates of the volume fractions of the three phases separately, and then combine the results to obtain the
desired comparison. As an alternate (incorrect) procedure, imagine taking counts
of b and g for each field, Pb and Pg. Add the b and g counts, (Pb + Pg). Next, take
54
Chapter 4
the ratio [Pg /(Pb + Pg)]. Average this ratio over a number of fields to estimate the
fraction of b cells occupied by g organelles. This procedure does not give the same
result as that described in the previous paragraph because the average of a sum of
ratios is not the same as the ratio of the averages. To obtain valid estimates it is
important to measure the volume fractions of the various phases with respect to the
structure as a whole, and subsequently manipulate these results to provide measures
of relative volumes among the phases.
Figure 4.6 illustrates the use of the point count to estimate the total volume
of a single feature in a three dimensional microstructure. A three dimensional array
of points is used to sample the feature. Visualize a box large enough to contain the
feature. N equally spaced planes are sliced through the feature; let h be the distance
between planes. N = 5 and h = 11.5 mm in Figure 4.6. Each plane has an (m ¥ n) grid
of points with a calibrated grid spacing l0 imposed on it. Grids with (5 ¥ 5) points
with spacing of 5.2 mm are shown in Figure 4.6). In this way a three dimensional
grid of points with (N ¥ m ¥ n) points {(5 ¥ 5 ¥ 5) = 125 in Figure 4.6} is constructed
which includes the entire feature. The box associated with each grid point has dimensions (v0 = l0 · l0 · h), {(5.2 · 5.2 · 11.5 = 311 mm3 in the figure}. Counts are made of the
number of points that hit the feature on each of the five sectioning planes. The sum
of these five counts is the total number of points in the three dimensional grid that
hit the particle. In Figure 4.6 the total number of hits is (1 + 3 + 2 + 3 + 1 = 10 hits).
Visualize a small box with the dimensions v0 = l0 · l0 · h at the upper left rear
corner of the large box that contains the feature. The upper, left, back corner of the
Figure 4.6. The Cavalieri principle is used to estimate the volume of a single feature
by counting points on a series of sectioning planes. The spacing between the planes
is h. (For color representation see the attached CD-ROM.)
Probe population:
Points in three dimensional space
This sample:
5 ¥ 5 ¥ 5 = 125 points in the grid
Calibration:
h = 11.5 mm, l0 = 5.2 mm
n0 = l0 · l0 · h = 5.2 · 5.2 · 11.5 = 311 mm3
Event:
Points lie in the feature
Measurement:
Count the points in the feature
This count:
1 + 3 + 2 + 3 + 1 = 10 points in the feature
Relationship:
Vb = PTv0·PPÒb = v0·P Òb
Geometric property:
Vb = 311 mm 3· 10 = 3110 mm3
Classical Stereological Measures
55
three dimensional grid will be located at some point Q = (x, y, z) within this small
box. For any given choice of Q, the positions of the remaining (N ¥ m ¥ n) points
in the grid is determined. They provide a systematic sample of the population points
within the containing box. As Q is moved to all of the points in the
small corner box, the grid of points samples all of the population of points within
the containing box. Thus, a random choice for the position of Q from its uniform
distribution of possible points in the small box produces a systematic random
sample of the population of points in the containing box. The fraction of points in
the three dimensional grid thus provides an unbiased estimate of the expected value
for the population of points in the containing box and equation (4.5) may be
applied:
PP
b
= VV b =
Vb Vb
=
VT PT v0
(4.6)
The number of points that hit the feature varies from trial to trial. Since the
total number of points PT is the same in all placements of the three dimensional
grid, the expected value of this point count estimates the point fraction
PP
b
=
P
PT
b
(4.7)
Insert this result into equation (4.6) and solve for Vb:
b
V b = PT v0 PP
b
= PT v0
P
= v0 P
PT
b
(4.8)
Thus, the volume of the b feature is the volume associated with an individual point (l0 · l0 · h) times the expected value of the number of hits that points
in the grid make with the feature. For a single position of the grid, v0Pb provides
an estimate of the volume of the feature. For the example illustrated in Figure
4.6, a total of 10 hits are noted. The volume associated with a grid point was
computed earlier to be 311 mm3. Thus the volume of the feature is estimated at
(10)(311) = 3110 mm3.
The structured random sample is a much more efficient procedure than
random sampling in which grid points are independently placed in the volume. In
the latter case, points will inevitably cluster in some areas producing oversampling,
while being sparse in others producing undersampling. It will take on the average
nearly three times as many independent random points to reach the same level of
precision as with the use of the structured approach, but the same answer will still
result, namely that the point fraction that hit the phase or structure measures the
volume.
The point counting procedure to estimate the volume of a three dimensional
object is an example of the oldest of the stereological procedures based upon the
“Cavelieri Principle” (Howard & Reed, 1998a). The object to be quantified is sliced
into a collection of slabs of known thickness. Some method is used to measure the
area of each slab, such as the point count described in an earlier section. An alternate procedure might use a planimeter, an area measuring mechanical instrument
56
Chapter 4
used by cartographers before the advent of the computer, to measure the individual cross section areas. If the slabs are of uniform thickness, and uniform content,
weighing each slab, together with a calibration of the weight per unit area, could be
used. The underlying principle takes the volume of each slab to be its cross sectional
area times its thickness, so that the volume of the object is the sum of the volumes
of the slabs
N
V = Â Ai h = A ( Nh)
(4.9)
i =1
where ·AÒ is the average cross section area of the slabs and (Nh) is the total height
of the object.
Two-Dimensional Structures; Feature Perimeter from the Line
Intercept Count
Each of the collection of features in the two dimensional structure shown
in Figure 4.7 has a boundary, and each boundary has a length commonly referred
to as its perimeter. The normalized parameter, LA, is the ratio of the total length
of boundaries of all of the features in the specimen divided by the area that the
specimen it occupies. This perimeter length per unit area can be estimated by
probing the structure with a set of line probes represented in Figure 4.7 by the
four horizontal lines in the superimposed grid. The event of interest in the measurement is “line intersects boundary”. A simple count of these events forms the
basis for estimating LA through the fundamental stereological formula (Underwood,
1970a):
PL =
2
LA
p
(4.10)
PL in this equation is called the “line intercept count”; it is the ratio of the
number of intersections counted to the total length of line probe sampled (only
the horizontal lines in the grid were used in this example). Each of the lines in
Figure 4.7 is found by calibration to be 17.7 micrometers (mm) long. The total
length of line probe sampled in this placement of the grid is 4 ¥ 17.7 = 70.8 mm.
Since there are 17 intersections marked and noted in Figure 4.7, for this example
PL is 17/70.8 = 0.24 counts/mm probed. The left hand side of equation (4.10) is the
expected value of this measurement for the population of lines that can be constructed in two dimensional space. Inverting equation (4.10) gives the estimate of
the perimeter length per unit area for the features shown in Figure 4.7:
LA =
p
p
Ê mm ˆ
PL = 0.24 = 0.38Á
˜
Ë mm2 ¯
2
2
(4.11)
The area within the grid shown in Figure 4.7 is (17.7)2 = 313 mm2. A rough
estimate of the boundary length of the features contained in the grid area is thus
0.38 (mm/mm2) · 313 mm2 = 119 mm. There are 8.5 particles in the area of the grid.
(Particles that lie across the boundary are counted as 1/2.) A rough estimate of the
Classical Stereological Measures
57
Figure 4.7. Use of the line intercept count to estimate the total length of the boundary
lines of a two dimensional feature set. (For color representation see the attached CDROM.)
Probe population:
Lines in two dimensional space
This sample:
Four horizontal lines in the grid
Calibration:
L0 = 17.7 mm, total probe length = 4 · 17.7 = 70.8 mm
Event:
Line intersects the feature boundary line
Measurement:
Count the intersections
This count:
17 intersections
Relationship:
·PL Ò = (p/2) · LA
Normalized count:
PL = 17 counts/70.8 mm = 0.24 counts/mm
Geometric property:
LA = (p/2)PL = (p/2) · 0.24 = 0.38 (mm/mm)
average perimeter of particles may be estimated by dividing the total boundary
length in the area by the number of particles: 119 mm/8.5 = 14 mm. Inspection of the
features in Figure 4.7 indicates that this result is plausible.
Each member of the population of lines in two dimensional space has two
attributes: position, and orientation. The set of lines in the grid used for a probe in
Figure 4.7 represent a few different positions in the population of lines, but only a
single orientation. In order for a sample of line probes to provide an unbiased estimate of a value for the population of probes it is necessary that the probe lines uniformly sample all orientations of the circle. A representative sample of the
population of orientations of lines could be obtained by rotating the stage by
58
Chapter 4
uniform increments between measurements. A much more direct strategy, which
guarantees uniform sampling of line orientations, uses test line probes in the shape
of circles to make the PL measurement.
Figure 4.8 shows a microstructure with phase boundaries that tend to be
aligned in the horizontal direction. In Figure 4.8a and 4.8b a square grid of points
is superimposed on this structure. Calibration shows that the grid is square 23.7 mm
on a side. The set of five horizontal lines are used to sample the population of lines
in two dimensional space in Figure 4.8a. The total length of lines probed in the
a
Figure 4.8a. Use of horizontal line probes to measure the total projected length of
boundaries of two dimensional features on the vertical axis LA,proj(vert). (For color representation see the attached CD-ROM.)
Probe population:
Horizontal lines in two dimensional space
This sample:
Five horizontal lines in the grid
Calibration:
L0 = 23.7 mm, total probe length = 5 · 23.7 = 118.5 mm
Event:
Line intersects the feature boundary line
Measurement:
Count the intersections
This count:
8 intersections
Relationship:
·PLÒ(horz) = LA,proj(vert)
Normalized count:
PL = 8 counts/118.5 mm = 0.068 counts/mm
Geometric property:
LA,proj(vert) = PL(horz) = 0.068 (mm/mm2)
Classical Stereological Measures
59
b
Figure 4.8b. Use of vertical line probes to measure the total projected length of boundaries of two dimensional features on the horizontal axis LA,proj(horx). (For color representation see the attached CD-ROM.)
Probe population:
Vertical lines in two dimensional space
This sample:
Five vertical lines in the grid
Calibration:
L0 = 23.7 mm, total probe length = 5 · 23.7 = 118.5 mm
Event:
Line intersects the feature boundary line
Measurement:
Count the intersections
This count:
62 intersections
Relationship:
·PLÒ(horz) = LA,proj(vert)
Normalized count:
PL = 62 counts/118.5 mm = 0.52 counts/mm
Geometric property:
LA,proj(vert) = PL(horz) = 0.52 (mm/mm2)
horizontal direction is (5 · 23.7 = 118.5 mm). The events, line intersects boundary, are
marked with red markers in Figure 4.8a. There are 8 intersections, giving a line intercept count in the horizontal direction of {8/118.5 = 0.068 (counts/mm)}. If the set
of vertical lines in the grid are used as a line probe (Figure 4.8b), the resulting count
is 62; for this direction, PL = 62/118.5 = 0.52 counts/mm. It is clear that the line intercept count is different in different directions in this aligned structure, and that this
reflects the anisotropy of the structure.
60
Chapter 4
c
Figure 4.8c. Use of circular line probes to measure the true total length of boundaries
in two dimensional features, LA. (For color representation see the attached CD-ROM.)
Probe population:
Lines in two dimensional space
This sample:
Set of lines equivalent to 8 circles
Calibration:
d = 23.7/4 = 5.9 mm, total probe length = 8 · p · d = 148.9 mm
Event:
Line intersects the feature boundary line
Measurement:
Count the intersections
This count:
55 intersections
Relationship:
·PLÒ = (p/2)LA
Normalized count:
PL = 55 counts/148.9 mm = 0.37 counts/mm
Geometric property:
LA = (p/2)PL = 0.58 (mm/mm2)
Indeed, counts in different directions may be used to characterize the
anisotropy quantitatively. Replications on a collection of fields produce mean and
standard deviations of the counts made in each direction. A polar plot of PL as a
function of q, the angle the line direction makes with the horizontal direction, as
shown in Figure 4.9, called the “rose of the number of intersections” [Saltykov, 1958;
Underwood, 1970b] provides a graphical representation of this anisotropy in a two
dimensional structure.
The true length of boundaries in this anisotropic structure may be estimated
by applying test lines that are made up of circular arcs, as shown in Figure 4.8c.
Classical Stereological Measures
61
Figure 4.9. Rose-of-the-number-of-intersections for a two-dimensional anisotropic
structure.
The population of orientations in this set of circular line probes provides a uniform
sample of the population of line orientations in two dimensional space. This line
grid is equivalent to eight circles, each with a calibrated diameter of 23.7/4 =
5.92 mm. The total length of this collection of line probes is thus 8 [p · (5.92 mm)] =
148.9 mm. The number of intersections with these line probes marked in Figure 4.8c
is 55, giving a line intercept count PL = 55/148.9 = 0.37 (counts/mm). Note that this
result is within the range between the result for horizontal lines (0.068) and the vertical lines (0.52). Equation (4.10) gives the corresponding estimate for the true
perimeter length: LA = (p/2) 0.37 = 0.58 mm/mm2.
The number of features per unit area in Figure 4.8 is counted to be
21/(23.7)2 = 0.037 (1/mm2). A rough estimate of the average perimeter of features in
this structure is LA/NA = 0.58/0.037 = 16 mm. Inspection of Figure 4.8 shows that
this is not an unreasonable estimate of the average particle perimeter.
Most image analysis programs provide a measurement of the perimeter of
individual features directly on the digitized image. Boundary pixels are identified in
the detected binary image. The boundary line is approximated as a broken line connecting corresponding points in adjacent boundary pixels. The perimeter of the
feature is then computed as the sum of the lengths of the line segments that make
up this broken line. The attendant approximation may produce significant errors.
The magnitude of these errors depends upon how many different directions are used
to construct the broken line boundary.
Consider the circle shown in Figure 4.10. Its digitized image is represented
by the blue plus red pixels. Identification of which pixels lie on the boundary may
vary with the choice of the number of neighbors that are required to be “non-b”.
The broken line used to compute the perimeter is illustrated for two different cases.
In Figure 4.10a only four directions are used in constructing the perimeter. It is easy
62
Chapter 4
Figure 4.10. Illustration of the bias in measuring perimeter of a circle in a digitized
image: a) four directions gives the perimeter of a square; b) eight directions gives that
of an octagon. (For color representation see the attached CD-ROM.)
to see that the line segments connecting neighboring pixels can be combined to make
the square that encloses the circle. If the diameter of the circle is d, then its actual
perimeter is (p d ). The reassembled broken lines make a square with edge length d;
its perimeter is (4 d ). This overestimates the perimeter of the circle by a factor of
4/p = 1.27. That is, use of this algorithm gives a built in bias for every particle
of about +27%. In Figure 4.10b the segments assemble into an octagon; the ratio
of perimeter of the broken line to the circle is computed to be 1.05. Thus, even for
this more sophisticated algorithm, evaluation of the perimeter of simple particles
has a bias of about +5%.
Constructing an n-sided polygon with sides running in 16 possible directions
(a 32-gon) further improves the accuracy of the perimeter. For example, measuring
circles varying from 5 to 150 pixels in diameter using this technique gives results
that are within 1% as shown in Table 4.2. However, the bias is always positive (the
measured value is longer than the true value). Another new method for perimeter
measurement is based on smoothing the feature outline and interpolating a superresolution polygon using real number values (fractional pixels) for the vertices (Neal
et al., 1998). This method has even better accuracy and less bias, but is not yet widely
implemented. The presence of any bias in measurements is anathema to stereologists and so they generally prefer to use the count of intercepts made by a line grid
with the outlines.
A simple experiment will permit the assessment of a given software in its
measurement of perimeter. Generate, by hand or computer graphics, a collection of
black circles or different sizes. Acquire the image in the computer. Detect the collection of circles. Instruct the software to measure the area A and perimeter P of
each feature and compute some measure of the “circularity” of the features, such
Classical Stereological Measures
63
Table 4.2. Perimeter measurements using a 32-sided polygon
Diameter (pixels)
5
10
15
20
30
40
50
75
100
125
150
Actual Perimeter
15.7078
31.4159
47.1239
62.8319
95.2478
125.6637
157.0796
235.6194
314.1593
392.6991
471.2389
Measured Perimeter
15.6568
31.8885
47.2022
63.4338
95.4562
126.6030
158.1100
236.6480
314.6880
393.1660
471.8700
Error (%)
-0.326
+1.504
+0.166
+0.958
+0.219
+0.747
+0.656
+0.437
+0.168
+0.119
+0.134
as the formfactor 4pA/P2. This parameter has a value of 1 for a perfect circle. Measured values may be consistently smaller than 1.00, and may vary with size of the
circle in pixels.
Use of the manual line intercept count to estimate perimeter does not suffer
from this bias.
Three-Dimensional Structures: Surface Area and the
Line Intercept Count
Figure 4.11 shows a specimen that contains internal interfaces. Usually interfaces of interest in microstructures are part of the boundaries of particles, e.g., the
ab interface in a two phase structure, but this is not a requirement for estimating
Figure 4.11. A three-dimensional specimen with internal surfaces. (For color representation see the attached CD-ROM.)
64
Chapter 4
their surface area, reported as the surface area of interface per unit volume of structure. This property, SV, is sometimes referred to as the surface density of the two
dimensional feature set imbedded in a three dimensional structure. Line probes are
used to sense the area of surfaces. A sample of the population of lines in three
dimensional space is usually constructed by sectioning the structure with a plane to
reveal its geometry, then superimposing a grid of lines on selected fields in that
plane. The events to be counted are the intersections of the probe lines with traces
of the surfaces revealed on the plane section. The line intercept count, PL, described
in the preceding section in the context of analyzing a two dimensional structure, i.e.,
the ratio of the number of intersection points to the total length of probe lines in
the sample, is made for the grid in each field examined. The governing stereological relationship is
PL =
1
SV
2
(4.12)
In the preceding section the structure under analysis was viewed as a two
dimensional microstructure, and the properties evaluated were two dimensional
(perimeter, area). If indeed the microstructures shown are plane probe samples of
some three dimensional microstructure, then the features observed have the
significance of sections through three dimensional features. To obtain an unbiased
estimate of the expected value of PL for the population of lines in three dimensional space, the probes in the sample must be chosen uniformly from the population of positions and orientations of lines in three dimensional space, and the
geometric properties estimated are those of the three dimensional microstructure
sampled.
In this context, the line intercept count obtained from Figure 4.7, PL = 0.24
(counts/mm) takes on new meaning. If the field chosen and the line probes are a
properly representative sample of the population of lines in three dimensions, then
this value is an unbiased estimate of the expected value ·PLÒ for that population.
This requires a carefully thought out experimental design that prepares and selects
fields and line directions to accomplish this task. For the single sectioning plane represented in Figure 4.7, this will only be valid if the surfaces bounding the features
in the figure are themselves distributed uniformly and isotropically. Then inversion
of equation (4.12) then gives SV = 2(0.24) = 0.48 mm2/mm3 for the surface density of
the ab interface in this structure.
It is difficult to visualize the meaning of this value of surface density for the
particles in Figure 4.7. As an aid in seeing whether this result is plausible, estimate
the mean lineal intercept for the b particles in the structure by applying equation
(3.2) in Chapter 3. The points in the grid may be used to estimate the volume fraction of the b phase by applying equation (4.5). For the placement shown, 3 points
hit the b phase, giving a rough estimate of VV b = 3/16 = 0.19. This result may be
combined with the surface density estimate to calculate a rough value of the mean
lineal intercept of the b particles:
l
b
=
4VV b 4 ◊ 0.19
=
= 1.6 mm.
0.48
SV ab
(4.13)
Classical Stereological Measures
65
The grid points in Figure 4.7 are about 6 mm apart. The mean lineal intercept averages longest dimensions with many small intercepts near the surface of the
particles. Thus, this very rough estimate appears to be reasonable.
The structure in Figure 4.8 may also be viewed as a section through a three
dimensional anisotropic microstructure. This structure is squeezed vertically and
elongated horizontally. However, a single orientation of sectioning plane is
insufficient to establish the nature of this anisotropy in three dimensions. Indeed,
out of the population of line probes in three dimensional space only those that
lie in the section plane of Figure 4.8 are sampled. The observation that PL is different in different directions takes on a three dimensional character, as does the
anisotropy that this difference measures. The concept of the rose of the number of
intersections as a quantitative description of the anisotropy in the microstructure
extends to three dimensional microstructure. The three dimensional rose is a plot
of the line intercept count as a function of longitude and latitude on the sphere of
orientation.
An understanding of the meaning of such a construction may be aided by
visualizing the meaning of the directed line intercept count. Consider the subpopulation of lines in three dimensional space that all share the same orientation,
specified by the longitude, q, and the co-latitude (angle between the direction and
the north pole), f, as shown in Figure 4.12. The line intercept count may be performed on a set of directed line probes taken from this subpopulation. This directed
count, PL(q, f) measures the total projected area, A(q, f) of surface elements in the
structure on the plane that is perpendicular to the direction of the test lines. Thus,
in Figure 4.8, the low value for counts for horizontal line reflects the fact that the
features have a small footprint when projected on the plane perpendicular to this
direction. The large number of counts for vertical line probes derives from the large
Figure 4.12. Orientation in three-dimensional space is specified by a point on a sphere
represented by spherical coordinates (q, f). The population of line orientations is
sampled without bias if the line probes used are cycloids on vertical sections (see text).
The dimensions of the cycloid can be expressed in terms of the height, h, of the circumscribed rectangle (sometimes referred to as the minor axis). (For color representation see the attached CD-ROM.)
66
Chapter 4
projected area of these particles on the horizontal plane. The governing stereological relationship is (Underwood, 1970b):
PL (q , f ) = SV cos a
(4.14)
where ·cos aÒ is the average value of the angle between the direction of the test
probes and the directions normal (perpendicular) to all of the surface elements in
the structure.
The circular test lines used for the two dimensional analysis of the structure
in Figure 4.8 provide an ingenious and economical mechanism for automatically
averaging over the population of line orientations in two dimensional space.
However in three dimensional space line probe orientations are uniformly distributed over the sphere of orientation. Averaging over the circle of orientations in a
given plane does not provide an unbiased sample of the population of line orientations in three dimensions, as is demonstrated by the shaded regions in Figure
4.12a.
Circular test probes in a plane sample the set of orientations in a great circle
on the sphere, such as the circle containing the pole in Figure 4.12a. This subset of
line orientations in a great circle is uniformly sampled in all directions by a circular line probe in the plane. This implies that the length of line probes with orientations within five degrees of the pole (green cap) is the same as then length sampled
within five degrees of the equator (red stripe). The green cap is clearly a very much
smaller fraction of the area of the sphere of orientation than is the red stripe. The
green cap represents a much smaller fraction of the set of line orientations in three
dimensions than does the red stripe. But the circular test probe samples these regions
with the same length of line probe. Thus, such a probe design over-samples orientations near the pole and under-samples them near the equator. This design does
not provide a uniform sample of the population of line orientations in three dimensions, and will produce a biased estimate of expected values for the line intercept
count.
An unbiased sample of this population is provided by a sample design known
as the method of vertical sections (Baddeley et al., 1986), illustrated in Figure 4.13.
A reference direction is chosen in the macroscopic specimen, shown in Figure 4.13a.
In the remainder of the analysis, this is called the “vertical direction”. Sectioning
planes are prepared which contain (are parallel to) this reference direction, as shown
in Figure 4.13b. These sections are chosen so that they represent the variations in
the structure with position, and with orientation in “longitude”, i.e., around the vertical direction.
In a typical experiment, a longitude direction is chosen at random and a
section is cut that contains this direction and the vertical direction. Two other vertical sections, 120° away from the first, are also prepared. This provides a random
systematic sample of the longitude orientations. Fields are chosen for measurement
in these sections that are uniformly distributed with respect to position on the vertical plane. The vertical direction is contained in each field, as shown in Figure 4.13c,
and is known.
In order to provide an unbiased sample of the population of line orientations on the sphere, a grid is constructed of line probes that have the shape of a
Classical Stereological Measures
67
Figure 4.13. The stereological experimental design based on vertical sections gives
an unbiased sample of the population of line orientations in three-dimensional
space. (For color representation see the attached CD-ROM.)
cycloid curve1, shown in Figure 4.13d. This curve has the property that the fraction
of its length pointing in a given direction decreases as the tangent rotates from the
vertical direction in a manner that is proportional to the sine of the angle from the
vertical. The need for this sine-weighting is indicated in Figure 4.12b. Two equal
intervals in the range of f values are shown, one at the pole (green) and the other
at the equator (red). The cycloid test line exactly compensates for the sine weighting required of the f direction with longer length of lie probe near the equator (red
segment) and shorter near the pole (green segment). This sine weighting of the line
length produces an unbiased sample of the population of orientations in three
dimensional space. A grid consisting of cycloid test lines with known dimensions,
so that the total length of lines probed is known, forms the basis for an unbiased
PL count in three dimensional space.
Figure 4.14 reproduces the anisotropic structure in Figure 4.8 assuming it is
a representative vertical section through a three dimensional structure. The height
of the box containing the grid is calibrated at 28.8 mm. The grid is made up of 12
cycloid segments joined together in rows of four. The height of each cycloid segment
is (28.8/6) = 4.8 mm. The length of a cycloid segment is twice its height, or 9.6 mm.
(Figure 4.12 has the dimensions of a cycloid segment.) Thus, the total length of lines
in the grid is (12 segments ¥ 9.6) = 115.1 mm. A total of 24 intersections with ab
boundaries are marked and counted. The line intercept count, PL = 24/115.1 = 0.21
1
The cycloid curve is generated by a point on the rim of a wheel as the wheel rolls along a
horizontal road. Cycloid segments used in the method of vertical sections as shown in Figure
4.13 correspond to the curve traced out by the first half turn of the wheel.
68
Chapter 4
Figure 4.14. Line intercept count made on a presumed vertical section using a cycloid
test line grid gives an unbiased estimate of the surface area per unit volume, SV. Arrow
indicates the vertical direction in the specimen. (For color representation see the
attached CD-ROM.)
Probe population:
Lines in three dimensional space
This sample:
Vertical section with cycloid line probes; 12 cycloid units joined
together in three lines
Calibration:
L0 = 28.8 mm. Each unit has height h = 28.8/6 = 4.8 mm length
of each cycloid = l = 2 · h = 9.6 mm total probe length =
12 · l = 115.1 mm
Event:
Cycloid probe intersects the feature boundary line
Measurement:
Count the intersections
This count:
24 intersections
Relationship:
·PLÒ = (1/2)SV
Normalized count:
PL = 24 counts/115.1 mm = 0.21 counts/mm
Geometric property:
SV = 2PL = 0.42 (mm2/mm3)
(counts/mm). This is an unbiased estimate of the expected value ·PLÒ for lines in
three dimensional space. It may thus be used to provide an estimate of the surface
area density through the equation:
SV ab = 2 PL
ab
Ê mm2 ˆ
Ê 1 ˆ
= 2 ◊ 0.21Á
˜ = 0.42 Á
˜
Ë mm ¯
Ë mm3 ¯
(4.15)
In the analysis of Figure 4.8 as a two dimensional structure, line intercept
counts were made in the vertical and horizontal directions (Figure 4.8). Results of
this analysis gave PL,h = 0.059 (1/mm) and PL,v = 0.45 (1/mm). Interpret these counts
as directed probes in the three dimensional population of lines. According to
Classical Stereological Measures
69
equation (4.14), these counts measure the total projected area of particles on a plane
perpendicular to the probe direction, and are related to the average of the cosine of
the angle that surface elements make with the probe direction. Measurement with
the cycloid test line grid provides an estimate of the unbiased value for SV. These
results may be combined to provide a first estimate of the average of the cosines of
surface element normals with the horizontal direction and the vertical direction in
Figure 4.8:
cos a
cos a
h
v
PL,h 0.059
=
= 0.14
0.42
SV
0.45
P
= L ,v =
= 1.09
0.42
SV
=
(4.16)
The maximum value of the cosine function is of course 1.00. The estimate
for the vertical projection violates this limting value. This impossible result arises
because counts were obtained from a single placement of the grids involved, and
these counts are only estimates of their expected values. Further, the variance of the
ratio of two statistics, like PL,v/SV is the sum of the variances of the individual statistics; the precision of the estimate from a single field is low. Nonetheless, it is clear
from these results (as from a casual inspection of the structures, that most of the
surface elements in this structure have normal directions near the vertical where a
is near zero and cos a is near 1.0.
In order to help visualize the geometric significance of the surface area value
extimated above it is useful to compute the mean lineal intercept of the features in
the structure. This requires an estimate of the volume fraction of the b phase. Apply
the point count to the grid in Figure 4.8: there are three hits on b particles. Thus, a
very rough estimate of VV is 3/25 = 0.12. Insert this result into the expression for
the mean lineal intercept given in equation (3.2), Chapter 3.
l =
4 ◊ 0.12
= 1.2 mm
0.42
(4.17)
For plate-like structures such as those shown in Figure 4.8, ·lÒ reports the
thickness of the plate. This value is plausible for the average plate thickness in this
structure.
Three-Dimensional Microstructures; Line Length and the
Area Point Count
Figure 4.15 shows a specimen with some lineal features as part of its
microstructure. The triple lines in a cell structure or grain structure described in
Chapter 3 provide a common example of such lineal structural features. Dislocations in crystals observed in the transmission electron microscope provide another
example. Edges of surfaces are also lineal features. Tubular structures, such as
capillaries, or plant roots, may be approximated as collections of space curves. The
total length of a one dimensional feature set may be estimated from observations
on a set of plane probes. Line features intersect plane probes in a collection of
points, as shown in Figure 4.16. These points are counted over a field of known
70
Chapter 4
Figure 4.15. Illustration of the use of a plane probe to intersect linear structures in a
volume. Lineal features in a three-dimensional specimen reveal themselves on the
plane probe as a collection of points. Counting the number of intersections provides a
tool to measure the total length of the structures as shown in Figure 4.16. (For color
representation see the attached CD-ROM.)
area. The area point count PA is the ratio of the number of points of emergence of
these line features in the area, normalized by dividing by the area. If the collection
of fields included in the measurement provides a uniform sample of the population
of positions and orientation of planes in three dimensional space, then this count
provides an unbiased estimate of LV, the length of line features in unit volume of
structure:
PA =
1
LV
2
(4.18)
Construction of a representative sample of the population of planes in three
dimensional space is a challenge, since sectioning the sample to form a plane probe
divides the sample. For the special situation in which the lineal features are contained in a transparent medium and can be viewed in projection a procedure similar
to the method of vertical sections is available (Gokhale, 1992).
Figure 4.17 shows a single phase cell structure (e.g., grains in a polycrystal,
or botanical cells). The triple points on this section result from the intersection of
the plane probe with the triple lines in the three dimensional structure. Calibration
shows that the square area to be sampled is 31.1 mm on a side; the area thus probed
in this field is (31.1 = 967 mm). The triple points in this area are marked and counted.
The grid facilitates counting the triple points by dividing the area into sixteen small
squares, which are counted systematically. There are 37 triple points in the area.
Thus, PA = 37/967 = 0.038 (counts/mm). Assuming this field is an unbiased sample
of the population of planes in the specimen, equation (4.18) may be used to estimate the length of triple lines in this cell structure:
Classical Stereological Measures
71
Figure 4.16. Point count made on a section through a structure containing linear features (shown here with finite cross-sectional area for clarity). (For color representation
see the attached CD-ROM.)
Probe population:
Planes in three dimensional space
This sample:
Square area contained within the measurement frame
Calibration:
L0 = 40.0 mm; probe area A0 = (40) = 1600 mm
Event:
Plane intersects the linear features (observed as a small
feature)
Measurement:
Count the intersections
This count:
13 intersections
Relationship:
·PAÒ = (1/2)LV
Normalized count:
PA = 13 counts/1600 mm2 = 0.0081 counts/mm2
Geometric property:
LV = 2PA = 2 · 0.0081 = 0.016 (mm/mm3) = 16 (km/cm)
LV = 2 PA = 2 ◊ 0.038 = 0.076
mm
mm3
(4.19)
In more familiar units, this length corresponds to 76 (km/cm). This at first
surprisingly large result is not unusual for line lengths in microstructures.
Figure 4.18 shows a two phase structure (a + b) with particles of the b phase
distributed along the aa interfaces and aaa triple lines. This is inferred from the
qualitative observation that the b particles are all situated at the former aaa triple
lines. One can visualize three different types of triple lines in this structure through
their corresponding triple points: aaa, aab, and aaa triple points that are not
72
Chapter 4
Figure 4.17. Section through a cell structure; a count of the triple points measures
length of triple lines in the three-dimensional structure, LV.. (For color representation
see the attached CD-ROM.)
Probe population:
Planes in three dimensional space
This sample:
Square area contained within the grid
Calibration:
L0 = 31.1 mm; probe area A0 = (31.1) = 967 mm2
Event:
Plane intersects the triple line (observed as triple point)
Measurement:
Count the intersections
This count:
37 intersections
Relationship:
·PAÒ (1/2)LV
Normalized count:
PA = 37 counts/967 mm2 = 0.038 counts/mm2
Geometric property:
LV = 2PA = 2 · 0.038 = 0.076 (mm/mm3) = 76 (km/cm3)
present because they are occupied by b particles (aaa¢). These three classes of triple
points are marked with respectively red, green and blue markers in the field be analyzed in Figure 4.18. These results are tabulated in Table 4.3.
The total length of aaa triple line, occupied plus unoccupied, is
0.36 (mm/mm). The fraction of the aaa triple line occupied by b particles is
0.16/0.36 = 0.44. Since the volume fraction of b is small (probably only a few percent, since there are no hits on b for the points in the grid in Figure 4.18) this observation that almost half of the cell edge network in three dimensional space is
occupied by the b phase measures the evidently strong tendency of b to be associated with the cell edge network. Further, although the b phase is sparse,
the triple line length in the category aab is about the same as the total length as
that of the cell edge network.
Classical Stereological Measures
73
Figure 4.18. A two phase microstructure in which the a phase is a cell structure. The
length per unit volume, LV, of three kinds of triple lines may be characterized: aaa, aab,
and aaa (occupied). (For color representation see the attached CD-ROM.)
Probe population:
Planes in three dimensional space
This sample:
Square area contained within the grid
Calibration:
A0 = 182.6 mm2
Event:
Plane intersects each class of triple line
Measurement:
Count the intersections
This count:
aaa = 19, aab = 39, aaa(occ) = 15
Relationship:
·PAÒ = (1/2)LV
Normalized count:
PAaaa = 19/182.6 mm2 = 0.10 counts/mm2
PAaab = 39/182.6 mm2 = 0.21 counts/mm2
PAaaa(occ) = 15/182.6 mm = 0.08 counts/mm2
Geometric property:
LVaaa = 2PAaaa = 2 · 0.10 = 0.20 (mm/mm3)
LVaab = 2PAaab = 2 · 0.21 = 0.42 (mm/mm3)
LVaaa(occ) = 2PAaaa(occ) = 2 · 0.08 = 0.16 (mm/mm3)
Table 4.3. Triple point counts in Figure 4.18. Calibrated probe area is 182.6 mm2
Category
aaa
aab
aaa(occ)
Mark
red
green
blue
Counts
19
39
15
PA (1/mm2)
0.10
0.21
0.08
LV (mm/mm3)
0.20
0.42
0.16
74
Chapter 4
Invoking equation (4.18) in these last two examples presumes that the fields
chosen are unbiased, though extremely limited, samples of the population of plane
probes in three dimensional space. The realization of this condition borders upon
intractable, particularly with respect to acquiring planes whose normals sample the
population of orientations on the sphere. The method of vertical sections devised
for line probes has the advantage that each vertical plane section contains all of the
line orientations from vertical to the equator. A plane probe through an opaque
specimen has a single orientation; each orientation contained in the sample requires
a different sectioning plane, and each section cuts the sample into two parts. Preparation of a series of plane probes with normals that are distributed in some sine
weighted fashion with respect to an overarching vertical direction in the specimen
may be possible if a large volume of specimen material is available (the Orientator,
shown in Chapter 7), but it requires a significant effort. A simpler sample design
that may accomplish this is shown in Chapter 6.
If the sample is transparent, so that the lineal features may be viewed as a
projected image, then a strategy similar to the method of vertical sections for lines
may be applied. Figure 4.19 shows the projected image of a thick slab of a
microstructure with internal lineal features. A probe line on the projection plane
corresponds to a probe plane, defined by that line and the projection direction, that
passes through the transparent structure. Intersection points between the probe line
and the lines in the projected image have a one-to-one correspondence to intersections of the lineal features in the structure with the associated probe plane. The area
probed is L0 · t, where L0 is the length of the line probe on the image and t is the
specimen thickness. The PA count is the number of intersections with the probe line,
divided by this area.
If the lineal structure is not isotropic, PA will be different for different directions of the probe line in the projection plane. Vertical and horizontal lines give different counts in the structure shown in Figure 4.19. The statistical test compares the
number of counts using the square root of the number as the standard deviation.
In the example, 30 vertical counts is clearly greater than 12 horizontal counts
(30 ± 30 or 30 ± 5.5 compared to 12 ± 12 or 12 ± 3.5). In order to obtain an
unbiased sample of the population of orientations of planes in space, choose a vertical direction for the specimen. Prepare slices of known thickness that contain this
vertical direction and uniformly sample the equatorial circle of longitudes; these are
called “vertical slices”. To obtain a uniform sampling of the latitude angle (angle
from the pole) it is again necessary to sample more planes with orientations near
the equator than near the pole. The orientation distribution of plane normals must
be sine weighted.
This can be accomplished by using cycloid shaped line probes on the projected image, as shown in Figure 4.20. In this application, the cycloids in the grid
must be rotated 90° from the direction used in the line probe sample design because
the normals to the cycloid curve (representing the plane normals in the structure)
must be sine weighted. Each cycloid curve, when combined with the projection direction, generates a cycloid shaped surface with normals that provide a sine weighting
of the area of the surface as a function of colatitude orientation. If PL is the line
intercept made on these cycloid line probes in the projected image, then the length
of lineal features in unit volume of the structure is given by
Classical Stereological Measures
75
Figure 4.19. A microstructure consisting of linear features viewed as a projection
through a section of thickness t = 2 mm. (For color representation see the attached CDROM.)
Probe population:
Planes in three dimensional space
This sample:
Planes represented by their edges which appear as lines on
the grid. (vertical and horizontal lines represent two sets of
planes)
Calibration:
L0 = 20 mm, A0 = L0 · T · 5 planes = 20 · 2 · 5 = 200 mm2
Event:
Linear features intersect the horizontal or vertical planes
Measurement:
Count the intersections with the horizontal and vertical grid
lines
This count:
Vertical (red arrows) = 30
Horizontal (green arrows) = 12
Relationship:
·PAÒ = (1/2)LV
Normalized Count:
PA = (30 + 12)/200 mm2 = 0.21 (counts/mm2)
Geometric Property:
LV = 2PA = 0.42 mm/mm3
LV = 2 PA =
2
PL
t
(4.20)
This result also forms the basis for estimating the total length of a single
lineal object suspended in three dimensional space, like a the whole root structure
of a plant, a wire frame object or the set of capillaries of an organ. In this application it must be possible to view the whole lineal structure from any direction.
Choose a vertical direction, then a vertical viewing plane. The plane of observation
must have an area A0 large enough to view the entire object. Visualize a box with
76
Chapter 4
Figure 4.20. A line probe on a projected image represents a surface probe in the
volume being projected (a). A cycloid-shaped line probe (b) produces a cycloidal surface
probe, which uniformly samples the latitudinal angle of the probe. Note that the cycloid
is rotated 90 degrees with respect to the orientation used in Figure 4.13. (For color representation see the attached CD-ROM.)
the area of the projection plane and depth t large enought to contain the whole
object. The volume of the box is A0 · t. Construct a grid of cycloid test lines with
major axis parallel to the vertical direction on the projection plane. The total length
LT of these probes is calibrated and known. Count intersections of the cycloid line
probes with the projected lineal features. If PL is the number of intersections per
unit length of line probes, then manipulation of equation (4.20) gives
L
2
2 P
= PL =
A0t t
t LT
A0
P
L=
LT
LV =
(4.21)
where ·PÒ is the expected value of the number of intersections that form between
the cycloid grid and the projected image of the structure. Thus, a simple count of
the number of intersections between the cycloid grid and lines in the projected
image, replicated and averaged over a number of orientations for the vertical projection plane, provides an unbiased estimate of the actual length of the feature in
three dimensions.
Summary
Normalized geometric properties, volume fraction VV, surface area density
SV and length density LV are accessible through simple manual counting measurements applied to interactions between grids that act as geometric probes and corresponding features in the three dimensional microstructure. Corresponding
measurements performed with image analysis software may automate these measurements if the features to be analyzed can be detected satisfactorily. Geometric
properties of single features can also be estimated if the observations can encompass the entire feature.
Classical Stereological Measures
77
The connecting relationships make no assumptions about the geometry of
the microstructure. It may be simple or complex, and exhibit anisotropies and gradients. Each of the relationships involves the expected value of counts of some event
that result from the interaction of probe and microstructure. The typical stereological experiment is designed to yield an unbiased estimate of the corresponding
expected value. This requires a sample design that guarantees that the population
of probes, encompassing all possible positions and, for lines and planes, all orientations on the sphere, will be uniformly sampled.
Uniform sampling of all positions requires selection of fields that accomplish that goal. Uniform sampling of orientations can be accomplished with appropriately configured test probes. In two dimensional structures, circular test lines
automatically sample all orientations uniformly. The method of vertical sections
uses cycloids to guarantee an unbiased sample of orientations of line probes in three
dimensional space. A similar procedure, with the cycloids rotated ninety degrees,
and applied to projected images, gives isotropic sampling of the population of orientations of planes in three dimensions.