Chapter 7
Procedures for IUR Sampling
Chapter 6 emphasized the requirement that sampling of a three-dimensional
structure with points, lines or planes be isotropic, uniform and random (IUR).
Accomplishing this is not easy, and much of the thrust of recent stereological development has been to explore sampling strategies that provide such results. It is useful
to summarize here just what IUR means (without recapitulating the detailed discussion in Chapter 6). Point probes do not have to be isotropic, because the points
themselves have no direction. It is sufficient to disperse a set of points through the
structure with an equal probability of sampling all regions (e.g., not points that are
concentrated near one end or near the surface), and which are randomly placed with
respect to the structure.
The idea of randomness is not always an easy one to test. If the
structure itself has no regular spacings or repetitive structure, a regular grid of
points can be used (and often is, for convenience in counting and for efficiency
of sampling). But if the structure does have some regularity, then the use of a
regular grid of points will produce a biased result. Figure 7.1 shows a cross
section of a man-made fiber composite. While not perfectly regular in their
arrangement, the fibers are clearly not randomly distributed in the microstructure.
Attempting to measure the volume fraction of fibers with a grid of points would
produce a biased result because that grid and the arrangement of the fibers would
interfere to produce a result in which points systematically hit or missed the fibers.
For an image such as this one a routine that placed points at random on the image
would be required. Typically this is done by generating two uniform random
numbers using a computer pseudo-random number generator (a mathematical
routine seeded by the system clock; we will not consider the intricacies of such routines here). The two numbers are typically provided as decimal fractions between 0
and 1, and can conveniently be scaled to the dimensions of the image to locate points
in X, Y coordinates.
Such random sampling can be used for any structure of course, even one
that has no regularities. But there is a more efficient procedure that produces equivalent results. It converges on the same answer as more and more points are sampled
and converges more rapidly than the random sampling approach (Gundersen &
Jensen, 1987). This is equivalent to saying that it has a better precision for a given
amount of effort. The method (described in earlier chapters) is called structured
random sampling, which at first seems to be an oxymoron. It can be applied to many
types of sampling with points, lines and planes, but will be explained first as it
applies to the point sampling of an image.
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Chapter 7
Figure 7.1. Cross section image of a man-made fiber composite, with an approximately
regular array of fibers.
Volume Fraction
To determine the volume fraction of a phase in a structure, assuming that
the sampling plane is representative (we will see below how to deal with that requirement), the fundamental stereological relationship is VV = PP, or the volume fraction
equals the point fraction. The precision of a counting measurement is a standard
deviation equal to the square root of the number of events counted. For instance,
if asked to estimate the number of cars per hour traveling along a highway we could
count the cars and measure the time required for 100 cars to pass. The number of
cars divided by the time T (expressed as a fraction of one hour) would give the
desired answer. The standard deviation of the number counted is 10, so the relative
standard precision would be 10/100 = 10%. By counting 1000 cars we could reduce
——
the standard deviation to ÷1000 = 31.6, or a relative standard precision of 3.16%.
Counting 10,000 cars would give a relative standard precision of 1%.
Usually in a given experiment we know what precision is required or desired.
Consider for the moment a desired precision of 10%. This requires counting 100
events as described above. For point counting, this means that there are 100 points
in the grid or random distribution that hit the phase of interest. If the phase represents (e.g.) 25% of the structure, this would require placing 400 points onto the
image to achieve 100 hits. However, there is one additional assumption hidden in
this procedure: each of the points is required to be an independent sample of the
structure. This means that the points should be separated from each other by enough
distance that they do not repeat the same measurement, which is usually taken to
mean that they do not lie closer together than a characteristic distance in the structure such as the mean intercept length.
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Figure 7.2. An example of 200 random points. Note the presence of clusters of points
that oversample some regions while large gaps exist elsewhere.
However, random points generated by the procedure outlined above will not
all conform to that assumption. As shown in Figure 7.2, random distributions of
points on an image inevitably produce some points that are quite close together.
These are not all independent samples, and so the actual precision of the measurement is not as good would be predicted based on the number of points. The clustering of points produced by a random (Poisson) generation process is used in
another context to study the degree to which features in images have tendencies
toward clustering or self-avoidance. The standard deviation of the PP value obtained
using random points will be greater (worse) than with a regular grid by a factor of
e (2.718). Notice that this does not mean that the result obtained with such a random
point array is biased or in error, just that it takes more work (more points have to
be sampled) to obtain an equivalent measurement precision.
The use of regular grids of points to count PP is widely used, of course. Grids
of points are available as reticles to fit within microscope eyepieces, or applied to
photographic prints as transparent overlays, or superimposed on images using computer displays. This is not so much a consequence of the greater measurement efficiency as a recognition of the ease of counting such points manually by having the
eye track along a regular grid.
The proper way to use such a grid while still obeying the requirements of
random sampling according to the structured random sampling method is as
follows:
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1. Determine, based on the desired level of precision and the estimated volume
fraction of the target phase, the number of points to be sampled in the image.
Note that in many real cases not all of the points will be sampled in a single
image—there will be many fields and many section planes in order to meet
the requirements of uniform sampling of the entire structure. There is also
the restriction against using too many points on one image based on the
requirement that the points must be separated far enough to be independent
samples, meaning as noted above that multiple points do not fall into the
same region of the structure.
2. The resulting number of points form a regular grid. Typically these grids are
square so that the number of points is a perfect square (25, 49, 100, etc.)
with fairly small numbers used for purely manual counting because larger
numbers foster human counting errors. In the example shown in Figure 7.3,
the grid is 5 ¥ 5 = 25 points.
3. Generate a pair of random numbers that define the X, Y coordinates for the
placement of the first point of the grid within a square on the image. This
will automatically place all of the other grid points at the same relative offset.
4. Perform the counting of PP.
5. For other fields of view, generate new random numbers for the offset and
repeat steps 3 and 4, until enough fields of view, section planes, etc. have
Figure 7.3. A 5 ¥ 5 grid placed on an image area. The upper left corner is displaced
randomly within the red square to accomplish structured random sampling of the area.
(For color representation see the attached CD-ROM.)
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131
been examined to produce uniform sampling of the structure and enough
points have been counted to produce the desired precision.
This procedure is a straightforward application of structured random sampling. Several examples will be detailed below that extend the same method to other
situations and types of probes. In all cases note that it is important to estimate
beforehand the number of images (sections planes and fields of view) that must be
examined in order to obtain the desired precision. This requires an estimate of the
quantity being measured, in this example the volume fraction. Typically this parameter is known at least approximately from previous sampling of the structure, prior
results on other similar specimens, or independent knowledge about the objects of
interest.
Sampling Planes
The same procedure as detailed above can be applied to the selection of the
sampling planes themselves. For example, consider the problem of determining the
volume fraction of a phase within an irregular 3D object as shown in Figure 7.4.
The measurement in this case must measure the volume of the object as a whole as
well as that of the phase in order to obtain the volume fraction. If the phase is
expected to be about 20% of the volume, and a measurement precision of 5% is
desired, this means that about 400 points should hit the phase and hence 2000 points
should hit the object. If the object is 5 inches long and about 3 inches in diameter,
a rough estimate of its volume can be made by assuming a more-or-less ellipsoidal
shape. The volume of an ellipsoid is about half (p/6 = 0.524) of the volume of the
bounding box. This means that grid should have about 4000 points. A cubical grid
spacing of 0.2 inches will produce a total of 15 ¥ 15 ¥ 25 = 5625 points, quite suitable for this measurement.
This sampling grid can be achieved by sectioning the specimen every 0.2
inches and placing a grid with that spacing onto each section for counting. The
placement of the grid uses the method described above. A similar method is used
to position the planes: A single random number can be used to determine the position of the first plane within the first 0.2 inches of the specimen, and then all of the
other planes are regularly spaced with respect to this first one. In this way structured random sampling is extended to all three dimensions.
This represents a fairly large number of points and a high sampling density;
most stereological experiments use much lower sampling density and achieve lower
precision on each specimen, but average over many specimens. In the example of
Figure 7.4, a lower sampling density has been used for clarity. The planes are spaced
1/2 inch apart and the grids have a 1/2 inch spacing, producing a total of 360 points,
of which about 180 can be expected to fall within the object and hence about 36
within the phase of interest. This corresponds to a standard deviation due to counting statistics of ±6, or a relative precision of 6/36 = 17%.
In many cases the sections of the specimen are obtained with a microtome,
in which case the procedure is to take every Nth section from the series to obtain
the desired spacing, starting with a randomized initial section. Likewise, if the
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a
b
Figure 7.4. A representative irregular object (potato) to be sampled as discussed in the
text, and one section to be used for counting with a portion of the grid superimposed.
(For color representation see the attached CD-ROM.)
examination procedure is to use M fields of view on each section, the procedure is
to create a grid of field positions which is offset by a random location on each
section. The procedure can also be extended to sampling specimens within a population in the same way.
Isotropic Planes
Unlike point probes, lines and planes have a direction and so it is also necessary to make their orientation isotropic within the structure. This is much more
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133
difficult to accomplish (and may be very important, as few natural structures are
themselves isotropic). Line probes are used to sample surfaces and volumes, either
by counting the intersections of the line probes with surfaces or measuring the intercept length with the volumes. Because counting is a much easier and hence preferred
technique, and the precision of counting experiments can be predicted as discussed
above, most modern stereological experiments emphasize counting of intercepts and
so we discuss here the use of line probes for that purpose.
Lines cannot in general be passed through structures without first cutting
planes on which to draw the lines. If all of the planes were cut as in the preceding
experiment as a series of parallel sections, it would not be possible to draw lines in
any direction except those included in the planes. Hence the most commonly used
and efficient method of cutting parallel plane sections through a specimen cannot
be used when line probes are to be employed for surface area measurement.
It is instructive to consider first the procedures for placing an IUR set of
planes through an object. This is required when the measurement probe is a plane,
for instance when measuring the size and shape of plane intersections with features
or counting the intersections of the plane with features in the volume. Plane probes
are primarily used for counting the number of intersections per unit area with linear
structures such as fibers, which are rarely isotropic in space, in order to measure the
total length of the linear structures. IUR planes can also be used for drawing line
probes through structures, although we will see later that there is an easier way to
accomplish this when the purpose is simply to count intersections of the lines with
surfaces.
In order for sampling to be uniform, it will still be necessary to obtain
a series of sections that cover the extent of the feature. There is a tendency in
visualizing the orientation of section planes to deal with those that pass through
the geometric center of the object being sectioned, which of course does not provide uniform sampling of the extremities. Since many real structures are not
only anisotropic but also have gradients, the need for uniform sampling is very
important.
Using a series of section planes (distributed as discussed above using the
principles of structured random sampling) naturally limits the sampled planes to a
single orientation, because once cut they cannot be reassembled to permit sampling
again in another orientation. This usually means that multiple specimens must be
used to obtain many different sampling orientations, which in turn requires that
specimens within a population must have some definable or discernible axis of orientation of their own that can be used as a basis for selecting orientations. In biological systems, which is the primary field of application of these techniques, this is
usually straightforward. For materials or geological specimens there are also natural
axes (typically perpendicular to deposited planes or parallel to a cooling gradient
or deformation direction) but these may be more difficult to ascertain before
sectioning.
In any case, assuming that there is some natural axis, one approach is to
define a different axis orientation for sampling each member of the population using
a scheme that distributes the orientations randomly and uniformly in three dimensions to achieve isotropic sampling. This is not so easy to do for an arbitrary number
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of orientations. We have already seen the efficiency of the structured random sampling approach, so a method based on random tumbling of the specimen is not
desired. Visualization of the problem may be assisted by realizing that each direction can be mapped as a point onto a half-sphere of orientations. Is there an
arrangement of such points so that they are regularly and equally spaced over the
surface of the sphere?
The arrangements which are exactly equal and regular correspond to the
surface normals of the regular polyhedra, of which there are only five (Figure 7.5):
the tetrahedron with four triangular faces, cube with size square faces, octahedron
with eight triangular faces, dodecahedron with twelve pentagonal faces, and icosahedron with twenty triangular faces. Taking just the faces whose normal directions
point into the upper half of the sphere of orientations, the number of orientations
is half of the number of faces (except for the tetrahedron). For instance, the dodecahedron has 12 faces but they provide only six sampling orientations. If one of these
solids corresponds to the desired number of orientations to be used for sampling,
a useful strategy would be to embed each entire object to be sectioned into such a
structure, the number of such objects corresponding to the number of faces (e.g.,
six organs from a population placed into six dodecahedra). This would place the
natural axis at some random orientation, which would however fall within the area
of one of the faces. Identify this face, and select a unique and different face for each
of the polyhedra. Figure 7.6 shows the normal directions for the dodecahedron. Use
the surface normal of the selected face as a sectioning axis for that object, using the
usual procedure as presented above to determine the number of sections to be used
and the location of fields of view within each section.
This procedure is fairly easy to implement but is appropriate only when
there are multiple objects representing a population which is to be measured.
When a single object must be sampled isotropically with planes, an equivalent procedure is to first subdivide the object into portions, for example cutting it into six
(for the dodecahedron) or ten (for the icosahedron) parts. Then each part can be
embedded into the regular polyhedron and sectioned as described above. The danger
in this approach is that orientations of structure within the object may vary
from place to place, and since the sampling orientation also varies this can result in
measurement bias. Correcting such bias requires performing the procedure on
several specimens with different orientations selected for each portion. In this case,
it is just as efficient to use the first approach with sectioning of each specimen in
one orientation.
If it is not required that the specimen be uniformly sampled at each orientation, then another sampling method can be employed to obtain section planes that
Figure 7.5. The five platonic solids (regular polyhedra).
Procedures for IUR Sampling
135
Figure 7.6. Orientation vectors for the dodecahedron, plotted on the sphere of
orientation. (For color representation see the attached CD-ROM.)
are isotropic. First cut a so-called “vertical” section that includes the specimen’s
natural axis (or some identifiable direction) and is uniformly random with respect
to orientation, as shown in Figure 7.7. Then cut a surface perpendicular to this
surface but at an angle that is not uniformly random but is instead sine-weighted.
Sine weighting is accomplished by generating a uniform random number in the
range -1 . . . 1 and then calculating the angle whose sine has that value = Arc Sin
(Random). This angle varies from -90 to +90 degrees and represents the angle from
Figure 7.7. Vertical sections cut to include the vertical axis of a specimen. Each section
includes the vertical direction (arrow) and is rotated randomly to a different azimuthal
angle. (For color representation see the attached CD-ROM.)
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Chapter 7
Figure 7.8. The “Orientator”. Place a horizontal cut surface (perpendicular to the
assigned vertical direction in a specimen) along the 0–0 line with a vertical cut (parallel to the vertical direction and uniformly random with respect to rotation) face down on
the diagram. Generate a random number from 0–99 to select a cutting direction as
shown. Repeating this procedure produces planes that are isotropic (uniformly sample
orientations). (For color representation see the attached CD-ROM.)
vertical. It is also easy to implement this procedure by creating a grid of radial lines
that are not uniformly spaced but instead have the appropriate sine weighting, and
then to select one using a uniform random number from 0 to 99 (Figure 7.8). This
procedure is called the Orientator (Mattfeldt, quoted in Gundersen et al., 1988).
Figure 7.9 shows the process applied to the object from Figure 7.4. It produces
section planes that are isotropically uniform. Of course, this produces only a single
section that passes through the object, and so does not provide spatially uniform
sampling.
Isotropic Line Probes
Fortunately, there are few stereological procedures that require IUR planes
as the sampling probes. IUR lines are much easier to generate. If the generated
planes in the specimen were themselves IUR then drawing lines on them that were
also IUR would produce line probes having an IUR distribution in 3D space. Such
a grid of lines would be produced for example by drawing a set of parallel lines with
an appropriate spacing, shifting them by a random fraction of that spacing, and
rotating them to an angle given by a uniform random number between 0 and 180
Procedures for IUR Sampling
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a
b
Figure 7.9. Implementation of the orientator: a) cutting a vertical section; b) cutting the
examination plane using the sine-weighted grid. (For color representation see the
attached CD-ROM.)
degrees. The structured random sampling approach lends itself directly to the positioning of the lines in this way. Note that a grid of radial lines with uniform angular
spacing drawn from the center of the field of view or of the specimen does not
satisfy the IUR requirement because it is not uniform across the area—more of the
lines sample positions near the center than at the periphery.
Given the difficulty of drawing the IUR planes in the first place, another
easier approach is generally used. It relies on the idea of a vertical section (Baddeley et al., 1986), the same as mentioned above. This is a plane that includes some
natural axis of the specimen or some readily identifiable direction within it. It does
not matter whether this orientation is actually “vertical” in a geocentric sense, and
the name comes from the fact that the placement of images of the vertical section
plane for viewing and measurement often positions this direction in the vertical
orientation.
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Figure 7.10. A series of planes cut parallel to one vertical section. (For color representation see the attached CD-ROM.)
A vertical section plane can be cut through a specimen parallel to the vertical direction without necessarily passing through the center of the specimen. In fact,
while most diagrams and illustrations of vertical sectioning tend to draw the plane
as passing through the center, for IUR sampling it is of course important that this
restriction not be present. A series of parallel vertical section planes (Figure 7.10)
with one uniformly random rotation angle about the vertical axis can be cut using
the same principles of uniform random sampling discussed above by calculating an
appropriate plane spacing and a random offset of a fraction of that spacing. The
uniform random sampling must then also be performed by cutting a similar set of
section planes on additional specimens (or portions of the same specimen) at angles
offset from the first chosen angle.
On all of the vertical section planes cut at different rotational angles, directions near the vertical direction are sampled. Plotted onto a sphere of directions (as
shown in Figures 7.11 and 7.12), it can be seen that these lines cluster near the north
pole of the figure while directions at low latitudes near the equator are sparsely
sampled. The compensation for this is to use the same sine-weighting as discussed
above. By generating lines with an angle from the vertical calculated as Arc Sine (R)
where R is a uniform random number between -1 and +1, the directions are spread
uniformly over the latitudes on the sphere (Figure 7.12).
This approach fits well with structured random sampling because the number
of angles in the horizontal (uniform) direction, the vertical (sine-weighted)
Procedures for IUR Sampling
139
Figure 7.11. Radial lines drawn on a vertical section; top) uniformly distributed
every 15 degrees (these lines are not isotropic in 3D space); bottom) the same
number of lines drawn with sine-weighting, which does produce lines isotropic in 3D
space.
direction, the number of lines and their spacing can all be calculated based on
the required precision. However, it requires more horizontal orientations and
hence more vertical section cuts than a modified procedure that uses cycloids
rather than straight lines. This obviously does not apply to experiments in which
intercept lengths are to be measured, but is quite appropriate for the counting
of intersections between the line probes (the cycloids) and surfaces within the
structure.
Cycloids are the correct mathematical curve for this application because they
have exactly the same sine weighting as used to generate the straight lines in the
method above. A cycloid is the path followed by a point on the rim of a rolling
circle, and can be generated using the mathematical procedure shown in Figure 7.13.
The curve has only a small fraction of line with vertical orientation and considerably greater extent that is near horizontal, and exactly compensates for the vertical
bias in the vertical section planes. Distributing a series of cycloidal arcs (in all four
possible orientations obtained by reflection) across the vertical section and shifting
the grid of these lines according to the usual structured random sampling guidelines produces isotropic uniform random sampling in three dimensions. Note that
the grid of cycloids (Figure 7.14) is not square, but wider than it is high; the shifting using random offsets must displace the grid by a random fraction of the grid
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a
b
Figure 7.12. Direction vectors in space mapped as points on a half-sphere of orientations: a) uniformly distributed angles at every 15 degrees along each vertical section
corresponding to Figure 7.11a; the vectors are close together near the north pole. b)
sine-weighted angles along the same vertical section corresponding to Figure 7.11b;
the same number of directions are now uniformly distributed. (For color representation
see the attached CD-ROM.)
size in each direction. As usual, the size of the grid must be such that in few cases
will more than one line intersect the same element of structure. In many examples
of this technique the statement is made that as few as three sets of planes at angles
of 60 degrees are sufficient for many purposes. Obviously, this premise can be tested
in any particular experiment by sampling at higher density and seeing whether the
results are affected.
For projected images through transparent volumes, it is possible to generate
IUR surfaces using cycloids. As described in detail in Chapter 14 on finite section
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141
Figure 7.13. A cycloidal arc and the procedure for drawing it. The are has a width of
p, a height of 2 and a length of 4 (twice its height).
thickness, a set of cycloidal lines drawn on the projected image represent surfaces
through the volume. Choosing a vertical direction (which can be arbitrary) for the
volume and rotating it about that axis while collecting projected images causes the
same bias in favor of directions near the vertical axis as that produced by vertical
sectioning. Elements of area of these surfaces have orientations that compensate for
the bias so that isotropic sampling of the structure is obtained. As shown in the
Figure 7.14. A grid of cycloidal arcs. Placed on vertical sections (the vertical direction
in the specimen is vertical on the figure), the lines isotropically sample directions in 3D
space.
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examples in that chapter, the result is an ability to obtain isotropic uniform random
sampling of intersections of linear features to measure NA, from which LV can be
calculated.
Volume Probes—The Disector
One of the first developments in the so-called “new” stereology that emphasizes design of probes and sampling strategies rather than “classical” methods
such as unfolding of size distributions based on shape is the Disector (Sterio,
1984). The disector requires comparing two parallel section planes to detect features
that are present in one but not the other. But while it is implemented using section
planes, it is actually a volume probe (the volume between the planes). Since volumes
have no orientation isotropy is not an issue, although requirements for uniform
random sampling remain (and can be satisfied using the same methods described
above).
The initial and still primary use of the disector is to determine the number
of objects per unit volume. Point, line and plane probes cannot accomplish this
without bias because they are more likely to intersect large features than small ones.
As noted in the first chapter, the number of objects present in a region is a topological property, and cannot be determined by probes of lower dimension than the
volume. The disector provides a surprisingly simple and direct way to count objects
that is unbiased by feature size and shape. It relies on the fact that features can be
counted by finding some unique characteristic that occurs only (and always) once
per feature. In this case that characteristic is taken to be the topmost point (in some
arbitrary direction considered to be “up”) of each feature.
For illustration, consider counting people in a room. For most of them the
topmost point would be the top of the head, but in some cases it might be the tip
of an raised hand, or the nose (someone lying on his back on the floor). Regardless
of what the point is, there is only one. Counting those points gives the number of
people. Of course, in this example the procedure must be able to look throughout
the volume of the room. When three-dimensional imaging is used, as discussed in
Chapter 15, this is the method actually used. The disector provides an efficient
method using just sets of two parallel planes.
The disector can be implemented either with physical sectioning (e.g.,
microtoming) to produce two thin sections, or by sequential viewing of two
polished surfaces of an opaque material, or by optical sectioning of a transparent
volume using confocal microscopy. In all cases the spacing between the two sections
must be known, which can be difficult particularly in the case of microtomed
sections. For polished opaque materials one method for accomplishing this is to
place hardness indentations (which have a known shape, typically either spherical,
conical or pyramidal) in the surface and measure the change in size with polishing,
from which the depth of polishing can be determined. These hardness indentations
also help to align the two images. Similar fiducial marks can be used for the
microtomed sections. Optical sectioning is generally the easiest method (when
applicable) because spacing can be measured directly and image alignment is
not required.
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143
The key to the disector is to have the two parallel images be close enough
together (small spacing between them) that the structure between them is relatively
simple. No entire feature of interest can be small enough to hide in that volume,
and as we will see no branching of networks can have branch lengths smaller than
that distance. It must be possible to infer what happens in the volume by comparing the two images. Only a small number of basic topological events can be allowed
to occur, which can be detected by comparing the images:
1. A feature may continue from one plane to the other with no topological
events (the size of the intersection can change, but this is not a topological
change).
2. A feature may end or begin between the planes, appearing in one but not the
other.
3. A feature may branch so that it appears as a single intersection in one plane
and as two (or more) in the other.
4. Voids and indentations within a feature may also continue, begin or end, or
branch.
Figure 7.15 shows a diagram illustrating several of these possibilities. The
critical assumption is made that familiarity with the structures will enable identification by a human to detect features in the two sections that are matched. For automatic analysis the planes must be spaced closely enough together that feature
overlaps can be used to identify feature matches. Features of type 6 (in the figure)
are considered to be continuations of the same object with no topological events
occurring. Events of type 3 and 4 represent the start or end of a feature (depending on which of the two planes is taken to be the floor and which the ceiling). Events
Figure 7.15. Diagram of the disector (cases illustrated are discussed in the text). (For
color representation see the attached CD-ROM.)
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Chapter 7
of type 1 and 2 represent simple branching. The type 5 event reveals the bottom of
an internal void.
If the spacing between the planes is small so that no complex or uninterpretable events can occur in the volume between them, then in real images most
objects will continue through the two planes (type 6) and these non-events are
ignored. For counting features, events of type 3 and 4 are of interest. The number
of these events divided by two since we are now counting both beginnings and
endings of features, and divided by the volume between the planes (the area of the
images times their spacing) gives the number of features per unit volume directly.
As noted above, this value is unbiased since feature size or shape does not affect the
count (Mayhew & Gundersen, 1996).
The method is only simple for convex objects. When features may branch,
or are long and slender so that they may cross and re-cross through the sampled
volume, it becomes necessary to keep track of all of the parts of the feature so that
it is only counted once. Since the images are finite in area, attention must be given
to the problems that the edges of the image introduce. As shown in Figure 7.16, this
is accomplished by defining “exclusion edges” and a guard region around the active
counting area so that features are ignored if they extend across the exclusion edge
(Gundersen et al., 1988). Of course, as noted above it is necessary to follow features
that branch or extend laterally to detect any crossing of the exclusion edges (this is
why the exclusion edges are extended as shown by the arrows in the figure). The
requirement for a small spacing between planes to eliminate confusion about connectivity means that only a few topological events are detected, so that a large area
or many fields of view are required to obtain enough counts for useful statistical
precision.
Figure 7.16. Guard frame and exclusion lines for the disector. Only the red features
are considered. (For color representation see the attached CD-ROM.)
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145
Networks
As noted above, the disector can be used to count the number per unit
volume (NV) when the objects are convex, or at least relatively compact and well
separated. Objects that are long, twisted, branched and intertwined create difficulties in identifying the multiple sections that occur as part of the same object. On
the other hand, the disector is also very useful for dealing with an extended network
such as the pore structure in a ceramic, blood vessels or neurons in the body, etc.
The topological property of interest for these structures is the connectivity or genus
of the network. This is a measure of how branched the structure is, or more specifically of the number of connections (per unit volume) or paths that are present
between locations.
Topological properties such as number of objects and connectivity of pore
networks require volume probes. The use of full three-dimensional imaging
(Chapter 15) offers one approach to this, but the disector offers another that is
elegant and efficient. It will be useful first to digress slightly and revisit some aspects
of topology and genus, and define some terms and quantities, which were introduced more comprehensively in earlier chapters.
Points on the surfaces of objects in three-dimensional space have normal
vectors (perpendicular to the local tangent plane) that identify their orientation.
Each such vector can be represented by a point on the surface of a sphere, which is
called the “spherical image” of the point. A patch or region on the curved surface
of the object corresponds to a patch of orientations on the sphere. For the surface
of a convex, closed object every direction on the sphere corresponds to one (and
only one) point somewhere on the surface of the object. Since the spherical image
of the convex surface exactly covers (or is mapped onto) the unit sphere, its spherical image value is 4p (the area of the unit sphere), independent of any details about
the shape of the body. The spherical image is an important topological property
since it has a value that is independent of size and shape.
For bodies that are not totally convex we must recall the idea of negative
spherical image introduced in Chapter 5. This is the projection of points on saddle
surfaces (Figure 7.17), which have two principle radii of curvature with different
signs. With this convention, the total spherical image of any simply-connected
closed surface is 4p (and the total spherical image of N objects would be 4Np). If
the body has a hole through it (for instance a donut or torus, Figure 7.18), the
surface around the hole is saddle surface and covers the sphere exactly once, and
the surface on the outside is convex and also covers the sphere once, so the new
spherical image of a donut is 0.
The hole in the donut changes the topological quantity called the genus of
the object. It becomes possible to make one cut through the object without disconnecting it into separate parts. Every hole introduced into the object adds a
spherical image of -4p, so the general result is that the spherical image of an
object is related to the genus or connectivity by the relationship Spherical Image =
4p · (1 - C). The total spherical image of a set of N objects with a total connectivity of C would be 4p · (N - C). We are usually primarily interested in the two extreme
cases when C = 0 (a set of separate, simply connected objects which need not be
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a
b
c
Figure 7.17. Convex, concave and saddle curvatures of surface elements. (For color
representation see the attached CD-ROM.)
Procedures for IUR Sampling
147
Figure 7.18. A torus; the green shaded area is saddle surface with a negative spherical image, the red surface is convex with a positive spherical image. (For color representation see the attached CD-ROM.)
convex but have no holes) or N = 1 (a single extended network whose genus we wish
to determine).
Imagine a plane sweeping through the volume of the structure and note the
occurrences when the plane is momentarily tangent to the surface. There can be
three different kinds of tangent events as illustrated in Chapter 5 (DeHoff, 1987;
Zhao & MacDonald, 1993; Roberts et al., 1997):
1. The plane may be tangent to a locally convex surface (both radii of curvature point inside the object). This is called a T++ event and the total number
of them per unit volume is denoted TV++.
2. The plane may be tangent to a locally concave surface (both radii of curvature point outside the object). This is called a T -- event and the total number
of them per unit volume is denoted TV--.
3. The plane may be tangent to a local patch of saddle surface (the two radii
of curvature lie on opposite sides of the plane). This is called a T +- event
and the total number of them per unit volume is denoted TV+-.
The disector can be used to count these tangent events. The appearance of
features in the two planes allows us to infer that a tangent event of one type or
another occurred in the volume between the planes. Referring back to the diagram
in Figure 7.15, types 1 and 2 correspond to T +- events, types 3 and 4 to T++ events,
and type 5 to a T -- event. The total number of counts per unit volume (the product
of the area of the image and the spacing between the planes) can be obtained by
counting.
The sum of these tangent counts (TV++ + TV-- - TV+-) is called the net tangent
count TV, and the total spherical image of the structure is just 2p · TV. Consequently
the difference between the number of features present N and the connectivity or
number of holes C is just N - C = TV/2. For a network structure with N = 1, this
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gives the connectivity of the structure directly. Because topological properties and
the volume probe used cannot be anisotropic, orientation considerations do not
arise in using the disector (but of course averaging of samples to obtain uniform
representation of the structure is still necessary).