Russ J.C. Image Analysis of Food Microstructure

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Similarly, the rotation angles around the vertical direction can be systematically

randomized. If three orientations are to be used, 360/3 = 120 degrees. So a random

number from 1 to 120 is generated and used for the initial angle of rotation, and

then 120 degree steps are used to orient the blocks that will be sectioned for the

other two directions. For the four sections to be examined from each block, it is

necessary to know how many total sections could be cut from each. For the purposes

of illustration, assume that this is 60. Sixty divided by four equals 15, so a random

number from 1 to 15 selects the ﬁrst section to be examined, and then the ﬁfteenth,

thirtieth and forty-ﬁfth ones past that are chosen.

For the F = 6 ﬁelds of view, the total area of the cut section or slide is divided

into 6 rectangles. Two random numbers are needed to specify coordinates of a ﬁeld

of view in the ﬁrst rectangle. Then the same locations are used in the remaining

rectangles for the additional ﬁelds of view. Figure 1.23 illustrates the method of

FIGURE 1.22 The principle of systematic random selection. Consider the task of selecting

5 apples from a population of 30. Dividing 5 into 30 gives 6, so generate a random number

from 1 to 6 to select the ﬁrst specimen. Then step through the population taking every sixth

individual. The result is random (every specimen has an equal probability of being selected)

and uniform (there are no bunches or gaps in the selection).

FIGURE 1.23 Systematic random location of ﬁelds of view on a slide. The total area of the

slide is subdivided into identical regions, six in this example, corresponding to the number

of ﬁelds of view to be imaged. Two random numbers are generated for the X and Y coordinates

of the ﬁrst ﬁeld of view in the ﬁrst region. The remaining ﬁelds of view come from the

identical positions within the other regions. Different random placement is used for each slide.

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systematic random location of ﬁelds of view on a slide. This method can be gener-

alized to handle any other sampling requirement, as well, such as the placement of

grid lines or points, and so forth.

TOPOLOGICAL PROPERTIES

The volume of 3D structures, area of 2D surfaces, and length of 1D lines, can

all be measured from section images. The most efﬁcient methods use counting

procedures with a point grid for volumes or a line grid for surfaces. The results of

these procedures, with a little trivial arithmetic, give the metric properties of the

structure. Of course, there may be many components of a typical food structure,

including multiple phases whose volumes can be determined, many different types

of surfaces (this includes potentially interfaces between each of the phases in the

material, but usually most of these combinations are not actually present), and all

sorts of linear structures. But with appropriate image processing to isolate each class

of structure, they can all be measured with the procedures described.

Consider the case in Figure 1.24. From the various sections through the structure

the total volume, the surface area, and the length of the tubular structure can be

determined. But the fact that the tube is a single object, not many separate pieces,

that it is tied into a knot, and that the knot is a right-handed overhand knot, is not

evident from the individual section images. It is only by combining the section

images, knowing their order and spacing, and reconstructing the 3D appearance of

the structure that these topological properties appear.

Topological properties of structures are often as important as the metric ones,

but require a volume probe rather than a plane, line or point probe as used for the

metric properties. The simplest and most familiar topological property is simply a

number, such as the number of points or features counted in a volume as described

above. The thick slice viewed in transmission is one kind of volume probe. Another

is a full three-dimensional imaging method, such as serial section reconstruction, or

magnetic resonance or CT imaging. In some cases, such as knowing that the knot

is right handed, full 3D imaging is necessary, but fortunately there is another easier

procedure that can usually provide basic 3D topological information.

The simplest volume probe consists of two parallel section images, a known

distance apart. These are compared to detect events that lie between them, which

means that they must be close enough together that nothing can happen that is not

interpretable from the comparison. In general, that means the plane separation should

be no more than one fourth to one ﬁfth of the size of the features that are of interest

in the structure.

The simplest kind of disector measurement (Figure 1.25) works to determine

the number per unit volume of convex, but arbitrarily shaped features. Each such

feature must have one, and only one, lowest point. Counting these points counts the

features, and they can be detected by observing all features that intersect one of the

two sections and thus appear in one image, but do not intersect the second section

and, hence, are not seen in that image. N

(number per unit volume) is measured

by the number of these occurrences divided by the volume examined, which is the

product of the image area times the distance between the sections.

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(a)

(b)

FIGURE 1.24 Twelve slices through a structure from which the metric properties can be

determined, and a three-dimensional reconstruction from the sections which reveals the

topological properties.

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The efﬁciency and precision of this procedure can be improved by counting tops

as well as bottoms, and dividing by two. So the features that appear in either section

image but are NOT matched in the other image are counted, and

(1.9)

where E is the number of ends.

We will see in Chapter 5 (Figure 5.7) that there are image processing techniques

that make it fairly quick to count the features that are not matched between the two

images. This is important because if the spacing between the planes is small (as it

must be), most of the features that intersect one plane will also intersect the second,

and it becomes necessary to examine a fairly large area of images (which must be

matched up and aligned) in order to obtain a statistically useful number of counts.

Because it is a volume probe and counts points, neither of which have any

directionality associated with them, the disector does not require isotropy in its

placement. Only the simpler requirements of uniformity and randomness are needed,

and so sectioning can be performed in any convenient orientation.

With the disector it is possible to overcome the limitations discussed previously

of assuming a known size or shape for particles in order to determine their number

or mean size. For example, the pores or cells in bread are clearly not spherical and

certainly not the same size or shape. The disector can count them to determine

number density N

. Figure 1.26 shows one simple way to do this: to cut a very thin

slice (e.g., about 1 mm thick) and count the number of pores that do NOT extend

all the way through. This can be facilitated by a little sample preparation. Applying

a colored ink to the top surface of the slice with an ink roller makes it easy to see

the pores, and placing the slice on a different color surface makes it easy to see

those that extend clear through.

FIGURE 1.25 Diagram of disector logic. Features that appear in either section plane that are

absent from the other represent ends (marked E). Features that are matched are not counted.

Area Distance

⋅⋅2

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(a)

(b)

FIGURE 1.26 (See color insert following page 150.) Application of the disector to counting

the number of pores in a baked product: (a) photograph of one cut surface; (b) result of inking

the cut surface and placing a thin slice on a contrasting color background. (Courtesy of David

Pechak, Kraft Foods Technology Center)

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Since this disector counts only bottoms of pores, and not their tops, the number

per unit volume is just the number of pores divided by the volume of the slice (area

times thickness). But there is more information available. The volume fraction of

the pores is readily measured by the area fraction of the top surface, for instance by

counting the number of inked pixels and dividing by the total area of the slice, or

if desired by applying a point grid and dividing the number that fall on the inked

regions to those that fall anywhere on the slice. Knowing the volume fraction of

pores and the number per unit volume allows a simple calculation of the mean

volume of a pore, without any assumptions about shape.

The disector method can be extended quite straightforwardly to deal with fea-

tures that may not be convex, and even to characterize the topological properties of

networks. The key property of a network is its connectivity, or the genus of the

network. This is the number of redundant connections that exist, paths that could

be cut without separating the network into two or more pieces. Figure 1.27 shows

one of the many types of high-connectivity networks that can exist.

The network in Figure 1.27 formed along the triple lines where bubbles met,

producing a very open and regular structure that can be visually comprehended from

SEM images, and can be characterized in several ways including the size of the original

bubbles and the lengths of the connections in the network. When the network structure

is more irregular and dense, it is more difﬁcult to visualize it using the SEM. Figure

1.28 shows an example. Even serial section reconstruction and visualization of this

FIGURE 1.27 Example of a network that formed along the triple lines between bubbles in

a foam, showing a high connectivity.

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type of structure does not convey a view that is very instructive and certainly is not

suitable for quantitative comparisons. The disector approach using two parallel

section images a known distance apart to count topological events is perhaps the

only way to gain a measurement that can be meaningfully used to correlate structural

changes with processing or performance variables. Of course, the section images can

also be used to determine the volume fraction of the solid material.

Counting the ends that occur between the two section images provides a way

to count the number of convex features per unit volume, regardless of shape. To

count features that are branched and complex in shape, or to learn about the con-

nectivity of networks, two additional types of events must also be identiﬁed and

counted. The end points as described above are now referred to as T

points,

meaning that they are convex tangent counts. If an end occurs between the planes,

then there must be a point where a tangent plane parallel to the section planes just

touches the end of the feature, and at this point the curvature of the feature must be

convex (the radii of curvature are inside the feature).

As shown in Figure 1.29, there are two other possibilities. A negative or T

– –

tangent count occurs when a hole within a feature ends between the planes. In

appearance this corresponds to a hollow circle in one plane with a matching ﬁlled-

in circle in the other. The tangent plane at the end of this hollow will locate a point

where both radii of curvature lie outside the body of the feature (and in the hollow).

These events are typically rather rare.

The other type of event is a branching. If a single intersection in one plane splits

to become two in the second plane it implies that between the two section planes

FIGURE 1.28 SEM image of the network structure in highly aerated taffy. (Courtesy of Greg

Ziegler, Penn State University Food Science Department)

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there must be a point where the tangent plane would ﬁnd saddle curvature — a point

where the two principal radii of curvature lie on opposite sides of the surface. Hence,

this is called a “mixed tangent” or “saddle point” and denoted by T

+–

Counting these events allows the calculation of the net tangent count per unit

volume or T

, where as before the volume is the product of the area examined and

the separation distance between the planes. The net tangent count is just (T

) + (T

– –

)

– (T

+–

), and the Euler characteristic is one half of the net tangent count. But the

Euler characteristic is also N

– C

, the difference between the number of discrete

(but arbitrarily shaped) features per unit volume and the connectivity per unit volume.

(1.10)

Many real samples of interest consist of discrete features that may be quite

irregular in shape with complex branching and protrusions. For a set of dispersed

features that are irregular in shape, the connectivity C

is zero. Consequently the

number of features per unit volume is calculated from the net tangent count. Sub-

tracting the mixed or saddle counts corrects for the branching in the feature shapes

as indicated schematically in Figure 1.30.

Another type of structure that is often encountered is an extended network that

may have many dead ends but is still continuous. In this case, N

= 1. In the previous

example of discrete features, the multiple branches and ends for a single feature

may add to the T

count, but the T

+–

count must increase along with it, as shown

in Figure 1.30. The net tangent count will still be two and the number of features

one. But in the case of a network, the T

+–

count will dominate, even though local

ends of network branches may produce some T

events. In Equation 1.10, T

+–

and

FIGURE 1.29 Diagram of extended disector logic. Features with ends between the section

planes are convex and correspond to positive tangent counts (T

). Pores that end are concave

and are labeled as negative tangent counts (T

– –

), while branches have saddle curvature and

are labeled as mixed tangent counts (T

+–

– –

+ +

+ –

TTT

Area Distance

−=

+−

⋅⋅

++ −− +−

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both appear with minus signs, so the net abundance of tangent counts measures

the network connectivity.

OTHER STEREOLOGICAL TECHNIQUES

This introductory chapter has covered the most commonly used stereological

calculations, but the subject is quite large, and a variety of measurement methodol-

ogies have been derived for speciﬁc circumstances. There is also in the modern

literature a wealth of information on sample preparation techniques that avoid bias,

and speciﬁc approaches that are appropriate for newer microscopies including the

confocal light microscope.

As a simple example of the latter, consider again the measurement of the total

length of ﬁbers per unit volume. The length can be measured by counting intersec-

tions with a surface, but as ﬁbers have an orientation, we would like to use as a

probe a surface that is anisotropic. One way to do this is to collect a set of parallel

optical section images with the microscope and then place a set of circles on them

that vary in diameter to correspond to a sphere in 3D space, as shown in Figure

1.31. Counting the number of intersections of the ﬁbers with these lines allows an

unbiased calculation of L

, the length of the ﬁbers, as L

= 2P

where A is the surface

area of the sphere.

There are also stereological models that apply to particular kinds of microstruc-

tures. One that is encountered with some frequency is a structure containing a layer,

or a thick membrane, or any other structure that can be described as a “muralia,”

“plate,” “sheet” or “blanket” in three-dimensional space. Measuring the thickness

FIGURE 1.30 Tangent counts on a simply connected nonconvex feature; the net tangent

count is still +2 (= +5 – 3 in this example) regardless of the number of branches and ends.

+ +

+ –

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of that layer is important, but it may not be directly accessible in the section images

cut through it at arbitrary angles. In general, as shown in the diagram, the apparent

width of the section will be greater than the true three-dimensional thickness.

This is a typical case in which stereology tells us what should be measured.

There are computer-based image analysis techniques that can measure the two-

dimensional width of the section, but this is a case where they should not be used,

as the result is meaningless. Instead, it is useful to consider the distribution of the

intersection lengths of line probes that pass through a thick layer (Figure 1.32). The

shape of this probability distribution, which falls off asymptotically as the intersec-

tion length increases, is hard to treat analytically. But replotting the data as the

frequency distribution of the reciprocal of the intersection length offers a dramatic

simpliﬁcation — a straight line distribution. In real structures, the ﬁnite extent and

curvature of the plates limits the number of extremely long intersections, which

correspond to the very small vales of the reciprocal, but this has little effect on the

distribution. The nice thing about this simple triangular shape is that the mean value

is just two thirds of the maximum, and the maximum is the reciprocal of the true

3D thickness of the layer.

So a measurement procedure would be to draw lines in many orientations on

the image and measure intercept lengths, to determine the average value of the

reciprocal. Chapter 4 (Figure 4.21) shows an example in which computer-generated

lines and automatic measurements are used. Note that these lines may be random

lines, or we can use the systematic random method to rotate a grid of parallel lines

to different orientations. Either way, the mean value of the inverse of the intercept

length λ is determined. Then the true thickness of the layer t, which may not even

be seen in any of the images, is calculated as

(1.11)

FIGURE 1.31 An isotropic sphere probe consists of circles in serial confocal sections that

form a sphere. Counting the number of linear features that cross the various circles gives the

number of counts P that can be combined with the surface area of the sphere A to calculate

length per unit volume.

mean

=⋅













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