Russ J.C. Image Analysis of Food Microstructure

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EDM could be calculated for the interior of the cells. The images were then combined

by a mathematical operation that keeps whichever pixel value is brighter in the two

images.

Measuring the grey scale value of each feature allows plotting the number of

features as a function of distance from the boundary. As shown in Figure 4.41(e),

an initial glance at the plot suggests that the features may be preferentially located

near cell walls, but this is not necessarily so. It is actually the number per unit area

that is of interest, and the size and shape of the cells affects the area as a function

of distance. The histogram of the EDM image provides this data as shown in Figure

4.41(f); note that it has the same shape as the plot of number of features. Since it

is the ratio of the two plots that measures the tendency of features to locate close

to or away from the boundary, in this example the features are uniformly distributed

within the cells.

(a) (b)

FIGURE 4.40 Measurement of the thickness of muscles in farm-raised salmon: (a) scanned

cross-section image; (b) binary image after thresholding and opening, with skeleton lines

(dilated for visibility) superimposed on the muscles; (c) histogram of the EDM values sampled

by the skeleton, which measures the thickness distribution of the muscles.

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It is also possible to measure the distance of features from each other. In many

instances features are distributed in non-random ways. Clustering of features pro-

duces groups separated by gaps, while a self-avoiding distribution has features that

are more-or-less uniformly spaced. The key to measuring these tendencies is the

random distribution, as will be discussed in detail in Chapter 5. The mean value of

the nearest neighbor distance for a random distribution can be calculated as

(4.1)

If a feature distribution is clustered, the mean nearest neighbor distance is lower

than that for a random distribution, and, vice versa, it is greater for a self-avoiding

distribution. For the example image, a plot of nearest neighbor distances is shown

in Figure 4.41(g). The mean value of the distribution (4.35 µm) is signiﬁcantly

greater than the calculated value expected for a random distribution (3.20 µm) of

the same number of features in the same area, indicating that the features are self-

avoiding. More examples of random, clustered, and self-avoiding distributions are

shown in Chapter 5. A plot showing all of the nearest neighbor distance and direction

vectors (Figure 4.41h) indicates that the spatial distribution is isotropic.

(c)

FIGURE 4.40 (continued)

01020304050

Half-width

Frequency

100

200

300

400

500

600

700

Mean Nearest Neighbor Distance

Number

Area

05.

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A very similar procedure makes it possible to measure the fractal dimension of

clusters. The characteristic of agglomerated clusters is that a plot of the number of

features (or the mass of agglomerated particles) as a function of their radial distance

from any point in the cluster, plotted on log-log axes, is a straight line whose slope

gives the cluster dimension. To measure this, combine the centroids or ultimate

eroded points of features such as the crystals in the image of shortening in Figure

4.42 with the EDM of the entire image ﬁeld. The latter is generated by selecting

any arbitrary point (but usually for convenience the center of the image) as a single

background point and then calculating the EDM for all other points. Assigning these

values to the ultimate eroded points and plotting the number of points as a function

of their grey scale value produces the required graph.

(a) (b)

FIGURE 4.41 Characterizing spatial distributions: (a) original image of toluidine blue stained

peanut (red channel from a color image, courtesy of David Pechak, Kraft Foods Technology

Center) showing stained protein bodies and starch granules (see Color Figure 1.12; see color

insert following page 150); (b) skeletonized cell walls and feature centroids; (c) EDM of cell

interior; (d) feature centroids with grey scale values assigned from the EDM; (e) plot of

number of features as a function of distance from cell wall; (f) plot of the area as a function

of distance; (g) plot of number of features as a function of nearest neighbor distance; (h) rose

plot of nearest neighbor distance and direction vectors showing isotropic distribution.

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The number of ways that the EDM, centroids or ultimate eroded points, outlines

(and dilated outlines), skeleton, and various points such as end points or nodes, can

be combined to perform subtle but important measurements on images is limited

only by the imagination. As one more example, consider the illustration in Figure

(e)

(f)

FIGURE 4.41 (continued)

Min = 0.125

Max = 25.50

Distance (µm), 20 bins, Total = 732

Mean = 8.77917, Std.Dev.=6.10416

Skew = 0.72642, Kurtosis = 2.59672

100

Number of Features

–

10 20 30

Distance (µm)

1000

2000

3000

4000

5000

6000

7000

8000

Pixels

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4.43 of a cell with dendritic processes. Visually, there is an appearance that the

length of the terminal branches (ones that have an end) is shorter for those branches

that are more distal from the cell body. How can that be veriﬁed by measurement?

In this case the skeleton segments can be disconnected by removing the nodes,

but only the terminal branches are wanted. So another copy of the skeleton is

processed to keep just the end points, which are then used in a feature-based AND

(g)

(h)

FIGURE 4.41 (continued)

Min = 1.25

Max = 12.33

Nearest Neighbor Distance (µm)

20 bins, Total = 732

Mean = 4.35004, Std.Dev.=1.45030

Skew = 0.91517, Kurtosis = 4.93289

128

Number of Features

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(a) (b)

(e)

FIGURE 4.42 Example of cluster fractal dimension measurement: (a) image of fat crystals

in shortening (courtesy of Diana Kittleson, General Mills); (b) thresholded image (after

closing); (c) EDM of the image ﬁeld; (d) ultimate eroded points with grey scale assigned

from the EDM; (e) log-log plot of cumulative count vs. radial distance.

100

distance

200

Count

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(a)

(b)

FIGURE 4.43 Measurement using skeleton, feature-AND, and EDM: (a) original image

illustrating a cell with dendritic extensions; (b) skeleton of the structure; (c) terminal branches

produced as described in the text; (d) EDM of the pixels outside the main cell body measuring

distance from the cell; (e) grey scale values from the EDM assigned to pixels in the terminal

branch skeletons (thickened for visibility); (f) correlation plot showing variation of branch

length with distance of the near end of each branch from the cell body.

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(c)

(d)

FIGURE 4.43 (continued)

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(e)

(f)

FIGURE 4.43 (continued)

min=39.0

Minimum Intensity

max=177.0

Equation: Y = -0.00110 *X + 0.22855

R-squared = 0.85995, 27 points

min=0.044

Skeleton Length

max=0.208

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to select only those branches that are marked by an end point. These are then

combined with the EDM calculated for all pixels outside the cell body. That assigns

distance values to the pixels in the skeleton segments that mark the terminal branches.

But what should be measured? The most reasonable choice is to plot the length of

each terminal segment of the skeleton (calculated as a smooth curve, as discussed

in the next chapter) versus the smallest value of grey scale for any pixel in the

skeleton. That corresponds to the distance of the nearest point from the cell body.

As shown in the ﬁgure, the plot indicates a signiﬁcant correlation between the two

variables, conﬁrming the visual impression.

SUMMARY

Morphological and Boolean procedures applied to binary images obtained by

thresholding serve multiple purposes. One is simply to clean up imperfect representa-

tions of the features or structure present, by removing extraneous noise, smoothing

boundaries, ﬁlling holes within features, separating touching features, and so forth.

Another is to extract skeletons or outlines from images either to use as characterizations

of shape and location or to use in conjunction with other images and operations for

measurement. A third very important class of operations uses Boolean logic to apply

various grids to images so that stereological measurements can be obtained. Many

of the global structural measurements (as opposed to the measurement of individual

features present in the image, which is covered in the next chapter) that characterize

foods (and other materials) can be very efﬁciently obtained by using these various

tools in the appropriate combinations. This chapter has illustrated some of those

combinations, but the inventive researcher will ﬁnd many more useful ones.

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