Russ J.C. Image Analysis of Food Microstructure

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(a)

(b)

FIGURE 4.3

Drawing of a panda: (a) original; (b) closing, neighbor count > 0, 1 iteration;

erosion, neighbor count > 7, 1 iteration.

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(c)

(d)

FIGURE 4.3 (continued)

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illustrates some of the possibilities. It contains isolated pixel noise (shot noise) in

both the white and black areas, as well as ﬁne lines and narrow gaps. Applying a

single iteration of classical open or closing can be used to ﬁll in the gaps (closing)

or remove the ﬁne lines (opening). In classical erosion and dilation, each pixel with

any touching neighbors of the opposite color is ﬂipped to that color. By changing

only those pixels that have more than seven (i.e., all eight touching neighbors) adjacent

pixels of the opposite color, it is possible to use erosion and dilation to selectively

remove just the isolated pixel noise, without altering anything else in the image.

While it is often possible to remove noise from an image by using erosion and

dilation after thresholding, in general this is not as good a method as performing

appropriate noise reduction on the grey scale image before thresholding. More

information is available in the grey scale image to reduce the noise, using tools such

as the median ﬁlter or maximum likelihood ﬁlter to evaluate the grey scale values

in the neighborhood and adjust pixel values accordingly. As shown in Figure 4.4,

this makes it possible to preserve ﬁne detail and local boundary shapes better than

applying morphological operations to the binary image after thresholding.

(e)

FIGURE 4.3 (continued)

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THE EUCLIDEAN DISTANCE MAP

The shape modiﬁcations introduced by erosion and dilation are of two kinds:

those that are intended and which are ideally dependent only on the shape of the

original thresholded feature, and those that occur because of the nature of the square

pixel grid from which the features are constructed. The latter alterations are unfor-

tunately not isotropic, and alter shapes differently in horizontal and vertical directions

as compared to 45 degree diagonal directions. Figure 4.5 shows a circle. Ideally, erosion

and dilation would act on this circle to produce smaller or larger circles, with a difference

in radius equal to the number of iterations (pixels added or removed).

(a) (b)

FIGURE 4.4

Noise reduction: (a) original X-ray map; (b) grey scale noise reduction before

thresholding; (c) thresholding applied to image (a); (d) closing (neighbor count > 5, 1 iteration)

applied to image (c).

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In fact, this is not the actual result. The dark ﬁgures show the effect of erosion

using twenty-ﬁve iterations. When classic erosion is used, pixels are removed that

touch any background pixel. Removing a pixel in the diagonal directions changes

the dimension by the diagonal of the square pixel, which is 41% greater than the

width of the pixel in the horizontal or vertical directions. Hence, the erosion proceeds

more rapidly in the diagonal directions and the original circle erodes toward the

shape of a diamond. Similarly, the light grey ﬁgure shows the result of dilation, in

which the feature grows more in the diagonal direction and evolves toward the shape

of a square.

Using other neighborhood criteria can alter the shape of the eroded or dilated

features, but it cannot produce an isotropic circle as a result. The ﬁgure shows the

effects of requiring more than 1, more than 2 or more than 3 neighbor pixels to be

of the opposite color. The shapes erode or dilate toward the shapes of squares,

diamonds or octagons, but not circles. Also the size of the resulting shapes varies,

so that the number of iterations cannot be used directly as a measure of the size

change of the feature. These problems limit the usefulness of erosion and dilation

based on neighboring pixels, and in most cases it is not wise to use them for the

addition or removal of more than 1 or 2 pixels.

There is another approach to erosion and dilation that solves the problem of

anisotropy. The Euclidean distance map (EDM) is used with binary images to assign

(a) (b)

FIGURE 4.5

Anisotropy of erosion/dilation (grey circle eroded 25 times to produce the black

inner shape, dilated 25 times to produce the light grey outer shape: (a) classic; (b) neighbor

count > 1; (c) > 2; (d) > 3.

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to every pixel that is black (part of the thresholded structure or features) a number

that measures the distance from that pixel to the nearest white (background) pixel,

regardless of direction. The construction of this grey scale image is very rapid, and

as shown in Figure 4.6 produces values that can be rendered as though they were a

surface to produce a smooth mountain with sides of uniform slope. Thresholding

this grey scale image is equivalent to eroding the original binary image. To perform

dilation, the image is inverted and the EDM is computed for the background, and

then thresholded. As shown in the ﬁgure, erosion and dilation using the EDM method

is isotropic and produces larger or smaller features that preserve shape. Because it

is not iterative, the method is also much faster than the traditional method.

In general, the EDM-based erosion and dilation methods should be used when-

ever more than 1 or 2 pixels are to be removed from or added to feature boundaries,

because of the improved isotropy and avoidance of iteration. However, the traditional

method using neighbor relationships offers greater control over the selective removal

of (or ﬁlling in of) ﬁne lines and points. Also, the EDM-based method requires that

the assigned grey scale pixels values that represent distance have more precision

than the integer values that grey scale images usually contain. The distance between

a feature pixel and the nearest point in the background is a real number whose

precision must be maintained for accurate results, and not all software systems do

(a) (b)

(c)

FIGURE 4.6

EDM of the circle, rendered as a surface to show values (plotted with isobrightness

contour lines) and isotropic erosion/dilation using the EDM (compare to preceding image).

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this properly. This becomes particularly important for some of the additional uses

of the EDM discussed below.

Just as for traditional morphological erosion and dilation, the EDM-based oper-

ations can be combined into openings and closings. In many cases, both are needed.

For instance, Figure 4.7 shows the distribution of stained fat particles in custards.

It is the clusters of particles that are important in determining the rheology, appearance,

and other properties of the product. Simply thresholding the red channel of the original

(a) (b)

FIGURE 4.7

Structural phase separation in custard: (a) grey scale rendering of original color

image (detail of Color Figure 4.7, nile red stain applied to fat globules; see color insert

following page 150); (b) fat droplets thresholded from red channel; (c) classic closing (six

iterations) and opening (three iterations) to delineate phase regions; (d) EDM-based closing

and opening, distance of 6.5 pixels. Images (c) and (d) are shown with original thresholded

image overlaid for comparison; the dark grey areas are the original thresholded pixels, the

light grey areas were removed by the opening, the black regions ﬁlled in by the closing).

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color image produces a good binary image of the distribution of nile red stain in the

sample. But there are gaps within the cluster regions that must be ﬁlled in, and a few

isolated particles in the other portions of the structure that should be erased.

Applying a closing to ﬁll in the gaps, and an opening to remove the isolated,

scattered features, produces an image in which the total volume fraction and size of

the clusters and the spacing between them can be measured. However, if done with

classic morphological methods this process produces suspicious shapes in which

90- and 45-degree boundary lines are prevalent. The EDM-based erosion and dilation

procedures generate shapes that are much more realistic and better represent the

actual structure present, so that more accurate measurements can be made.

SEPARATING TOUCHING FEATURES

The Euclidean distance map has other uses as well. Watershed segmentation

uses the EDM as the basis for separating touching, convex shapes. This is primarily

useful for section images or surface images in which features are adjacent or overlap

only slightly. It is less useful for arrays of three-dimensional particles because

particles of different sizes may overlap signiﬁcantly, and small particles may be

hidden by larger ones. The method also has difﬁculties with jagged or irregular

shapes, and depending on the quality of the EDM used to control the separation,

may cause the breakup of ﬁbers.

Figure 4.8 shows the basic logic behind watershed segmentation. Thresholding the

image of touching particles produces a binary representation in which all of the pixels

touch, and would be considered to be part of a single feature. The grey scale values

generated by the EDM, when represented as a rendered surface, produce a mountain

peak for each of the particles. Imagine that rain falls onto this terrain; drops that strike

the mountains run downhill until they reach the sea (the background). Locations on the

saddles between the mountains are reached by drops running down from two different

mountain peaks. Removing those points separates the peaks, and correspondingly sep-

arates the binary images of the various particles as shown in the ﬁgure. Notice that

some boundaries around the edge of the image are not separated, because the informa-

tion is missing that would be required to deﬁne the mountain peak beyond the edge.

The edge-touching features cannot be measured anyway, as discussed in the chapter on

feature measurements, so this does not usually create analysis difﬁculties.

The watershed method works well when particles are convex, and particularly

when they are spherical (producing circular image representations). It happens that

in many cases, due to membranes or physical forces such as surface tension, particles

are fairly spherical in shape and, as shown in Figure 4.9, the separation of the

particles for counting or individual measurement can use the watershed segmentation

method.

There is an older technique that is also used with spherical particles that has

found its principal application in ﬁelds other than food science, but which can

certainly be used for food structure images as shown in Figure 4.10. The two methods

for erosion and dilation described earlier in this chapter apply to binary images after

thresholding. A third erosion/dilation procedure was described in an earlier chapter

that is applied to grey scale images. Used, for instance, to remove features from an

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(a) (b)

(d) (e)

FIGURE 4.8

Watershed separation of touching particles: (a) original image; (b) thresholded;

background; (d) watershed separation; (e) feature outlines superimposed on the original image.

(c)

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image and produce a background that can be used to level an image by subtraction

or division, grey scale erosion and dilation replace each pixel with the brightest or

darkest (respectively) pixel values in a neighborhood.

When applied to an image such as the one shown in the ﬁgure, the process

shrinks all of the features by one pixel on all sides. If the particles are convex and

not overlapped by more than 50%, they will eventually separate before disappearing

completely as the procedure is repeated. By counting the number of features that

disappear at each iteration of erosion or dilation, a distribution of particle sizes can

be obtained. Since each iteration reduces the radius of the particles by 1 pixel, the

distribution is the number of features as a function of the inscribed radius. Performing

the counting operation requires some additional Boolean logic that has not yet been

introduced, but is relatively straightforward and very fast.

(a) (b)

FIGURE 4.9

Separation of touching spherical particles: (a) red channel of original color

image of mayonnaise; (b) thresholded fat globules; (c) watershed segmentation of (b);

(d) overlay of separate particle outlines on original image.

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