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124 Chapter 3 ■ Digital Morphology

Of course, this is only half of a sphere and is only approximately spherical
because of truncation error and sampling. Nonetheless, imagine this sphere
being rolled over the underside of the surface represented by the image being
opened. Whenever the center of the sphere is directly beneath an image pixel,
the value of the opened image at that point is the highest (maximum) point
achieved by any part of the sphere. The closing would be modeled by rolling
the structuring element over the top of the surface and taking the lowest point
on the sphere at all pixels as the value of the corresponding pixel in the closed
image.
Figure 3.25 shows this process in two dimensions, as if viewing a cross
section of the image. An opening, in this case, can be seen as a smoothing
process that decreases the average level of the pixels, whereas closing appears
to increase the levels.

(a) (b)

(c) (d)
Figure 3.25: Geometric interpretation of grey-level opening and closing. (a) A ‘‘slice’’
through the image being opened, showing four positions of the structuring element. (b)
The opened slice — the highest points of the circle at all pixels. (c) Rolling the circle over
the top of the slice. (d) The closed slice — the lowest points of the circle at all pixels.
These figures are approximations.

Figure 3.26 shows grey-level opening and closing applied to the keys image
of Figure 3.24.
One interesting application of opening and closing is in the visual inspection
of objects. For example, when an object is cut or polished there can be scratches
left in the material. These become more visible if light is reflected off of the
surface at a low angle and the object is seen from the side opposite the lighting
source. Figure 3.27 shows an example of this, using a pair of disk guards from
3 / 2 inch floppy disks. The guard on the right (3.27b) is scored, which can be
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seen clearly in the image.