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90 Chapter 3 ■ Digital Morphology

The result of A +{(0, 1)} is:
(3, 3) + (0, 1) = (3, 4) (3, 4) + (0, 1) = (3, 5)
(4, 3) + (0, 1) = (4, 4) (4, 4) + (0, 1) = (4, 5)
The set C is the result of the dilation of A using structuring element B,and
consists of all the pixels listed above (some of which are duplicates). Figure 3.3
illustrates this operation, showing graphically the effect of the dilation. The
pixels marked with an X, either white or black, represent the origin of each
image. The location of the origin is really quite important. In the example
above, if the origin of B were the rightmost of the two pixels the effect of the
dilation would be to add pixels to the left of A, rather than to the right. The
set B in this case would be {(0, −1)(0, 0)}.

A =

(a) (b) (c)

B = = structuring element

(d)
Figure 3.3: Dilation of the set A (Figure 3.2a) by the set B. (a) The two sets. (b) The set
obtained by adding (0,0) to all elements of A. (c) The set obtained by adding (0,1) to all
elements of A. (d) The union of the two sets is the result of the dilation.

Moving back to the simple binary dilation that was performed in Figure 3.2,
one question that remains is: What was the structuring element that was
used? Note that the object increases in size in all directions, and by a single
pixel. From the example just completed, it was observed that if the structuring
element has a set pixel to the right of the origin, then a dilation that uses
that structuring element ‘‘grows’’ a layer of pixels on the right of the object.