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96 Chapter 3 ■ Digital Morphology

First, a simple example. Consider the structuring element

B ={(0, 0) (1, 0)}

and the object image
A ={(3, 3) (3, 4) (4, 3) (4, 4)}
The set A B is the set of translations of B that align B over a set of black
pixels in A. This means that not all translations need to be considered, but only
those that initially place the origin of B at one of the members of A.There are
four such translations:
B (3,3) ={(3, 3) (4, 3)}
B (3,4) ={(3, 4) (4, 4)}
B (4,3) ={(4, 3) (5, 3)}

B (4,4) ={(4, 4) (5, 4)}
In two cases, B (3,3) and B (3,4) , the resulting (translated) set consists of pixels
that are all members of A, and so those pixels will appear in the erosion of A
by B. This example is illustrated in Figure 3.7.
Now consider the structuring element B 2 ={(1, 0)}; in this case the origin is
not a member of B 2 .The erosion A B can be computed as before, except that
now the origin of the structuring element need not correspond to a black pixel
in the image. There are quite a few legal positions, but the only ones that result
in a match are:
B (2,3) ={(3, 3)}
B (2,4) ={(3, 4)}
B (3,3) ={(4, 3)}
B (3,4) ={(4, 4)}
This means that the result of the erosion is

{(2, 3) (2, 4) (3, 3) (3, 4)}
which is not a subset of the original.
It is important to realize that erosion and dilation are not inverse operations.
Although there are some situations where an erosion will undo the effect of a
dilation exactly, this is not true in general. Indeed, as will be observed later,
this fact can be used to perform useful operations on images. However, erosion
and dilation are duals of each other in the following sense:

ˆ
c
c
(A B) = A ⊕ B (EQ 3.12)