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Chapter 3 ■ Digital Morphology 89

two columns in this case. This is a translation in the same sense that is seen in
computer graphics — a change in position by a specified amount.
The reflection of the set A is defined as:

ˆ
A ={c|c =−a, a ∈ A} (EQ 3.2)
This is really a rotation of the object A by 180 degrees about the origin.
The complement of the set A is the set of pixels not belonging to A This would
correspond to the white pixels in the figure, or in the language of set theory:
c
A ={c|c /∈ A} (EQ 3.3)
The intersection of the two sets A and B is the set of elements (pixels)
belonging to both A and B:
A ∩ B ={c|((c ∈ A) ∧ (c ∈ B))} (EQ 3.4)
The union of the two sets A and B is the set of pixels that belong to either A
or B,ortoboth:
A ∪ B ={c|(c ∈ A) ∨ (c ∈ B)} (EQ 3.5)
Finally, completing this collection of basic definitions, the difference between
the set A and the set B is:

A − B ={c|(c ∈ A) ∧ (c /∈ B)} (EQ 3.6)
which is the set of pixels that belong to A but not also to B.Thisisreally just
the intersection of A with the complement of B.
It is now possible to define more formally what is meant by a dilation. A
dilation of the set A by the set B is:

A ⊕ B ={c|c = a + b, a ∈ A, b ∈ B} (EQ 3.7)

where A represents the image being operated on, and B is a second set of
pixels, a shape that operates on the pixels of A to produce the result; the set B
is called a structuring element, and its composition defines the nature of the
specific dilation. To explore this idea, let A be the set of Figure 3.2a, and let
B be the set {(0, 0)(0, 1)}. The pixels in the set C = A + B are computed using
Equation 3.7, which can be re-written in this case as:

A ⊕ B = (A +{(0, 0)}) ∪ (A +{(0, 1)}) (EQ 3.8)
There are four pixels in the set A, and because any pixel translated by (0,0)
does not change, those four will also be in the resulting set C after computing
C = A +{(0, 0)}:
(3, 3) + (0, 0) = (3, 3) (3, 4) + (0, 0) = (3, 4)
(4, 3) + (0, 0) = (4, 3) (4, 4) + (0, 0) = (4, 3)