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88 Chapter 3 ■ Digital Morphology

(a) (b) (c)
Figure 3.2: The effects of a simple binary dilation one a small object. (a) Original image.
(b) Dilation of the original by one pixel. (c) Dilation of the original by two pixels (dilation
of (b) by one pixel).

The object in Figure 3.2b is also square, but is one pixel larger in all directions.
It was obtained from the previous square by simply setting all white neighbors
of any black pixel to black. This amounts to a simple binary dilation,so named
because it causes the original object to grow larger. Figure 3.2c shows the result
of dilating Figure 3.2b by one pixel, which is the same as dilating Figure 3.2a
by two pixels; this process could be continued until the entire image consisted
entirely of black pixels, at which point the image would stop showing any
change.
This is a very basic example of digital morphology, one that can be imple-
mented directly by first marking all white pixels having at least one black
neighbor, and then setting all the marked pixels to black. This is not, how-
ever, how morphological operators are usually implemented. In general, the
object is considered to be a mathematical set of black pixels; because each
pixel is identified by its row and column indices, a pixel is said to be a
2
point in two-dimensional space (E ). Theset of pixels comprising theobjectin
Figure 3.2a can now be written as {(3, 3)(3, 4)(4, 3)(4, 4)} if the upper-left pixel
in the image has the index (0,0). This set is too awkward to write out in full all
the time, so it will simply be called A. The operation shown in Figure 3.2 will
be called a ‘‘simple’’ binary dilation, because it is the most basic of what will
soon be seen to be a large set of possible dilations.

3.3.1 Binary Dilation
Now some definitions of simple set operations can be stated, with the goal
being to define dilation in a more general fashion in terms of sets. The translation
of the set A by the point x is defined, in set notation, as:

(A) x ={c|c = a + x, a ∈ A} (EQ 3.1)
For example, if x were at (1,2), then the first (upper left) pixel in A x would
be (3, 3) + (1, 2) = (4, 5); all the pixels in A shift down by one row and right by