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116 Chapter 3 ■ Digital Morphology

3.3.8 Identifying Region Boundaries
The pixels on the boundary of an object are those that have at least one
neighbor that belongs to the background. Because any background neighbor
is involved it cannot be known in advance which neighbor to look for, and a
single structuring element that would allow an erosion or dilation to detect
the boundary can’t be constructed. This is in spite of the fact that an erosion
by the simple structuring element removes exactly these pixels!
On the other hand, this fact can be used to design a morphological boundary
detector. The boundary can be stripped away using an erosion and the eroded
image can then be subtracted from the original. This should leave only those
pixels that were eroded — that is, the boundary.
A MAX program for this is:

// Boundary extraction
image a, b, c;
begin
doa<< ˝ $1˝;
b := {[3,3], [1,1], ˝ 111111111˝}; // Simple structuring element
c := (a - (a--b));
message c;
doc>> ˝ boundary.pbm˝;
end;
Figure 3.19 shows this method used to extract the boundaries of the
‘‘squares’’ image of Figure 3.18a. A larger example, that of a quarter rest
scanned from a page of sheet music, also appears in the figure.

(a) (b) (c) (d) (e)
Figure 3.19: Morphological boundary extraction. (a) The squares image. (b) The squares
image after an erosion by the simple structuring element. (c) Difference between the
squares image and the eroded image: the boundary. (d) A musical quarter rest, scanned
from a document. (e) The boundary of the quarter rest as found by this algorithm.

3.3.9 Conditional Dilation

There are occasions when it is desirable to dilate an object in such a way that
certain pixels remain immune. If, for example, an object cannot occupy certain