Shih_Image_Processing_and_Mathematical_Morpholo.pdf

26 Image Processing and Mathematical Morphology
3.1 Grayscale Dilation and Erosion
Grayscale images can be represented as binary images in a three- dimensional
space, with the third dimension representing brightness. Grayscale images
can then be viewed as three-dimensional surfaces, with the height at each
point equal to the brightness value. We begin with defi nitions of the top
surface of a set and the umbra of a surface. Given a set A in E N , the top surface
of A is a function defi ned on the projection of A onto its fi rst (N – 1) coordinates,
and the highest value of the N-tuple is the function value (or gray
value). We assume that for every x � F, {y |(x, y) � A} is topologically closed,
and also that the sets E and K are fi nite.
Defi nition 3.1: Let A Õ E N and F = {x � E N-1 |for some y � E, (x, y) � A}. The
top or top surface of A, denoted by T[A]: F Æ E, is defi ned as
T[A](x) = max{y | (x, y) � A}. (3.1)
Defi nition 3.2: Let F Õ E N-1 and f : F Æ E. The umbra of f, denoted by U[ f ],
U[ f ] Õ F × E, is defi ned as
U[ f ] = {(x, y) � F × E | y £ f(x)}. (3.2)
Example 3.1: Let f = {2, 1, 0, 2}. Show the umbra U[ f ].
0 1 2 3
0 2 1 0 2
f
3 0 0 0 0
2 1 0 0 1
1 1 1 0 1
0 1 1 1 1
. 1 1 1 1
. 1 1 1 1
. 1 1 1 1
–∞ 1
0 1 2 3
1 1 1
U[ f ]
The top surface of a set and the umbra of a function are essentially inverses
of each other. In the above example, the top surface T[A] is equal to the input
function f where A = U[ f ]. That is, the one-dimensional grayscale input
function f maps the one-dimensional coordinates to its function values. The
umbra extends the value “1” from the function value (as coordinates) of