Image Processing Reference
In-Depth Information
X
0
X
1
X
2
The hashed box indicates a don't care term.
This approach is directly equivalent to Mathematical morphology.
1,2,3
The
shapes above that are tested against the image are in fact structuring elements.
In general, any logical function may be implemented through a sum of products
expression. An increasing function is implemented when none of the variables are
negated. Similarly in the world of mathematical morphology, any morphological
operator may be written as a
union of erosions
.
4
These are in fact one and the same
thing. The erosions are equivalent to the products (or the ANDs) and the union is
equivalent to the sum. A number of different sub-components are tested against the
image. If one or more of them fits the image, the overall result is true. In the mor-
phological representation, a set of structuring elements are used. These are equiva-
lent to the
minterms
in the logical representation. In set theory, this would be
written as
iB
Θ
Ib
i
,
(5.7)
∈
is the erosion operator,
I
is the image, and
b
i
are the structuring elements equivalent to
X
0
X
1
and
X
2
shown above.
In mathematical morphology literature, there are few clues to selecting the best
set of structuring elements. In Soille's book of applications of morphological im-
age processing, most of the structuring elements are designed heuristically (in other
words, by guesswork).
5
There are some explanations in the literature but these are
buried within other more involved texts.
6
For comparison, an example of a 3-variable nonincreasing function, i.e., a
function for which the increasingness property does not hold, is shown in Fig. 5.4.
The inputs
x
= (0, 0, 1) and
x
= (0, 1, 1) prevent this function from being in-
creasing. It cannot be represented in the same way as the increasing function. The
function may still be minimized using a K map as shown in Fig. 5.5.
where U represents the union operator,
Θ
FXX XX XX
=
+
+
(5.8)
01
02
12
The resulting basis inputs of the minimized function are shown below.
X
0
X
1
X
0
X
2
X
12
