Mathematical Morphology and Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 69

the medium A, where B⊂A (remember erosion- or opening) and A∩B 6= φ(dilation or
closing). We clarify this idea by citing what G. Matheron wrote in his book in 1975,page
xi (Matheron, 1975)(Serra also referred to this part in his book (Serra, 1983, p.84)):

”In general, the structure of an object is defined as the set of relationships existing be-
tween elements or parts of the object. In order to experimentally determine this structure,
we must try, one after the other, each of the possible relations and examine whether or not
it is verified. Of course, this image constructed by such a process will depend to the greatest
extent on the choice made for the system<of relationships considered as possible. Hence
this choice plays a priori a constructive role in (in the Kantian meaning) and determines
the relative worth of the concept of structure at which we will arrive.
In the case of a porous medium, let A be a solid component(union of grains), and A c the
porous network. In this medium, we shall move a figure B, called the structuring pattern,
playing the role of a probe collecting information. This operation is experimentally attain-
able.”

We can get more information about the object if we gather more information about it.
The information can be obtained through the relations, whether it is false or true. Assume
that we have a family of structuring elementsB, each B∈Band each relation (B⊂A,
B∩A 6= φ)gives us some information about A. As an example of a set of structuring
elements, consider a sequence{B i }made up of compact disks of radius r i = r o + 1/i, which
tend toward the compact disk of radius r 0 .

Topology and Mathematical Morphology

Topological properties of mathematical morphology have been introduced and studied by
Matheron and Serra in (Matheron, 1975) and (Serra, 1983), respectively. Here we briefly
mention some of them. We start with opening and closing. The concepts of topological
closure and interior are comparable with morphological closing and opening. The only
difference is that in morphology the closing and opening are obtained with respect to a
given structuring element (Serra, 1983) but in topology closure and interior are defined in
terms of closed and open sets of the topology (Engelking, 1989). To blur this difference
and obtain a closer relation between mathematical morphology operators and topological
interior and closure, consider the following proposition.

Proposition 4.3.1 Let (X, d) be a metric space. Then the collection of open balls is a
basis for a topology τ on X (Gemignani, 1990).
The topology τ induced by the metric d and (X, τ ) is called the induced topological space.
If d is the euclidean metric on , then a set of open balls is a basis for the topology τ induced

by a metric d. Let B r and B r be an open ball and closed ball of radius r, respectively.
Consider the sets of structuring elementsBand ˚ Bas follows:
B={B r |r > 0}. (4.7)
˚ B={B r |r > 0}. (4.8)
Based on the above proposition, ˚ Bform a basis for the topology on the euclidean space
or image plane. The following equations are showing the relations between mathematical
morphology's opening and closing to topological closing and opening for a set A⊂X in
euclidean space, let

R

A and A be interior and closure of a set A. Then
A = [
B∈ ˚ B

o B (A) = [
B∈B

o B (A). (4.9)