Image Processing Reference

In-Depth Information

4.5

Mathematical Morphology and Rough Sets

There are two main papers connecting mathematical morphology to rough set theory. One

by Polkowski(Polkowski, 1999), who uses the language of topology to connect the two fields

and Bloch's work that is mainly based on the language of relations,(Bloch, 2000).

We begin this section with some examples from (Polkowski, 1999). The first one is

partitioning Z⊆E
n
into a collection{P
1
, P
2
, ..., P
n
}, where P
i
is the partition of the i−th

axis E
i
into intervals of (j, j +1]. In the second example, a structuring element B = (0, 1]
n
is

selected. It is easily seen that o
B
(X) = X
−
for each X⊆Z. Also, c
B
(X) = X
−
. These two

examples clarify the relation between mathematical morphology and rough sets in a cogent

way. Mathematical morphology is mainly developed for the image plane and a structuring

element has a geometrical shape in this space. L. Polkowski defined a partition in this space

(not necessarily through the definition of equivalence relation) and then obtained the upper

and lower approximation of a set X based on these partitions. Equivalence classes have the

same characteristics and they are in the form of (j, j + 1]
n
. At the same time, a structuring

element B with the same characteristics of equivalent classes is defined, (0, 1]
n
. Then B
x
,

translation of B by x, can hit (overlap) any of the equivalence classes.

In classical rough set theory, objects are perceived by their attributes and classified into

equivalence classes based on the indiscernibility of the attribute values. In the above exam-

ples, the geometrical position of the pixels in the image plane, act as attributes and form

the partitions. In (Polkowski, 1993), the morphological operators are defined on equivalence

classes.

In her article, I. Bloch tried to connect rough set theory to mathematical morphology

based on relations (Bloch, 2000). She uses general approximation spaces, where, instead of

the indiscernibility relation, an arbitrary binary relation is used. She suggests that upper

and lower approximation can be obtained from erosion and dilation,(Bloch, 2000). The

binary relation defined in her work is xRy⇔y∈B
x
. Then from the relation R, r(x) is

derived in the following way.

∀x∈X, r(x) ={y∈X|y∈B
x
}= B
x

Consider∀x∈X, x∈B
x
and let be B be symmetric. Then erosion and lower approximation

coincide:

∀A⊂X, A
−
={x∈X|r(x)⊂A}={x∈X|B
x
⊂X}= e
B
(A)

The same method is used to show that upper approximation and dilation coincide. I. Bloch

also extends the idea to dual operators and neighbourhood systems.

The common result of both (Polkowski, 1999) and (Bloch, 2000) is the suggestion that upper

and lower approximations can be linked to closing(dilation) and opening(erosion). Topology,

neighbourhood systems, dual operators and relations are used to show the connection.

In the following subsection we tried some experiments to demonstrate the ways of incorpo-

rating both fields in image processing.

Lower Approximation as Erosion Operator

When mathematical morphology is used in image processing applications, a structuring

element mainly localized in the image plane is used.

The result is the interaction of the

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