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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