Mathematical Morphology and Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 78

partitions formed on the images inside the universe. The empty set φ is also added to the
basis (Steen and Seebach, 1995). Let (X, τ ) be the partition topology on X. We consider
each category of images as a subspace topology. Let C be a non-empty subset of X. The
collection τ C ={T∩C : T∈τ}of subsets of C is called the subspace topology. The
topological space (C, τ C ) is said to be a subspace of (X, τ ), (Gemignani, 1990). Let I∈X
be an image. We want to find the interior of the set I in terms of open sets of different

subspaces.

I C ={c∈τ c |c⊆I}

(4.23)

where (C, τ c ) is a subspace topology for a category of images. Based on equation 4.19, we
could say that

I C = I C−

(4.24)

In other words, the interior of a set I relative to subspace C is equal to the lower approxi-
mation of the set I with respect to subspace C.
Figures 4.6 and 4.7 demonstrate the results of lower approximation of a query image
in terms of the specified subspace. As it is obvious in the examples, the more similar the
query image to the subspace in terms of the predefined features, the more complete is the
lower approximation. The black pixels in the lower approximation image are those parts of
the image that are filtered. In these examples the features are three color components in
RGB space. Notice that we used 4 features to form the partition topology.

4.6 Conclusion

In summary, this chapter presents the basic definitions of mathematical morphology and
rough set theory. Mathematical morphology is defined and expanded in the image process-
ing domain. Rough set theory is first introduced for image archives. Although the initial
domains and applications of these two fields are different, there are connections between
the two. This chapter brings together the common aspects of mathematical morphology
and rough set theory. The lower approximation of rough set theory is analogous to open-
ing/erosion of mathematical morphology. The same is true for upper approximation and
closing/dilation.
We have proposed a method to use the idea of lower approximation to find the similarity
between images. A partition topology is defined on images gathered as a universe of im-
ages. Four features including color information and image indices are used to form image
partitions. Subspace topologies are used to model each category of images. An interior of
a query image is then calculated based on different subspaces. In other words, we find the
lower approximation of the query image in terms of different subspaces. We are proposing
that the closer the lower approximated image is to the query image, the more similar the
query image is to the subspace.