Image Processing Reference

In-Depth Information

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.

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