Mathematical Morphology and Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 75

structuring element with the image underneath. On the other hand in rough set theory, the
universe is partitioned or covered by some classes(indiscernibility, tolerance or arbitrary)
and the objects in the universe are perceived based on the knowledge available in these
classes. The interaction of the structuring element with the image underneath is localized
in space; In other words, the underneath image is being seen(characterized) through the
small window opened by the structuring element. In rough set theory a set is approximated
by the knowledge gathered in equivalence classes from the whole universe. This is the main
difference in mathematical morphology and rough sets.
In mathematical morphology, especially when the concept of lattice is introduced in this
field, the universe consists of all the possible images. But when we are applying morpho-
logical operators on images, there is only one image and a set of structuring elements. The
result is a new image belonging to the universe. In the application of rough set theory,
the available data in databases forms the universe and all the approximations are based on
the available data. To have almost the same milieu, we also consider a set of finite images
to be our universe (this is possible, if we view the universe X as a set of points and each
image A in the universe to be a subset of the universe, i.e., particular set of points A⊂X).
Ten sets of images in different categories are considered. Each set consists of 100 images.
Figure 4.4 shows samples of some of the categories. A category of images is defined as a
collection of images with visual/semantic similarities. For instance, the category of seaside
images, mountain images, dinosaurs or elephants can be derived from images in the Sim-
plicity image archive (Group, 2009). The categorization is done by an individual and it is
not unique. Each image may belong to different categories. For instance, in figure 4.5a
the elephant pictures are categorized into elephant category; but in figure 4.5b, they are
in animal and/or nature categories. Therefore the categorization is completely application
dependent and subjective.
FIGURE 4.4: Some image samples from different categories
The aim is to approximate a query image, I∈X, based on one and/or several of these