Image Processing Reference

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Defining entropy measures based on rough set theory has been considered by researchers

in the past decade. Probably, first such work was reported in (Beaubouef, Petry, and

Arora, 1998), where a 'rough entropy' of a set in a universe has been proposed. This

rough entropy measure is defined based on the uncertainty in granulation (obtained using

a relation defined over universe (Pawlak, 1991)) and the definability of the set. Another

entropy measure called the 'rough schema entropy' has been proposed in (Beaubouef et al.,

1998) in order to quantify the uncertainty in granulation alone. Other entropy measures of

granulation have been defined in (Duntsch and Gediga, 1998; Wierman, 1999; Liang, Chin,

Dang, and Yam, 2002). Later, entropy measures of fuzzy granulation have been reported

in (Bhatt and Gopal, 2004; Mi, Li, Zhao, and Feng, 2007). It is worthwhile to mention

here that (Yager, 1992) and (Hu and Yu, 2005) respectively present and analyze an entropy

measure, which, although not based on rough set theory, quantifies information with the

underlying elements having limited discernibility between them.

Incompleteness of knowledge about a universe leads to granulation (Pawlak, 1991) and

hence a measure of the uncertainty in granulation quantifies this incompleteness of knowl-

edge. Therefore, apart from the 'rough entropy' in (Beaubouef et al., 1998) which quanti-

fies the incompleteness of knowledge about a set in a universe, the other aforesaid entropy

measures quantify the incompleteness of knowledge about a universe. The effect of incom-

pleteness of knowledge about a universe becomes evident only when an attempt is made to

define a set in it. Note that, the definability of a set in a universe is not always affected

by a change in the uncertainty in granulation. This is evident in a few examples given in

(Beaubouef et al., 1998), which we do not repeat here for the sake of brevity. Hence, a mea-

sure of incompleteness of knowledge about a universe with respect to only the definability

of a set is required.

First attempt of formulating an entropy measure with respect to the definability of a

set was made in (Pal, Shankar, and Mitra, 2005), which was used for image segmentation.

However, as pointed out in (Sen and Pal, 2007), this measure does not satisfy the necessary

property that the entropy value is maximum (or optimum) when the uncertainty (in this

case, incompleteness of knowledge) is maximum.

In this section, we propose classes of entropy measures, which quantify the incompleteness

of knowledge about a universe with respect to the definability of a set of elements (in the

universe) holding a particular property (representing a category). An inexactness measure of

a set, like the 'roughness' measure (Pawlak, 1991), quantifies the definability of the set. We

measure the incompleteness of knowledge about a universe with respect to the definability

of a set by considering the roughness measure of the set and also that of its complement in

the universe.

3.2.1

Roughness of a Set in a Universe

Let U denote a universe of elements and X be an arbitrary set of elements in U holding a

particular property. According to rough set theory (Pawlak, 1991) and its generalizations,

limited discernibility draws elements in U together governed by an indiscernibility relation

R and hence granules of elements are formed in U . An indiscernibility relation (Pawlak,

1991) in a universe refers to the similarities that every element in the universe has with the

other elements of the universe. The family of all granules obtained using the relation R is

represented as U/R. The indiscernibility relation among elements and sets in U results in

an inexact definition of X. However, the set X can be approximately represented by two

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