Image Processing Reference

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The different names of
< RX, RX >

TABLE 3.1

X

R

< RX, RX >

Crisp

m
Y
(u)∈{0, 1}(crisp equivalence )

rough set of X

Fuzzy

m
Y
(u)∈{0, 1}(crisp equivalence)

rough-fuzzy set of X

Crisp

m
Y
(u)∈[0, 1](fuzzy equivalence)

fuzzy rough set of X

Fuzzy

m
Y
(u)∈[0, 1](fuzzy equivalence)

fuzzy rough-fuzzy set of X

Crisp

S
R
: U×U→{0, 1}(crisp tolerance)

tolerance rough set of X

Fuzzy

S
R
: U×U→{0, 1}(crisp tolerance)

tolerance rough-fuzzy set of X

Crisp

S
R
: U×U→[0, 1](fuzzy tolerance)

tolerance fuzzy rough set of X

Fuzzy

S
R
: U×U→[0, 1](fuzzy tolerance)

tolerance fuzzy rough-fuzzy set of X

and Stepaniuk, 1996). When R is a tolerance relation, we consider the expressions for the

membership values corresponding to the lower and upper approximations (see (3.5)) of an

arbitrary set X in U as

M (u)

=

inf

ϕ∈U

max(1−S
R
(u, ϕ), µ
X
(ϕ))

M (u)

=

sup

ϕ∈U

min(S
R
(u, ϕ), µ
X
(ϕ))

(3.6)

where S
R
(u, ϕ) is a value representing the tolerance relation R between u and ϕ. Note

that, two different notions of expressing the upper and lower approximations of a set, exist

in literature pertaining to rough set theory (Radzikowska and Kerre, 2002). Among the

two notion, one is based on concept of similarity and the other is based on concept of

granulation due to limited discernibility. We use the first aforesaid notion in (3.6) and the

second aforesaid notion in (3.5), considering aspects of their practical implementation for

measuring ambiguity.

We refer the pair of sets < RX, RX > differently depending on whether X is a crisp or

a fuzzy set; the relation R is a crisp or a fuzzy equivalence, or a crisp or a fuzzy tolerance

relation. The different names are listed in Table 3.1.

As mentioned earlier, the lower and upper approximations of a vaguely definable set X in

a universe U can be used in the expression given in (3.3) in order to get an inexactness

measure of the set X called the roughness measure ρ
R
(X). The vague definition of X in U

signifies incompleteness of knowledge about U .

Here we propose two classes of entropy measures based on the roughness measures of

a set and its complement in order to quantify the incompleteness of knowledge about a

universe. One of the proposed two classes of entropy measures is obtained by measuring

the 'gain in information' or in our case the 'gain in incompleteness' using a logarithmic

function as suggested in the Shannon's theory (Shannon, 1948). This proposed class of

entropy measures for quantifying the incompleteness of knowledge about U with respect to

the definability of a set X⊆U is given as

2
h
ρ
R
(X) log
β
ρ
R
(X)

+ ρ
R
(X
{
) log
β
ρ
R
(X
{
)

i
(3.7)

H
R
(X) =−
1

β

β

where β denotes the base of the logarithmic function used and X
{
⊆U stands for the

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