Image Processing Reference

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objects definitely belonging to the vague concept, whereas the upper approximation is the

set of objects possibly belonging to the same.

An information system is a pair (U, A), where U represents a non-empty finite set called

the universe and A a non-empty finite set of attributes. Let B⊆A and X⊆U . Taking

into account these two sets, it is possible to approximate the set X making only the use

of the information contained in B by the process of construction of the lower and upper

approximations of X.

Let IN D(B)⊆U×U be defined by

(x, y)∈IN D(B) if and only if, for all a∈B a(x) = a(y)

An approximation space AS
B
= (U, IN D(B)).

For X⊆U , the sets

LOW (AS
B
, X) ={x∈U : [x]
B
⊆X},

and

U P P (AS
B
, X) ={x∈U : [x]
B
∩X 6=∅}.

where [x]
B
denotes the equivalence class of the object x relative to B (the equivalence

relation IN D(B)), are called the B-lower and B-upper approximations of X in U.

It is possible to express numerically the roughness R(AS
B
, X) of a set X with respect to

B (Pawlak, 1991) by assignment

R(AS
B
, X) = 1−
card(LOW (AS
B
, X))

card(U P P (AS
B
, X))
.

In this way, the value of the roughness of the set X equal 0 means that X is crisp with

respect to B, and conversely if R(AS
B
, X) > 0 then X is rough (i.e., X is vague with respect

to B). Detailed information on rough set theory is provided in (Pawlak, 1991; Pawlak and

Skowron, 2007; Stepaniuk, 2008).

Let U be a non-empty set of objects called the universal set and P (U ) be the power set of

U as described in previous subsection. Binary relation R on U is referred to as reflexive if

for all x∈U, xRx. Relation R is referred to as symmetric if for all x, y∈U, xRy implies

yRx. Relation R is referred to as transitive if for all x, y, z∈U, if xRy and yRz then xRz.

Relation R is an equivalence relation if it is reflexive, symmetric and transitive.

Parameterized approximation space

In (Skowron and Stepaniuk, 1996; Stepaniuk, 2008) parameterized approximation space

has been defined as a system AS
#,$
= (U, I
#
, ν
$
) where

•U non empty set of objects

•I
#
: U→P (U ) uncertainty function

•ν

: P (U )×P (U )→[0, 1] rough inclusion function

$

and #, $ describe parameter indices ( #, $ may be omitted if the context is clear).

In this way, each predefined number of cluster centers defines for each data object different

uncertainty function I
#
: U→P (U ). Detailed description is given in (Stepaniuk, 2008).

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