Image Processing Reference

In-Depth Information

be inaccurate in a noisy environment (Krishnapuram and Keller, 1993, 1996). In real

data analysis, noise and outliers are unavoidable. Hence, to reduce this weakness of fuzzy

c-means, and to produce memberships that have a good explanation of the degrees of

belonging for the data, Krishnapuram and Keller (Krishnapuram and Keller, 1993, 1996)

proposed a possibilistic approach to clustering which used a possibilistic type of membership

function to describe the degree of belonging. However, the possibilistic c-means sometimes

generates coincident clusters (Barni, Cappellini, and Mecocci, 1996). Recently, the use of

both fuzzy (probabilistic) and possibilistic memberships in a clustering has been proposed

in (Pal, Pal, Keller, and Bezdek, 2005).

2.2.2

Rough Sets

The theory of rough sets begins with the notion of an approximation space, which is a pair

< U, R >, where U be a non-empty set (the universe of discourse) and R an equivalence

relation on U , i.e., R is reflexive, symmetric, and transitive. The relation R decomposes

the set U into disjoint classes in such a way that two elements x, y are in the same class iff

(x, y)∈R. Let denote by U/R the quotient set of U by the relation R, and

U/R ={X
1
, X
2
,, X
m
}

where X
i
is an equivalence class of R, i = 1, 2,, m. If two elements x, y∈U belong

to the same equivalence class X
i
∈U/R, then x and y are called indistinguishable. The

equivalence classes of R and the empty set∅are the elementary sets in the approximation

space < U, R >. Given an arbitrary set X∈2
U
, in general it may not be possible to

describe X precisely in < U, R >. One may characterize X by a pair of lower and upper

approximations defined as follows (Pawlak, 1991):

R(X) =

X
i
⊆X

R(X) =

X
i
∩X

X
i
;

X
i

=

∅

That is, the lower approximation R(X) is the union of all the elementary sets which are

subsets of X, and the upper approximation R(X) is the union of all the elementary sets

which have a non-empty intersection with X. The interval [R(X), R(X)] is the representa-

tion of an ordinary set X in the approximation space < U, R > or simply called the rough

set of X. The lower (resp., upper) approximation R(X) (resp., R(X)) is interpreted as the

collection of those elements of U that definitely (resp., possibly) belong to X. Further,

•a set X∈2
U
is said to be definable (or exact) in < U, R > iff R(X) = R(X).

•for any X, Y∈2
U
, X is said to be roughly included in Y , denoted by X
˜
⊂Y , iff

R(X)⊆R(Y ) and R(X)⊆R(Y ).

•X and Y is said to be roughly equal, denoted by X≃
R
Y , in < U, R > iff

R(X) = R(Y ) and R(X) = R(Y ).

In (Pawlak, 1991), Pawlak discusses two numerical characterizations of imprecision of a

subset X in the approximation space < U, R >: accuracy and roughness. Accuracy of X,

denoted by α
R
(X), is simply the ratio of the number of objects in its lower approximation

to that in its upper approximation; namely

α
R
(X) =
|R(X)|

|R(X)|

Search WWH ::

Custom Search