Image Processing Reference

In-Depth Information

dist(C
k
, C
p
) =

min

u∈C
k
,w∈C
p

d(u, v)

Mean Squared Error

A mean squared error value between all data object assigned to the cluster and this cluster

center C
i
presents objective function minimized during k -means clustering procedure and at

the same time in numerous partitioning clustering schemes. The formula is given as follows

X

k

X

n
i

(X
ij
−X
i
)
2
,

M SE =

i=0

j=1

where X
i
denotes center of the C
i
cluster and X
ij
object with index j from objects assigned

to the cluster C
i
.

Within-Class and Between-Class Variance

Within-class variance is defined in the following way

n
i

X

k

X

p
i
(X
ij
−X
i
)
2
,

wV ar =

i=1

j=1

and between-cluster variance is defined as follows

X

k

p
i
(X−X
i
)
2
.

cV ar =

i=1

11.4

RECA Algorithms

11.4.1

Rough Set Theory

Information granules (Zadeh, 1997; Pedrycz, Skowron, and Kreinovich, 2008; Stepaniuk,

2008) are viewed as linked collections of objects (data points, in particular) drawn together

by the criteria of indistinguishability, similarity or functionality. Information granules and

the ensuing process of information granulation is a vehicle of abstraction leading to the

emergence of high-level concepts.

A granule is most often defined as a closely coupled group or clump of objects (for

example pixels in image processing setting), in the examined space that are interpreted

as an indivisible entity because of its indistinguishable character, similarity, proximity or

functionality. The granulation process basically consists in and subsequently results in

compression and summarization of information.

In the last decades, rough set theory has been attracted growing attention as a robust

mathematical framework for granular computing. Rough set theory has been introduced

by (Pawlak, 1991) in the 1980s, creating a comprehensive platform for discovering hidden

patterns in data with extensive applications in data mining. It has recently emerged as an

important mathematical tool for managing uncertainty that arises from granularity in the

domain of discourse, i.e., from the indiscernibility between objects in a set. The intention

is to approximate a rough (imprecise) concept in the domain of discourse by a pair of exact

concepts, called the lower and upper approximations. These exact concepts are determined

by an indiscernibility relation on the domain, which, in turn, may be induced by a given set

of attributes ascribed to the objects of the domain. The lower approximation is the set of

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