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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