Image Processing Reference
In-Depth Information
where we assume that det(B) = 1 to avoid a minimisation of the objective function by
matrices with zero or almost zero entries.
The elements of matrix B for cluster v are
calculated according to
n
q
det(G
v
)∗G
−1
B =
(6.26)
v
where G
v
is an n×n diagonal matrix, the fuzzy covariance matrix defined as
G
v
=
X
x∈X
µ
v
(x)(x−v)(x−v)
T
(6.27)
In this concept only the cluster shapes can vary now, while the cluster size is kept constant,
which in turn allows for a more intuitive partition of the data. The cluster algorithm
determines prototype locations, prototype parameters and a matrix of memberships using
the iterative procedure as described in (Gustafson and Kessel, 1979). The membership
matrix contains the membership values µ(x
j
)(v) of the j-th input pattern x
j
for cluster v
and these values satisfy the following conditions:
0≤µ(x
j
)(v)≤1, j = 1, 2, . . . , m, v = 1, 2, . . . , M
M
X
X
m
µ(x
j
)(v)
=
1, 0 <
µ(x
j
)(v) < m
v
=1
j
=1
If the data set is representative for the system, we can assume that additional data could
cause only slight modifications of the clusters shape. First of all, we might want to determine
the memberships of all possible data. Therefore, we have to extend the discrete membership
matrix to a continuous membership function.
It is commonly known that membership functions can often be assigned linguistic labels.
In particular, in image understanding tasks we very often use attributes such as “rather
colorful”, “well contrasted”, etc. This would make fuzzy system easy to read and interpret
by humans. But it is often very di
cult to specify meaningful labels in a multi-dimensional
feature space. Assigning labels is typically easier in one-dimensional domains. We therefore
project the discrete membership values µ(x
j
)(v) to the respective axes related to image
features and denoted as x
1
, . . . , x
n
.
To obtain continuous membership functions from projected membership various methods
have been proposed (Gustafson and Kessel, 1979; Sugeno and Yasukawa, february 1993;
Zheru, Hong, and P., 1996). In our approach we suggest to approximate each generated
cluster by a hyper-ellipsoid described by its cluster prototype. The lengths of hyper-ellipsoid
axes are defined by the variance of the cluster projected on each dimension. The projec-
tion produces a triangular membership function with a peak pointing to the corresponding
cluster prototype v ={v
(1)
, . . . , v
(n)
}. The size of the triangular base depends on variances
In our case we have described the triangular function as the special case of the trapezoidal
function, Π defined as
8
>
>
>
>
<
0
x≤a
(x−a)(y−b)
a < x≤b
Π(x, a, b, c, d) =
1
b < x≤c
(6.28)
>
>
>
>
:
(d−x)(d−c)
c < x≤d
0
x > d
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