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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