Biomedical Engineering Reference
In-Depth Information
In fuzzy segmentation, each pixel is assigned a membership value in each
of the
c
regions. If the memberships are taken into account while computing
properties of regions, we oftain obtain more accurate estimates of region prop-
erties. One of the known techniques to obtain such a classification is the FCM
algorithm [40, 41]. The FCM algorithm is an unsupervised technique that clus-
ters data by iteratively computing a fuzzy membership function and mean value
estimates for each class. The fuzzy membership function, constrained to be be-
tween 0 and 1, reflects the degree of similarity between the data value at that
location and the prototypical data value, or centroid, ot its class. Thus, a high
membership value near unity signifies that the data value at that location is close
to the centroid of that particular class.
FCM has been used with some success in image segmentation in general
[45, 46], however, since it is a point operation, it does not preserve connectivity
among regions. Furthermore, FCM is highly sensitive to noise. In the following
sections, we will present a new system to segment digital images using a modified
Fuzzy c-means algorithm. Our algorithm is formulated by modifying the objec-
tive function of the standard FCM algorithm to allow the labeling of a pixel to be
influenced by the labels in its immediate neighborhood. The neighborhood ef-
fect acts as a regularizer and biases the solution toward piecewise-homogeneous
labelings. Such a regularization is useful in segmenting scans corrupted by scan-
ner noise. In this paper, we will present the results of applying this algorithm to
segment MRI data corrupted with a multiplicative gain field and salt and pepper
noise.
9.4.1 Standard Fuzzy-C-Means
N
The standard FCM objective function for partitioning
{
x
k
}
k
=
1
into
c
clusters is
given by
c
N
u
ik
||
x
k
−
v
i
||
2
J
=
,
(9.23)
i
=
1
k
=
1
N
c
where
{
x
k
}
i
=
1
are the prototypes of
the clusters and the array [
u
ik
]
=
U
represents a partition matrix,
U
∈
U
, namely
k
=
1
are the feature vectors for each pixel,
{
v
i
}
c
U
{
u
ik
∈
[0
,
1]
|
u
ik
=
1
∀
k
i
=
1
