Biomedical Engineering Reference
In-Depth Information
problems. Second, the EM algorithm requires a good initial guess for either
the bias field or the classification estimate. Otherwise, the EM algorithm could
be easily trapped in a local minimum, resulting in an unsatisfactory solution
[31].
Another approach based on the FCM [40, 41] clustering technique has been
introduced lately [42-44]. FCM has been used with some success in image seg-
mentation in segmenting MR images [42, 47, 50]. Xu
et al.
[42] proposed a new
adaptive FCM technique to produce fuzzy segmentation while compensating
for intensity inhomogeneities. Their method, however, is also computationally
intensive. They reduced the computational complexity by iterating on a coarse
grid rather than the fine grid containing the image. This introduced some er-
rors in the classification results and was found to be sensitive to a considerable
amount of salt and pepper noise [43].
To solve the problem of noise sensitivity and computational complexity of
the Pham and Prince method, we will generalize the MFCM algorithm to segment
MRI data in the presence of intensity inhomogeneities.
9.4.4.1 Signal Modeling
The observed MRI signal is modeled as a product of the true signal generated
by the underlying anatomy and a spatially varying factor called the gain field:
Y
k
=
X
k
G
k
∀
k
∈
[1
,
N
]
(9.35)
where
X
k
and
Y
k
are the true and observed intensities at the
k
th voxel, respec-
tively,
G
k
is the gain field at the
k
th voxel, and
N
is the total number of voxels
in the MRI volume.
The application of a logarithmic transformation to the intensities allows the
artifact to be modeled as an additive bias field [28]
y
k
=
x
k
+
β
k
∀
k
∈
[1
,
N
]
,
(9.36)
where
x
k
and
y
k
are the true and observed log-transformed intensities at the
k
th
voxel, respectively, and
β
k
is the bias field at the
k
th voxel. If the gain field is
known, it is relatively easy to estimate the tissue class by applying a conventional
intensity-based segmenter to the corrected data. Similarly, if the tissue classes
are known, we can estimate the gain field, but it may be problematic to estimate
