Image Processing Reference
In-Depth Information
combination of low-frequency discrete cosine transformation (DCT) basis functions
as the inhomogeneity field model [34].
In parallel with EM-based approaches, a number of methods based on exten-
sions of the K-means algorithm [35,36] were developed that also iteratively
alternate between segmentation and inhomogeneity field correction. In order to
explain the original K-means algorithm, let us ignore the inhomogeneity field
artifact for now, and model MR images as consisting of K tissue types, each
governed by a Gaussian intensity distribution with mean
µ k and a variance that
is assumed equal for all tissue types. Both the tissue labels
l
=,
[
l
l
,
…l N
,
]
T
and
12
the tissue means
µµµ
=,
[
12
,
,
µ
]
T
can be estimated simultaneously from an
K
image y using the MAP principle
arg maxlog
pl
(
µ
,
|
y
)
.
µ
,
l
Assuming a uniform prior for the tissue means and for the configuration of the
tissue labels in the image, this is equivalent to minimizing the objective function
∑∑
Quy
(
µ
,
|
)
=
uy
(
µ
)
2
(5.9)
1
ik
i
k
i
k
where
#
1
0
if
otherwise
l
=
k
%
%
u
=
i
,
ik
.
Equation 5.9 can be optimized by iteratively performing a crisp classification
that assigns each voxel exclusively to the tissue type that best explains its intensity
#
1
if
|
y
− | <|
µ
y
− |,
µ
j
=,
1
… K
,
,
j
k
%
%
i
k
i
j
u
=
(5.10)
ik
0
otherwis
e
.
and updating the tissue means accordingly
uy
ik
i
µ k
=
i
.
(5.11)
u
ik
i
A well-known generalization of the K-means algorithm is the fuzzy C-means
algorithm [37,38], which aims at minimizing*
∑∑
Quy
(
µ
,| =
)
(
uy
) (
µ
)
m
2
m
ik
i
k
i
k
(5.12)
with
m
∈,∞,
[
1
)
u
∈∈,,
[]
01
and
u
=.
1
ik
ik
k
* Unequal tissue-specific covariances can also be taken into account.
 
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