Image Processing Reference
In-Depth Information
in the model, Van Leemput et al. aimed at finding the MAP model parameters
ˆ
Φ
=
arg max
py
(
Φ
|
)
.
Φ
In the absence of prior knowledge about the parameters
Φ
, a uniform prior
distribution
p
(
)was assumed, reducing the MAP principle to a maximum-
likelihood (ML) approach
Φ
ˆ
Φ
=
arg max
py
(
|
Φ
)
.
Φ
), Van Leemput
et al. derived an EM algorithm that iteratively repeats three consecutive steps.
In the E step, the image voxels are classified based on the current inhomogeneity
field and Gaussian distribution estimates, using Equation 5.7. The M step
involves two separate steps, in which first the inhomogeneity field is updated
according to the current classification and Gaussian distribution parameters,
followed by an update of the Gaussian distributions. The inhomogeneity field
estimation is given by
Using a zero-gradient condition on the logarithm of
p
(
y
|
Φ
ψ
()
x
ψ
221
()
x
ψ
()
x
…
ˆ
ˆ
c
c
1
11
31
=
(
AWA
T
)
−
1
AWr
T
with
A
=
ψ
()
x
ψ
()
x
ψ
()
x
…
.
2
12
22
32
where the weight matrix
W
and the residue image
r
are the same as in the Wells
algorithm. In other words, the inhomogeneity field is estimated as a weighed
least-squares fit to the residue
r
(
Figure 5.2d
), using the confidence weights
w
(Figure 5.2e).* The Gaussian distribution parameters are updated according to
∑
py
(
−
β
)
ik
i
i
µ
=
i
∑
k
p
ik
i
and
∑
py
(
−−
βµ
)
2
ik
i
i
k
σ
2
=
i
∑
k
p
ik
i
* Note that by modeling the inhomogeneity field as a linear combination of smooth basis functions,
rather than by the Gaussian prior proposed by Wells et al., no approximations are required to compute
the inhomogeneity estimation.
Search WWH ::
Custom Search