Biomedical Engineering Reference
In-Depth Information
where:
•
J
obs
(
o
(
)) is the data fidelity term. It corresponds to the term Pr(
i|o
) from the
noise distribution (discussed in previous section), and
x
•
J
reg
(
o
(
)) is the prior object energy. It corresponds to the penalty term Pr(
o
) on
the object that ensures smoothness of the solution.
x
In Bayesian statistics, a MAP estimate is a mode of the posterior distribution. It
is closely related to Fisher's method of ML that was mentioned earlier, but employs
an augmented optimization objective which incorporates the prior distribution over
the quantity one wants to estimate. MAP estimation can therefore be seen as a
regularization of an ML estimation. In MAP, there is at least one variable parameter,
λ ∈
R
+
, which usually controls the algorithm output by regulating the noise and
the high frequencies. This controls the trade-off between the data fidelity term and
the regularization term in Eq. (
4.21
).
If the Power spectral density (PSD) of the noise (denoted by
P
w
(
f
))andthe
object (denoted by
P
o
(
)) respectively are known, then with the AWGN likelihood
in Eq. (
4.12
) the Wiener deconvolution is [
11
,
80
]:
f
⎛
⎞
OTF
∗
(
f
)
⎝
⎠
,
F
−
1
o
(
x
|
2
+
λ
P
w
(
f
)
P
o
(
f
)
F
(
i
(
x
)=
))
(4.22)
|
OTF(
f
)
F
−
1
is the 3-D inverse Fourier transform, OTF(
where
f
) is the incoherent optical
transfer function, and OTF
∗
(
) is its complex conjugate. We can see that when
λ
=0in Eq. (
4.22
) or in Algorithm
6
, the deconvolution is by a simple inverse
filter.
The RLS filter [
60
,
79
]inTable
4.1
is another approach that uses a pseudo-
inverse and an a priori on
o
to find a smooth solution. The choice of regularization
functionals
f
)) in Eq. (
4.21
) is important because it integrates the a priori
knowledge about the expected solution into the reconstruction process. In this
respect, the quadratic regularization terms with
2
norms (as in Tikhonov-Miller
[
48
,
79
]) attracted most attention. This is primarily because they were the first to
be introduced and also due to their computational simplicity. Nevertheless, such
regularization techniques cannot generate reconstructions with sharp edges, and as
a result singular regularization energies, especially those of
1
norm type (described
in next section), have become quite popular.
J
reg
(
o
(
x
Algorithm 6
The Wiener deconvolution algorithm
Input:
Observation
i
(
x
)
∀
x
∈ Ω
s
, PSDs
P
w
(
f
)
P
o
(
f
)
, parameter
λ ∈
R
+
.
Output:
Restored specimen
o
(
x
)
.
1: Calculate PSF
h
(
x
)
(Eq.
4.5
)andthe
OTF(
f
)
by FFT,
2: Deconvolve:
o
(
x
)
by Eq. (
4.22
).
3: Sub-space projection (real):
o
(
x
)
←
Real(
o
(
x
))
.
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