Biomedical Engineering Reference
In-Depth Information
4.2.2.2
Methodological Developments
The earliest application of a deconvolution algorithm in microscopy was the nearest-
neighbor algorithm developed by Agard et al. [
2
]. It is based on the assumption
that the most of the blur within the focal plane is due to the light emitted from
its two neighboring planes. The focal plane intensities are restored by comparing
the observed intensity with these two planes and using a parameter to control the
amount that each contributes. The advantage is that computation is fast (only 2
N
z
number of 2-D forward and inverse Fourier transforms). The disadvantage is that,
in the presence of an analytical PSF/OTF model, it only uses partial information,
and hence the deconvolution could be said to be axially incomplete for the volume.
Since 1989, there were several developments in this field and most of them could
be overall classified under two large categories: direct inversion approaches and
iterative approaches. A survey on deconvolution algorithms was carried out earlier
by Cannell et al. [
14
], Meinel [
47
], Sarder and Nehorai [
64
] and more recently
in [
53
]. For the convenience of the reader, we have listed in Table
4.1
the most
significant methods that have been applied to fluorescence microscopy. In this
chapter, we will present some of these approaches, but by adopting a general
probabilistic framework.
If the number of photons are large, we can assume that observation model
follows a Gaussian distribution hypothesis as in Eq. (
4.10
), and the likelihood of
the observation
i
(
x
) given the specimen
o
(
x
) is:
N
x
N
y
N
z
2
Pr(
i|o
)=
1
2
πσ
n
exp
,
2
−
i
(
x
)
−
(
h ∗ o
)(
x
)
(4.12)
2
σ
n
∈Ω
s
X
where
σ
n
is the variance of the Gaussian noise. The negative logarithm of the
likelihood function in Eq. (
4.12
)is:
2
,
x
∈ Ω
s
,
J
obs
(
o
(
x
)) =
i
(
x
)
−
(
h
(
x
)
∗ o
(
x
))
(4.13)
where the terms independent of
i
and
o
were dropped from Eq. (
4.12
). Here,
J
obs
:
Ω
s
is a measure of fidelity to the data and it has the role of pulling
the solution towards the observation data. It specifies as well the penalty paid by the
system for producing an incorrect estimate of the scene. The specimen function can
be estimated by maximizing the likelihood in Eq. (
4.12
) or equivalently minimizing
the function
→
R
)) in Eq. (
4.13
). We remark that the minimization of this cost
function is equivalent to the minimization of the Mean-squared error (MSE). It is
straightforward to show that the gradient of this cost function can be written as:
J
obs
(
o
(
x
∇
o
J
obs
=
h
∗
(
− h
∗
(
x
)
∗ h
(
x
)
∗ o
(
x
)
x
)
∗ i
(
x
)
,
(4.14)
where
h
∗
(
x
)=
h
(
−
x
) is the Hermitian adjoint of
h
(
x
). As the functional
J
obs
(
o
(
x
))
is convex w.r.t
o
(
x
), a minimum of the function
J
obs
(
o
(
x
)) is calculated at the point
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