Biomedical Engineering Reference
In-Depth Information
In Eq. (
4.27
), if we choose
h
(
x
) to be a Dirac function, then the minimization
of the above energy function gives a denoised image in the successive iterations,
while a non-Dirac PSF leads to simultaneous deconvolution and denoising. The
main challenge in minimizing this functional is the strong nonlinearity of the
Poisson based negative log-likelihood functional in Eq. (
4.16
). We use the 3-D
numerical implementation of the regularization algorithm for CLSM as described
by Dey et al. [
24
], Pankajakshan [
52
], Pankajakshan et al. [
54
]. There are many
approaches proposed in literature to minimize the above cost function. In the
following paragraph, we will discuss one such method based on the RL algorithm
with TV regularization
As in the case of the RL algorithm, the Euler-Lagrange equation for minimizing
J
(
o|i
),givenbyEq.(
4.27
), with respect to
o
(
x
) is
− λ
div
=0
,
i
(
x
)
∇o
(
x
)
∗ h
∗
(
1
−
x
)
(4.28)
(
h ∗ o
)(
x
)+
b
(
x
)
|∇o
(
x
)
|
ε
where
h
∗
(
) and div stands for the
divergence operator. Equation (
4.28
) can be solved for the object
o
by adopting
the RL algorithm with TV regularization:
x
) is the Hermitian adjoint operation on
h
(
x
)=
)
o
(
n
)
(
i
(
x
)
x
)
o
(
n
+1)
(
)
∗ h
∗
(
− λ
div
∇
o
(
n
)
(
x
)
|∇o
(
n
)
(
x
)
|
ε
,
(4.29)
x
x
·
(
h ∗
o
(
n
)
)(
x
)+
b
(
x
1
where (
) denotes the Hadamard multiplication (component-wise) and
n
, the itera-
tion number for the deconvolution algorithm. We stop the deconvolution algorithm
as soon as the difference measure between two successive iterations is smaller than
·
a specified threshold. The term div
∇
|
ε
can be numerically
discretized with the central finite differences and the
minmod
scheme [
23
,
52
].
Unlike in the RL algorithm, the introduction of regularization does not always
ensure that the successive estimates are positive even if the initial estimate is
positive. But, a fluorescence distribution is known to be greater than or equal to
zero at all points in space. In order to constrain the reconstructed sample to positive
values, we force values below a minimum intensity to that minimum intensity after
each iteration step. Another alternative suggested in [
83
] is to operate on the real-
valued square-root of the image rather than the intensity value. These steps are given
in the Algorithm
7
.
In order to test this algorithm, we simulated a fluorescent bead of size 100 nm
with a radial sampling of 47 nm and axial sampling of 166 nm to respect the Nyquist
criterion. This bead was assumed to be observed using a 1
.
3
/
40X oil objective lens
and then corrupted by the photon statistics with a
γ
=5. The three line profile
intensities of the original object, the blurred object, and the observed object are
plotted in the Fig.
4.10
along with the intensity profile of the restored bead after
using the RL-TV algorithm. We noticed that the FWHM of the observed bead is
much larger than that of the original intensity, while the restoration has almost the
same intensity and FWHM as that of the original.
o
(
n
)
(
o
(
n
)
(
x
)
/|∇
x
)
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