Biomedical Engineering Reference
In-Depth Information
improvement by deconvolution as the addition of an auxiliary computational lens
because the system works almost like a virtual lens. The aim of this section is
to firstly describe the challenges, and then the methodological developments to
highlight the importance of post-processing of images from confocal microscopes.
4.2.2.1
Challenges
If the process of imaging (the forward problem) is expressed by convolving a given
specimen with a PSF (Eq.
4.7
), deconvolution (the inverse problem) aims to obtain
the specimen from the image. This assumes that the PSFs used in the forward and
in the inverse problems are the same, and also that both the signal and the PSF data
are free from noise.
As mentioned in the previous section, the PSF is not known a priori and needs to
calculated or measured. In addition, as the convolution is a linear operation (Eqs.
4.7
and
4.8
), deconvolution involves multiplying by the inverse of the matrix
H
(if it
H
−
1
exists, applying it to the noisy
observations will only amplify the noise as the eigenvalues in
exists). The difficulty lies in the fact that even if
are very small
[
11
]. For example, if the observation in matrix notation is written as
H
i
=
Ho
+
w
,
where
w
is the AWGN term, then, the specimen function can be estimated as ˆ
o
≈
H
−
1
i
−
H
−
1
w
is ill-conditioned,
7
the term
H
−
1
w
.As
H
will drastically damage
the unknown image
we are looking for by amplifying the noise.
The ill conditioning of the matrix
o
H
is due to very small eigenvalues of
H
H
−
1
. The fact that
that implies very high eigenvalues of
has always very small
eigenvalues (when they are non-zero) comes from the fact that
H
H
models a low pass
filtering. As
is block circulant, it is diagonalized by the 3-D Fourier transform,
and the eigenvalues of
H
matrix are exactly given by the 3-D Fourier transform of
the PSF. As the PSF is a low pass filter, the eigenvalues corresponding to its high
frequency coefficients are quite null or exactly null. So, there are two cases that are
probable:
H
•
The PSF cuts off certain spatial frequencies, and
is not invertible. In this case,
the lost frequency information cannot be regained by inversion.
H
•
There are only non-null PSF spatial frequencies, but these are very small at
the higher frequencies. In this case, the inversion is dominated by the noise
amplification.
In the last case, the noise amplification can be prevented and the lost high frequen-
cies can be restored, by introducing a priori constraints (for example by appropriate
penalty functions or information on the search specimen
o
as in Sect.
4.2.3
), during
the inversion process.
7
A given problem is said to be ill-conditioned when it has a high condition number or the solution
changes by a very significant amount in proportion to very small changes in the input data.
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