Deconvolution and Denoising for Confocal Microscopy - Modeling in Computational Biology and Biomedicine - page 130

equation could be inverted with the knowledge of the scanning system properties
(here the PSF h ( )).

When the imaging system has been a priori calibrated, there is almost negligible
offset in the detector and the illumination is uniform; i.e., no misalignment in the
laser. For an aberration-free model, the convolution is LSI, and can be written in a
matrix notation,

x

( h ∗ o )( x )= Ho , (4.8)

where the matrix is block-circulant (if periodic boundary conditions are used)

[ 11 ]. It can be shown that if the matrix is block-circulant and when

H

is a vector
containing all the pixel in the lexicographical order, it is diagonalized by the
Discrete Fourier transform (DFT). In addition, we will assume that the encircled or
volumetric energy is constant with respect to the blurring process. This assumption
is valid because blurring is a passive process and hence

o

)=
x ∈

o ( x

( h ∗ o )(

x

) .

(4.9)

x ∈ Ω s Ω s

Imaging Statistics. The CLSM is a low-photon imaging technique and it's PMT
works as a photon counter. That is, it collects the photons, converts the electrical
impulse to a discrete intensity value, after analog to digital conversion [ 58 ]. There
are several parameters that influence the final intensity. These variables include the
quantum yield, 6 bleaching [ 25 ], saturation of the fluorescent molecules, the limited
transmittance of signal throughout the entire optical path, and the QE of the detector.
In confocal microscopy, the detected signal is “quasi-random”, as statistics
determine both how many photons are emitted on any particular trial but also what
fraction of these are detected. If we denote by N , the total number of photons
collected, then the counting process follows a Poisson distribution so that [ N

∼

( ( q d ))],where q s is the total flux due to the specimen fluorescence, while

q d is the dark current flux [ 52 ]. Due to the pixel-wise scanning of the specimen
using a laser, the image acquisition process is slow compared to wide-field based
methods in which the entire image is acquired at once. This can be compensated by
increasing the scan speed, but will inevitably lead to the reduction in the photons
per pixel producing noisy images. The pinhole aperture is a key cause why finally
very few photons reach the PMT from the specimen. For example, in Fig. 4.3 b,
the maximum intensity drops to about 3,000 IU while in Fig. 4.3 a the maximum
intensity is about 20,000 IU. As the intensity is linearly proportional to the number
of photons collected, low intensity means lower photon counts. Although the flux
q s can be increased by averaging, for many applications this is not recommended as
multiple scans of the same region are required, resulting in increased phytotoxicity
(the degree of toxic effect produced by chemical compounds) and photo-bleaching.

P ( q s )+ P

6 The quantum yield of a radiation-induced process is the number of times that a photon is
emitted per photon absorbed by the system. This is essentially the emission efficiency of a given
fluorophore.