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A quite natural way of discretizing Eq. (1) is as follows:
where 
and 
are the 
image and object, respectively; the
are quadrature weights.
Eq. (3) can also be
written in the form 
where now 
is a 
matrix.
It can be diagonalized by the Discrete Fourier Transform (DFT) and becomes
where 
, 
, and 
are the DFT's of 
, 
,
and 
, respectively. The matrix is, in general, ill-conditioned and
therefore the solution
of (3) 
, as well as the solution of (4),
are affected by numerical instability.
Then one has to use regularized inversion methods in order to obtain
approximate and sensible solutions.
The methods we have considered are the following:
- (
)
- Tikhonov regularization (TIKH),
- (
)
- Landweber iteration (LW),
- (
)
- Landweber iteration with imposed positivity (LWPOS),
- (
)
- Conjugate gradients (CG),
- (
)
- Conjugate gradients with imposed positivity (CGPOS), and
- ()
- Richardson-Lucy method (R-L).
All these methods contain a free parameter which is the so-called
regularization parameter; in the case of method () this is a smoothing of
the higher spatial frequencies; for the methods () - (), it is the number
of iterations. In the case of a simulated experiment, the typical behavior of
the approximate solutions as a function of the free parameter is the
following: if we compute the distance between the approximate solution and the
known true solution (for instance the Euclidean norm of their difference) and
we consider its behavior as a function of the free parameter, then this
distance will exhibit a minimum. The value of the parameter corresponding to
this minimum can be considered as the optimum one since it provides the best
approximation to the solution of the problem. Obviously, in the case of real
data, the solution of the problem is not known and therefore this method can
not be used for the estimation of the free parameter. For this reason several
criteria have been investigated for its choice (Bertero 1988). However, in
general they are not very practical and very often the choice they provide is
not satisfactory, so that one has to use some empirical criterion which depends
on the problem one is investigating. In the case of simulated data, we will
choose the value of the parameter corresponding to the minimum of the distance
between the approximate and the true solution.
For the convenience of the reader
we recall that the approximate solution given by method () is
where 
is the regularization parameter. More explicitly in the case of
Eq. (4) we have
In the case of method () the iteration scheme is given by
where 
is the stationary relaxation parameter and
We point out that this iterative method is equivalent to a filtering since
the final result of 
iterations can be explicitly written and in Fourier
space it is given by
This expression cannot be used in method () because the negative part of
the reconstruction of 
obtained from (7) must be zeroed at the end
of each iteration stage. Since 
must be computed in Fourier space at each
iteration, two FFTs are required: the first for computing 
; the
second one for coming back to the physical space in order to impose positivity.
On the practical side, for both methods () and () the relaxation
parameter 
can be set as follows:
where 
.
Finally, the conjugate gradients iteration scheme is
where
with 
, 
and the
non-stationary relaxation parameter 
.
This iteration scheme can also be easily written in Fourier space.
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Up: Assessment of Methods Used
Previous: Introduction