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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:
All these methods contain a free parameter which is the so-called regularization parameter; in the case of method (
- (
)
- 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).
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.