Among the various iterative methods that can be implemented
for finding an approximation to
the image (or the object) to be reconstructed,
there exists a very slow algorithm
which is based on a matching pursuit strategy.
As will be clarified in this section,
this algorithm is nothing but
an aborted version of a particular algorithm
minimizing q on 
(see the introduction of Sect. 1.1).
The corresponding iterative process must never be used in practice
for solving the problem.
Its slow convergence may however be of interest
for initializing the choice of
the object representation space E .
It is therefore important to analyse its principle
(Sect. 2.1),
and in particular, to show that CLEAN is an algorithm of this type
(Sect. 2.2).
Let 
be the virtual data vector
corresponding to the object atom 
(cf. Sect. 1.1),
and
be the projection (operator) of 
onto
the space generated by 
:
where
The guiding idea
is to determine the projection of 
onto 
via
the elementary projections 
.
Let us consider the iteration
in 
:
is a relaxation parameter to be defined.
At each iteration,
is chosen so that
If
,
then
(the projection of 
onto 
)
and the problem is solved.
Let us set
and
.
As
,
we have from Eq. (21):
It follows that
hence
Likewise,
.
Provided that 
lies in the open interval (0, 2) ,
is strictly positive.
Then,
.
The sequence 
,
where 
,
therefore converges towards some nonnegative
number 
.
As shown in Appendix 2,
proves to be equal to 0.
As a result,
.
The iterates (21) then converge
towards 
.
The maximal value of 
is attained for
.
To increase the convergence speed of the projection 
,
may be set equal to this optimal value.
The corresponding algorithm,
,
is nothing but a traditional matching pursuit process
(see Mallat & Zhang 1993).
As 
,
we have from Eq. (19),
hence
where
.
The relaxed matching pursuit iteration (21)
can therefore be written in the form
Clearly, this sequence is the image by A
of the object sequence (in 
):
where
According to its definition,
the residue 
is obtained via the iteration:
As, from Eqs. (24) and (26),
,
we have (cf. Eq. (20)):
On setting (cf. Eq. (1))
it follows from Eq. (23) that 
is obtained
through the iteration:
Provided that 
lies in the open interval (0, 2) ,
the iterates 
converge towards
the minimal value of q on 
.
Sequence (25) then converges towards a solution 
of the problem; 
is the
unique solution 
,
if and only if 
is a one-to-one map.
In our formulation of ÇLEAN|,
which essentially follows that of Högbom (1974),
the object space is the 
space 
introduced in Sect. 1.2.
The vectors 
are then
translated versions of the clean beam
(see Fig. 3 (click here)b).
More precisely, the elements of 
are the clean beams 
centred on the nodes of
the
"clean box" 
:
The data space 
coincides with the experimental data space 
,
and A with
the experimental Fourier sampling operator:
on 
.
As the image to be reconstructed is defined as the
convolution of the original object by the clean beam (Eq. (13)),
the data vector 
must be defined as the experimental data vector 
damped by the Fourier transform of the clean beam:
(cf. Eq. (14)).
We then have 
with (cf. Eqs. (1), (12) and (17)):
Figure 5: Image reconstruction via CLEAN with 
;
a) dirty beam;
b) dusty map;
c) image to be reconstructed (Fig. 4 (click here));
d) clean map for 
(the definition of the fit criterion 
is given in Eq. (35)).
In the conditions of this simulation
(see Fig. 3 (click here)),
the optimal fit threshold of CLEAN is of the order of 1.75.
For a lower threshold,
the support of the clean map
is no longer contained in that of the image to be reconstructed.
In the framework of the analysis presented in this paper,
the residual maps 
or 
must not be added to the clean map
As explicitly shown in Appendix 3,
the "dirty map" is the map of the scalar components
of 
in the basis of the elementary particles 
.
In this context,
may be referred to as the "dusty map".
For clarity, we set
and
.
Likewise,
the action of 
corresponds to a
"discrete convolution" by the "dirty beam" 
:
(the precise definition of this operation is given in Appendix 3).
Thus, from Eq. (20),
the parameters 
are all equal to:
The relaxed matching pursuit iteration (25) can then be written in the form
where (from Eq. (26))
Clearly, 
(the map of the 
)
is nothing but the "discrete intercorrelation" of 
with CB.
The residue 
and the quadratic errors 
are respectively obtained via the iterations (27) and (28):
and
Note that
.
In the classical presentation of ÇLEAN|,
the convolution by the clean beam is performed
a posteriori, whence some small differences in
these iterations
(cf. Appendix 4).
In particular,
in the version of ÇLEAN| presented here,
is chosen (at each iteration) so that
.
The process is interrupted as soon
as 
is less than a threshold value
related to the level of the noise in the Fourier domain.
In our implementation of ÇLEAN|,
we introduce the "fit criterion" (cf. Eqs. (18) and (29)):
As soon as 
is less than 2 (for example),
the matching pursuit process is interrupted;
is the corresponding "clean map".
In the simulation presented in Fig. 5 (click here),
we show the clean map corresponding to the fit threshold 2 .
The relaxation parameter 
was set equal to 0.2 ,
and the clean box was defined as the support of 
at a lower level of resolution (twice as low).
In the conditions of this simulation,
the optimal fit threshold of CLEAN is of the order of 1.75.
For a lower threshold, the support of the clean map
is no longer contained in that of the image to be reconstructed.
Let E be the object representation space
generated by the 
selected by ÇLEAN|.
Clearly,
the clean map 
does not minimize 
on E .
The same matching pursuit
algorithm (with 
)
can be confined to E for performing
the complete minimization on this space.
This corresponds to the principle of what is referred to as "Window ÇLEAN|"
(Schwarz 1978).
The algorithms presented in Sects. 3 and 4
are much more efficient for this purpose,
but as specified in Sect. 5,
they only reveal that
(in situations of astrophysical interest)
the solution thus obtained is without any interest:
the problem is ill-conditioned.