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).
be the virtual data vector
corresponding to the object atom (cf. Sect. 1.1),
be the projection (operator) of onto
the space generated by :
The guiding idea is to determine the projection of onto via the elementary projections .
Let us consider the iteration
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
we have from Eq. (21):
It follows that
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).
we have from Eq. (19),
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 ):
According to its definition,
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
"clean box" :
The data space
coincides with the experimental data space ,
and A with
the experimental Fourier sampling operator:
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
in the basis of the elementary particles .
In this context,
may be referred to as the "dusty map".
For clarity, we set
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),
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):
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.