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2. Reconstruction via matching pursuit methods

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 tex2html_wrap_inline3198 (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).

2.1. Reconstruction principle

Let tex2html_wrap_inline3212 be the virtual data vector corresponding to the object atom tex2html_wrap_inline3214 (cf. Sect. 1.1), and tex2html_wrap_inline3216 be the projection (operator) of tex2html_wrap_inline3218 onto the space generated by tex2html_wrap_inline3220:

The guiding idea is to determine the projection of tex2html_wrap_inline3222 onto tex2html_wrap_inline3224 via the elementary projections tex2html_wrap_inline3226.

Let us consider the iteration in tex2html_wrap_inline3228:
tex2html_wrap_inline3230 is a relaxation parameter to be defined. At each iteration, tex2html_wrap_inline3232 is chosen so that

If tex2html_wrap_inline3234, then tex2html_wrap_inline3236 (the projection of tex2html_wrap_inline3238 onto tex2html_wrap_inline3240) and the problem is solved.

Let us set tex2html_wrap_inline3242 and tex2html_wrap_inline3244. As tex2html_wrap3312, we have from Eq. (21):
It follows that

Likewise, tex2html_wrap_inline3248. Provided that tex2html_wrap_inline3250 lies in the open interval (0, 2) , tex2html_wrap3314 is strictly positive. Then, tex2html_wrap3316. The sequence tex2html_wrap_inline3258, where tex2html_wrap_inline3260, therefore converges towards some nonnegative number tex2html_wrap_inline3262. As shown in Appendix 2, tex2html_wrap_inline3264 proves to be equal to 0. As a result, tex2html_wrap3318 tex2html_wrap3320. The iterates (21) then converge towards tex2html_wrap_inline3270.

The maximal value of tex2html_wrap_inline3272 is attained for tex2html_wrap3322. To increase the convergence speed of the projection tex2html_wrap3324, tex2html_wrap_inline3276 may be set equal to this optimal value. The corresponding algorithm, tex2html_wrap_inline3278, is nothing but a traditional matching pursuit process (see Mallat & Zhang 1993).

As tex2html_wrap_inline3280, we have from Eq. (19),
where tex2html_wrap_inline3282. 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 tex2html_wrap_inline3286):

According to its definition, the residue tex2html_wrap_inline3288 is obtained via the iteration:

As, from Eqs. (24) and (26), tex2html_wrap_inline3290, we have (cf. Eq. (20)):

On setting (cf. Eq. (1))
it follows from Eq. (23) that tex2html_wrap_inline3292 is obtained through the iteration:

Provided that tex2html_wrap_inline3294 lies in the open interval (0, 2) , the iterates tex2html_wrap_inline3298 converge towards the minimal value of  q on tex2html_wrap_inline3302. Sequence (25) then converges towards a solution tex2html_wrap_inline3304 of the problem; tex2html_wrap_inline3306 is the unique solution tex2html_wrap_inline3308, if and only if tex2html_wrap_inline3310 is a one-to-one map.

2.2. Presentation of CLEAN as a matching pursuit algorithm

In our formulation of ÇLEAN|, which essentially follows that of Högbom (1974), the object space is the tex2html_wrap3438 space tex2html_wrap_inline3328 introduced in Sect. 1.2. The vectors tex2html_wrap_inline3330 are then translated versions of the clean beam tex2html_wrap3440 (see Fig. 3 (click here)b). More precisely, the elements of tex2html_wrap_inline3334 are the clean beams tex2html_wrap_inline3336 centred on the nodes of the "clean box" tex2html_wrap3444:

The data space tex2html_wrap_inline3340 coincides with the experimental data space tex2html_wrap_inline3342, and  A with the experimental Fourier sampling operator: tex2html_wrap3448 on tex2html_wrap_inline3348. As the image to be reconstructed is defined as the convolution of the original object by the clean beam (Eq. (13)), the data vector tex2html_wrap_inline3350 must be defined as the experimental data vector tex2html_wrap_inline3352 damped by the Fourier transform of the clean beam: tex2html_wrap_inline3354 (cf. Eq. (14)). We then have tex2html_wrap_inline3356 with (cf. Eqs. (1), (12) and (17)):

Figure 5: Image reconstruction via CLEAN with tex2html_wrap_inline3358; a) dirty beam; b) dusty map; c) image to be reconstructed (Fig. 4 (click here)); d) clean map for tex2html_wrap_inline3360 (the definition of the fit criterion tex2html_wrap_inline3362 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 tex2html_wrap_inline3364 or tex2html_wrap_inline3366 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 tex2html_wrap_inline3368 in the basis of the elementary particles tex2html_wrap_inline3370. In this context, tex2html_wrap_inline3372 may be referred to as the "dusty map". For clarity, we set tex2html_wrap3452 and tex2html_wrap3454. Likewise, the action of tex2html_wrap_inline3378 corresponds to a "discrete convolution" by the "dirty beam" tex2html_wrap_inline3380: tex2html_wrap3458 (the precise definition of this operation is given in Appendix 3). Thus, from Eq. (20), the parameters tex2html_wrap3460 are all equal to:

The relaxed matching pursuit iteration (25) can then be written in the form
where (from Eq. (26))

Clearly, tex2html_wrap_inline3390 (the map of the tex2html_wrap_inline3392) is nothing but the "discrete intercorrelation" of tex2html_wrap_inline3394 with CB.

The residue tex2html_wrap_inline3396 and the quadratic errors tex2html_wrap_inline3398 are respectively obtained via the iterations (27) and (28):

Note that tex2html_wrap_inline3402.

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, tex2html_wrap_inline3404 is chosen (at each iteration) so that tex2html_wrap3506.

The process is interrupted as soon as tex2html_wrap_inline3408 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 tex2html_wrap_inline3410 is less than  2 (for example), the matching pursuit process is interrupted; tex2html_wrap_inline3414 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 tex2html_wrap_inline3418 was set equal to  0.2 , and the clean box was defined as the support of tex2html_wrap_inline3422 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 tex2html_wrap_inline3426 selected by ÇLEAN|. Clearly, the clean map tex2html_wrap_inline3428 does not minimize tex2html_wrap_inline3430 on  E . The same matching pursuit algorithm (with tex2html_wrap3510) 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.

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