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5. How WIPE wipes CLEAN clean

We now have all the tools for analysing the weak points of CLEAN as well as the tricks of WIPE| allowing the corresponding difficulties to be overcome.

In situations of astrophysical interest, CLEAN is implemented with a value of the relaxation parameter tex2html_wrap_inline3982 much less than 1 (say  0.2 ). The basis vectors tex2html_wrap_inline3986 selected in the matching pursuit process then define an acceptable object representation space  E . Unfortunately, the problem is often ill conditioned; tex2html_wrap_inline3990 is a one-to-one map, but its condition number is very large. For example, in the simulation presented in Fig. 5 (click here)d, tex2html_wrap_inline3992 is equal to  45.08 . As a result, tex2html_wrap_inline3996 is a very perturbed version of the clean map. This is unsatisfactory. Indeed, in this situation, the clean map can only be regarded as an image for which the fit criterion tex2html_wrap_inline3998 (introduced in Eq. (35)) is of the order of the threshold value (say 2). In other words, the clean map must be accepted as it is, without any reference to a stable optimization process. The interpretation of the results may then be doubtful. As indicated in Sect. 5.1, the regularization principle of WIPE| remedies this lack of robustness, but the regularized version of CLEAN thus obtained is still different from WIPE|. Indeed, CLEAN  has another weak point: the boundaries of the structuring entities of the image may not be correctly represented in the clean map. In such situations, the matching pursuit strategy of CLEAN is not well suited for solving the problem. For example, in the particular case of the Fourier data of our simulation, the best possible fit corresponds to a value of tex2html_wrap_inline4000 of about  0.9 (see Fig. 7 (click here) further on). Even with a good support constraint (a reasonable "clean box"), ÇLEAN| cannot reach this fit threshold with a satisfactory representation of the image support. In the same situation, as shown in Sect. 5.2, WIPE| reaches this threshold without tex2html_wrap4004.

5.1. How WIPE regularizes CLEAN

In the basic version of CLEAN presented in Sect. 2.2, the functions tex2html_wrap_inline4014 are linear combinations of atoms tex2html_wrap_inline4016. In the sense defined in Sect. 1.2, the "energy" of the Fourier transform of each atom tex2html_wrap_inline4018 is concentrated in tex2html_wrap_inline4020 (cf. Eq. (15)). The intrinsic instability of CLEAN is related to the fact that this property does not necessarily hold for any linear combination of such atoms. This difficulty arises any time the distances between these atoms are much smaller than the corresponding resolution limit. This is precisely the case when dealing with extended objects and a small relaxation parameter tex2html_wrap_inline4022.

To overcome this difficulty, WIPE| suggests that CLEAN should define the reconstructed image as the function minimizing on  E the functional
where (see Eq. (29))

The experimental criterion tex2html_wrap_inline4026 constrains tex2html_wrap_inline4028 (the model) to be consistent with the damped Fourier data, while the regularization criterion tex2html_wrap_inline4030 penalizes the high-frequency components of tex2html_wrap_inline4032. The elements of the regularization frequency list tex2html_wrap_inline4034 are located outside tex2html_wrap_inline4036, at the nodes of grid tex2html_wrap_inline4038 where

In the traditional version of ÇLEAN|, q reduces to tex2html_wrap_inline4042. The minimization of tex2html_wrap_inline4044 on  E then reveals the ill-conditioned character of the problem.

The regularized version of CLEAN corresponding to criterion (42) can be formulated in the theoretical framework presented in Sect. 1.1. To clarify this point, let us introduce the data vector:

This vector lies in a data space tex2html_wrap_inline4052 endowed with the scalar product:

The Fourier sampling operator A is then the operator:

According to Eqs. (42), (29) and (43), tex2html_wrap_inline4060 can then be effectively written in the form tex2html_wrap4102 (see Eq. (1)). The problem is then stated in terms of Fourier interpolation (Lannes et al. 1987a and 1994). This is why q is of the form tex2html_wrap_inline4066 with tex2html_wrap_inline4068. In this context, it is important to note that in the definition of tex2html_wrap_inline4070 (Eq. (29)), where tex2html_wrap_inline4072 is defined in Eq. (14), the weighting function tex2html_wrap_inline4074 takes into account the local redundancy of tex2html_wrap_inline4076 up to tex2html_wrap_inline4078 (Eq. (17)).

The algorithms described in Sects. 3 and 4 can be used for minimizing q on  E . The action of tex2html_wrap_inline4084 is that of a convolutor. The corresponding point spread function, the "dusty beam", has two components: the dirty beam and the "regularization beam". The latter is induced by the regularization list tex2html_wrap_inline4086. With regard to the dusty map, note that tex2html_wrap_inline4088 (cf. Sect. 2.2).

In the simulation presented in Fig. 6 (click here), we compare the clean map of Fig. 5 (click here)d (for which tex2html_wrap_inline4090) with the image provided by this opt(tex2html_wrap_inline4092). The condition number is now reasonable: tex2html_wrap_inline4094. As shown in the next subsection, the construction of the object representation space  E can, however, be refined.

Figure 7: Image reconstruction through WIPE|; a) image to be reconstructed; b) neat map (reconstructed image). The latter, for which tex2html_wrap_inline4098 and tex2html_wrap_inline4100 (and which was obtained without any clean box), is to be compared to image a) and to the maps presented in Fig. 6 (click here). The boundaries of the structuring entities of the image are now correctly restored, hence a better intensity distribution. The unreliable character of the oscillating perturbation along the main structuring entity of the reconstructed images is revealed by the image-eigenmode analysis provided by WIPE| (see Fig. 6 (click here) of Lannes et al. 1996)

5.2. How WIPE relaxes the matching pursuit process of CLEAN

In the regularized version of CLEAN described in Sect. 5.1, the calculation of the condition number tex2html_wrap_inline4176 requires the action of the projection tex2html_wrap_inline4178 at each iteration of algorithm 3. This projection is performed at the cost of the tex2html_wrap4478 iterations of algorithm 2. This first remark suggests that  E  should be redefined as the linear space generated by the elementary particles tex2html_wrap_inline4182 of all the atoms tex2html_wrap_inline4184 selected in the matching pursuit process of ÇLEAN|. The projection onto  E is then trivial since these elementary particles form an orthogonal set. As the resolution limit of the reconstruction process is then controlled by the regularizer tex2html_wrap_inline4188 (cf. Eq. (43)), the choice of such an object representation space proves to be very natural. Moreover, the definition of  E can then be refined by continuing (or even by conducting) the matching pursuit process at the level of the elementary particles. As specified below, this is what is precisely done in WIPE|.

Let D then be the subset of  G corresponding to the choice of tex2html_wrap_inline4196. (The elementary particles generating  E are centred on the nodes of tex2html_wrap4480.) We say that D  is the "discrete field (or support)" associated with the definition of  E . Depending on the particular problems to be solved, this discrete field may be fixed from the outset (for example, in an interactive manner), or constructed step by step in a matching pursuit strategy.

In this last case, which corresponds to the basic version of WIPE|, let us denote by tex2html_wrap_inline4206 the discrete field obtained at the end of the tex2html_wrap_inline4208 step of the construction of the object representation space. Let tex2html_wrap_inline4210 then be the solution of the problem in the corresponding object representation space tex2html_wrap_inline4212. In the basis of the elementary particles tex2html_wrap_inline4214 (the interpolation basis of tex2html_wrap_inline4216), the scalar components of the residue tex2html_wrap_inline4218 are the quantities:

According to the definition of tex2html_wrap_inline4220, these coefficients vanish on tex2html_wrap_inline4222 (see Eq. (2) with tex2html_wrap_inline4224, tex2html_wrap_inline4226). One then has to decide whether the current field has to be extended. The current values of tex2html_wrap_inline4228 and tex2html_wrap_inline4230 play an essential role in this decision. When the reconstruction procedure is not interrupted at this stage, WIPE| uses algorithm 3 for computing the solution of the problem in the object representation space relative to the union of tex2html_wrap_inline4232 with some set tex2html_wrap4482:

There exist many ways of selecting D' . All are based on the examination of the distribution of the coefficients tex2html_wrap_inline4238 outside tex2html_wrap_inline4240. For example, one may try to define  D' as a connected region containing the "pixel" tex2html_wrap_inline4244 for which the maximum of these coefficients is attained. The simplest choice is then to define D' as the discrete field of the atom  s centred on this pixel. With regard to the construction of the object representation space, the corresponding version of WIPE is then very similar to that of CLEAN.

In the matching pursuit steps where the field of the reconstructed image must be refined, it is natural to choose the nodes of D' along the boundaries of the structuring entities of the image. Let tex2html_wrap_inline4252 be the number of particles involved in the linear combination defining the neat beam  s (the number of nodes in tex2html_wrap_inline4256). In the basic version of WIPE|, the size of  D' , expressed in number of nodes, is defined as a fraction of tex2html_wrap_inline4260 (say tex2html_wrap_inline4262), and the selected nodes are those for which the coefficients tex2html_wrap_inline4264 are the largest above some given threshold (half of the maximal value, for example). The field of the image (or object) to be reconstructed can thus be obtained in a natural manner.

The construction of the object representation space is interrupted as soon as the fit criterion tex2html_wrap_inline4266 is sufficiently small, for instance, less than or of the order of  0.85 . The current field is then refined by a morphological smoothing of its connected entities. In this classical operation of mathematical morphology, the discrete support of the neat beam, tex2html_wrap_inline4270, is of course used as structuring element. The boundaries of the effective field of the "neat map" (the reconstructed image) are thus defined at the appropriate resolution. In particular, the connected entities of size smaller than that of tex2html_wrap_inline4272 are eliminated. As illustrated in Fig. 7 (click here), it is thus possible to reach the optimal value of tex2html_wrap_inline4274 ( 0.88 in the simulation under consideration) with a satisfactory representation of the image field.

Let E be the object representation space at the end of the action of WIPE|, and D  be the corresponding discrete field. There exists a variant of WIPE|, in which the object representation space is a particular subspace of  E , that generated by all the atoms tex2html_wrap_inline4284 whose discrete field is contained in  D . In the conditions of the simulation presented in Fig. 7 (click here), the corresponding solution is very close to that provided by WIPE|. As expected, the condition number is then slightly smaller (here, 3.36  instead of  3.83 ).

From the outset, the discrete field D may be taken equal to that of the clean box. One then uses the global version of WIPE| in which the nonnegativity constraint is imposed (cf. Sect. 4.3 of Lannes et al. 1996). At the end of the corresponding reconstruction process, the fit criterion tex2html_wrap_inline4294 is often smaller than its optimal value. As a result, the support of the image (or object) to be reconstructed is not well restored. A similar remark can be made for the Fourier synthesis methods in which the regularization principle is based on the concept of entropy. Moreover, the relative weights of the experimental and regularization criteria must then be carefully chosen (Cornwell 1983; Maréchal & Lannes 1997). The strategy adopted in the basic version of WIPE is therefore preferable; its implementation is simpler and more efficient.

The condition "tex2html_wrap_inline4296 of the order of 1 with tex2html_wrap_inline4298 less than say 5, with a sufficiently small value of tex2html_wrap_inline4300'' often suffices to ensure a good solution to the problem, but strictly speaking, this is not a sufficient condition. The complete control must be based on a multiresolution strategy. The corresponding developments will be presented in a forthcoming paper.

Appendix 1. Notion of condition number

For any tex2html_wrap_inline4302 in E , we have from the definitions of tex2html_wrap_inline4306 and tex2html_wrap_inline4308 given in Eq. (7):

For tex2html_wrap_inline4310, the first inequality gives
whereas for tex2html_wrap_inline4312 the second yields

By combining these inequalities, it follows that

The square root of tex2html_wrap_inline4314 is referred to as the condition number of tex2html_wrap_inline4316.

Appendix 2. Convergence property

Using the notation introduced in Sect. 2.1, we have
i.e., from Eq. (19):

Thus (cf. Eq. (22)),

where tex2html_wrap_inline4318. As the relaxation parameter tex2html_wrap_inline4320 is supposed to lie in the open interval (0, 2) , C  is strictly positive. As shown in remark A2, there exists a positive constant C' such that for all z in tex2html_wrap_inline4330, we have:

As a result,
with tex2html_wrap_inline4332.

Let us now assume that tex2html_wrap_inline4334 is different from 0. There then exists n such that




This is impossible, since tex2html_wrap_inline4342 must be greater than tex2html_wrap_inline4344. Consequently, tex2html_wrap4484.

REMARK| A2. The property in question can be established as follows. Consider the operator on tex2html_wrap_inline4348:

For any z and z' in tex2html_wrap_inline4354, we have:

This identity shows that R is self-adjoint. Moreover, as
the condition tex2html_wrap4486 implies tex2html_wrap4488; R is therefore positive definite. The fact that tex2html_wrap_inline4364 is of finite dimension then implies that the smallest eigenvalue of R is strictly positive. Consequently, for any tex2html_wrap4490,

It follows immediately that
with tex2html_wrap_inline4370.

Appendix 3. Dirty map and dirty beam

We first show that the dirty map is the map of the scalar components of tex2html_wrap_inline4372 in the basis of the elementary particles tex2html_wrap_inline4374. According to Eqs. (11) and (10), tex2html_wrap_inline4376 can be expanded in the form tex2html_wrap4492, where

it then follows from Eq. (12) that
hence, since tex2html_wrap_inline4380:

This explicitly shows that tex2html_wrap_inline4382 can be identified with the dirty map (see for example Fig. 5 (click here)b).

The action of tex2html_wrap_inline4384 corresponds to a "back Fourier sampling operation". The dirty map looks like the inverse Fourier transform of tex2html_wrap_inline4386, but from a mathematical point of view, it isn't. Indeed, tex2html_wrap_inline4388 is a vector in the experimental data space tex2html_wrap_inline4390 and not the distribution tex2html_wrap_inline4392. When considering the basic versions of CLEAN and WIPE|, this distinction may seem to be a "mathematical stylishness", but this is not the case, for example, in multifrequency Fourier synthesis (see the context of Eq. (68) in Lannes et al. 1996).

Let us now consider the action of tex2html_wrap_inline4394 on any tex2html_wrap_inline4396. Setting tex2html_wrap_inline4398, and expanding tex2html_wrap_inline4400 and tex2html_wrap_inline4402 in the forms
we have:

By using the same arguments as above, it then follows that tex2html_wrap_inline4404 can be written in the form

Note that tex2html_wrap_inline4406 and tex2html_wrap4506. Let G' be the grid twice as large as  G :

The map of the coefficients tex2html_wrap_inline4414 on  G' defines what is referred to as the dirty beam DB (see for example Fig. 5 (click here)a). An expression such as
then denotes the vector (lying in tex2html_wrap_inline4420) whose scalar components are given by the discrete convolution:

As a result, in the general case where the nonzero components of tex2html_wrap_inline4422 are distributed all over grid  G , the operation tex2html_wrap_inline4426 is performed by implementing the FFT algorithm on grid  G' .

When N is large and the experimental frequency list very long, the direct calculation of the dirty map and the dirty beam may be very time-consuming. To save computer time, it is then preferable to use appropriate Fast Fourier Sampling techniques. The complete description of these FFS algorithms is given in Sect. 3 of Lannes et al. (1996).

Appendix 4. On the traditional version of CLEAN

In the classical presentation of ÇLEAN| (Högbom 1974), the Fourier data are not damped by tex2html_wrap_inline4432, and the convolution by the clean beam is performed a posteriori. More precisely, the successive clean maps of the traditional version of CLEAN are given by the convolutions
where the iteration in tex2html_wrap_inline4436 is defined by the formula:
tex2html_wrap_inline4438 denotes the scalar component of the "experimental residue"
at pixel tex2html_wrap_inline4442, and tex2html_wrap_inline4444 that of the dirty beam at the origin:

The relaxation parameter tex2html_wrap_inline4446 is referred to as the "loop gain". At each iteration, tex2html_wrap_inline4448 is chosen so that

The experimental residue tex2html_wrap_inline4450 is obtained iteratively according to the formula:

Note that tex2html_wrap_inline4452.

The objective of the iteration in tex2html_wrap_inline4454 is to minimize the quadratic functional:

This iteration is nothing but the matching pursuit iteration (25) in which the vectors tex2html_wrap_inline4456 are basis vectors tex2html_wrap_inline4458, tex2html_wrap_inline4460 lying in the clean box (cf. Eqs. (20) and (26); see also Appendix 3). Let us note in passing that in the traditional version of ÇLEAN|, "these basis vectors are more or less dealt with as tex2html_wrap_inline4462 functions'' centred on the nodes of grid  G .

As far as the connection with our formulation of CLEAN is concerned, the related iterations in tex2html_wrap_inline4466 and r are of the form

These iterations slightly differ from those introduced in our presentation of CLEAN (Eqs. (31) and (33)). The main difference is that the selected successive pixels tex2html_wrap_inline4472 are not necessarily the same: in our version of ÇLEAN|, tex2html_wrap_inline4474 is chosen, at each iteration, so that tex2html_wrap_inline4476 (see Eq. (32)). According to the analysis presented in Sect. 5, the weak points of CLEAN related to its intrinsic instability unfortunately remain the same. As the objective of CLEAN is to find an approximation to the object function convolved by the clean beam, it is therefore more natural to consider that, basically, ÇLEAN| is the matching pursuit process described in Sect. 2.2.

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