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 much less than 1 (say 0.2 ). The basis vectors selected in the matching pursuit process then define an acceptable object representation space E . Unfortunately, the problem is often ill conditioned; 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, is equal to 45.08 . As a result, 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 (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 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 .
In the basic version of CLEAN presented in Sect. 2.2, the functions are linear combinations of atoms . In the sense defined in Sect. 1.2, the "energy" of the Fourier transform of each atom is concentrated in (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 .
To overcome this difficulty,
suggests that CLEAN
should define the reconstructed image as the function
minimizing on E the functional
where (see Eq. (29))
The experimental criterion
constrains (the model) to be consistent
with the damped Fourier data,
while the regularization criterion
penalizes the high-frequency components of .
The elements of the regularization frequency list
are located outside ,
at the nodes of grid where
In the traditional version of ÇLEAN|, q reduces to . The minimization of 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
endowed with the scalar product:
The Fourier sampling operator A is then the operator:
According to Eqs. (42), (29) and (43), can then be effectively written in the form (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 with . In this context, it is important to note that in the definition of (Eq. (29)), where is defined in Eq. (14), the weighting function takes into account the local redundancy of up to (Eq. (17)).
The algorithms described in Sects. 3 and 4 can be used for minimizing q on E . The action of 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 . With regard to the dusty map, note that (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 ) with the image provided by this opt(). The condition number is now reasonable: . 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 and (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)
In the regularized version of CLEAN described in Sect. 5.1, the calculation of the condition number requires the action of the projection at each iteration of algorithm 3. This projection is performed at the cost of the iterations of algorithm 2. This first remark suggests that E should be redefined as the linear space generated by the elementary particles of all the atoms 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 (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 . (The elementary particles generating E are centred on the nodes of .) 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
the discrete field
obtained at the end of the step
of the construction of the object representation
Let then be
the solution of the problem in the corresponding object representation
In the basis of the elementary particles
(the interpolation basis of ),
the scalar components of the residue
are the quantities:
According to the definition of ,
these coefficients vanish
(see Eq. (2) with , ).
One then has to decide
the current field has to be extended.
The current values of and
play an essential role in this decision.
When the reconstruction procedure is not interrupted at this stage,
uses algorithm 3
for computing the solution of the problem in
the object representation space relative to
the union of
with some set
There exist many ways of selecting D' . All are based on the examination of the distribution of the coefficients outside . For example, one may try to define D' as a connected region containing the "pixel" 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 be the number of particles involved in the linear combination defining the neat beam s (the number of nodes in ). In the basic version of WIPE|, the size of D' , expressed in number of nodes, is defined as a fraction of (say ), and the selected nodes are those for which the coefficients 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 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, , 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 are eliminated. As illustrated in Fig. 7 (click here), it is thus possible to reach the optimal value of ( 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 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 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 " of the order of 1 with less than say 5, with a sufficiently small value of '' 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 in E , we have
from the definitions of and
given in Eq. (7):
For , the first inequality gives
whereas for the second yields
By combining these inequalities, it follows that
The square root of is referred to as the condition number of .
Appendix 2. Convergence property
Using the notation introduced in Sect. 2.1, we have
i.e., from Eq. (19):
Thus (cf. Eq. (22)),
As the relaxation parameter 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'
for all z in ,
As a result,
Let us now assume that is different from 0. There then exists n such that
This is impossible, since must be greater than . Consequently, .
The property in question can be established as follows.
Consider the operator on :
For any z and z' in ,
This identity shows that
R is self-adjoint.
the condition implies ; R is therefore positive definite. The fact that is of finite dimension then implies that the smallest eigenvalue of R is strictly positive. Consequently, for any ,
It follows immediately that
Appendix 3. Dirty map and dirty beam
We first show that
the dirty map is the map of the scalar components
in the basis of the elementary particles .
According to Eqs. (11) and (10),
can be expanded in the form
it then follows from Eq. (12) that
hence, since :
This explicitly shows that can be identified with the dirty map (see for example Fig. 5 (click here)b).
The action of corresponds to a "back Fourier sampling operation". The dirty map looks like the inverse Fourier transform of , but from a mathematical point of view, it isn't. Indeed, is a vector in the experimental data space and not the distribution . 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
on any .
and expanding and in the forms
By using the same arguments as above,
it then follows that
can be written in the form
Let G' be the grid
twice as large as G :
The map of the coefficients
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 ) whose scalar components are given by the discrete convolution:
As a result, in the general case where the nonzero components of are distributed all over grid G , the operation 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 ,
and the convolution by the clean beam is performed
the successive clean maps of the traditional
version of CLEAN are given by the convolutions
where the iteration in is defined by the formula:
denotes the scalar component of the "experimental residue"
at pixel , and that of the dirty beam at the origin:
The relaxation parameter is referred to as
the "loop gain".
At each iteration,
is chosen so that
The experimental residue
is obtained iteratively according to the formula:
Note that .
The objective of the iteration in
is to minimize the quadratic functional:
This iteration is nothing but the matching pursuit iteration (25) in which the vectors are basis vectors , 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 functions'' centred on the nodes of grid G .
As far as the connection with our formulation of CLEAN
the related iterations in 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 are not necessarily the same: in our version of ÇLEAN|, is chosen, at each iteration, so that (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.