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4. Optimization with control of robustness

For clarity, let us now assume that tex2html_wrap_inline3768 is a one-to-one map. The method presented in Sect. 3 then yields a solution tex2html_wrap_inline3770 of Eq. (38), and thereby the solution of the problem: tex2html_wrap_inline3772. Unfortunately, as already mentioned, this method does not provide any information on the robustness of the reconstruction process. The most natural way of obtaining this information is to find tex2html_wrap_inline3774, directly, as the solution of the normal Eq. (4):

In this section, we present the corresponding developments. To conduct our analysis, the eigenvalues of B are ordered so as to form a nondecreasing sequence (cf. Eq. (7)):

As tex2html_wrap_inline3778 is assumed to be a one-to-one map, tex2html_wrap_inline3780 is strictly positive. In the general case where the tex2html_wrap_inline3782 generating  E do not form an orthogonal set, the reader must keep in mind the fact that the action of tex2html_wrap_inline3786 can be performed with the aid of algorithm 2.

4.1. Reconstruction algorithm

The problem is solved by using the conjugate-gradients method (cf. Sect. 2.3 of Lannes et al. 1987b). Starting from any tex2html_wrap_inline3794 in  E , the iterates tex2html_wrap_inline3798 converge to tex2html_wrap_inline3800 in at most m  iterations, tex2html_wrap_inline3804 getting closer to tex2html_wrap_inline3806 at each iteration. In this algorithm, tex2html_wrap_inline3808 is the "direction of research" in tex2html_wrap3876, whereas tex2html_wrap_inline3812 is the corresponding "parameter of exact line search"; tex2html_wrap_inline3814 is the residue of the normal Eq. (39) for tex2html_wrap3878:
As tex2html_wrap_inline3818, we have tex2html_wrap_inline3820, hence:

Denoting by tex2html_wrap_inline3822 an estimate of tex2html_wrap_inline3824, we therefore have:

Let us introduce an acceptable error threshold tex2html_wrap_inline3834 for tex2html_wrap_inline3836. Clearly, the iterative process can be interrupted as soon as tex2html_wrap_inline3838 is less than tex2html_wrap_inline3840; tex2html_wrap_inline3842 therefore plays the role of a convergence estimator. The estimate of tex2html_wrap_inline3844 is refined throughout the iterative process as indicated in Sect. 4.2. The corresponding algorithm can then be summarized as follows.


#&# Step 0: &Set tex2html_wrap_inline3846 (for example) and n=0 ; &choose a natural starting point tex2html_wrap_inline3850 in E ; &compute tex2html_wrap_inline3854; &set tex2html_wrap_inline3856. Step 1: &Compute &tex2html_wrap_inline3858, 5pt&tex2html_wrap_inline3860, 5pt&tex2html_wrap_inline3862, 5pt&tex2html_wrap_inline3864, 5pt&tex2html_wrap_inline3866; 6pt&if tex2html_wrap_inline3868, termination. 6pt &Compute &tex2html_wrap_inline3870, 5pt&tex2html_wrap_inline3872. 5pt&Increment n and loop to step 1. 101

4.2. Effective object representation space

In the conjugate-gradients method, the n -dimensional subspace of  E generated by the conjugate directions tex2html_wrap_inline3898,

is referred to as the Krylov space of order  n . According to a well known property (cf. properties 2 and 3.1 of Lannes et al. 1987b), tex2html_wrap_inline3902 minimizes tex2html_wrap_inline3904 on tex2html_wrap_inline3906.

Provided that  n  is sufficiently large, the least-squares solutions in  E and tex2html_wrap_inline3912 are very close to one another. At the end of the reconstruction process, tex2html_wrap_inline3914 is therefore the effective object representation space. The dimension of this space, as well as the robustness of the reconstruction process, depends upon the localization of the eigenvalues of  B , and more precisely, on the relative weight of the projections of tex2html_wrap_inline3918 onto the corresponding eigenspaces. We are thus led to consider the operator
where tex2html_wrap_inline3920 is the projection (operator) onto tex2html_wrap_inline3922.

The residues tex2html_wrap3976 form an orthogonal basis for tex2html_wrap_inline3926 (see Appendix 4 of Lannes et al. 1987b). As established in Appendix 2 of Lannes et al. (1996), the matrix of tex2html_wrap_inline3928 expressed in this basis is tridiagonal (this matrix is of course symmetric). Its diagonal and upper-diagonal elements are respectively given by the relationships

The eigenvalues of tex2html_wrap_inline3934 can therefore be calculated very easily with the aid of the QR algorithm (cf. Sect. 11.3 of Press et al. 1992). Let us order these eigenvalues as those of  B (see Eq. (41)):

By referring to the eigenvalue analysis based on the notion of "minmax numbers" (cf. Appendix 5 of Lannes et al. 1987a), it is easy to show that

Figure 6: Image reconstruction via the regularized version of CLEAN; a) traditional clean map for tex2html_wrap_inline3938; b) improved clean map tex2html_wrap_inline3940 provided by the regularized version of CLEAN (tex2html_wrap_inline3942). These images have to be compared with the image to be reconstructed (Fig. 5 (click here)c). As shown in Fig. 7 (click here)b, the matching pursuit process of WIPE can still refine image (b)

Provided that the projections of tex2html_wrap_inline3944 onto the eigenspaces corresponding to tex2html_wrap_inline3946 and tex2html_wrap_inline3948 are different from zero, a condition which is always numerically satisfied in practice, tex2html_wrap_inline3950 and tex2html_wrap_inline3952 respectively tend tex2html_wrap3980 as  n tends to  m (see Fig. 3 (click here) of Lannes et al. 1996).

In our reconstruction processes, the eigenvalues of tex2html_wrap_inline3962 are computed at each iteration. (The cost for this is negligible compared to that of the action of  B .) As soon as tex2html_wrap_inline3966 is less than say tex2html_wrap_inline3968,
are very good approximations to tex2html_wrap_inline3970 and tex2html_wrap_inline3972, respectively. In most cases, the termination test of the basic algorithm is then satisfied (see Fig. 3 (click here) of Lannes et al. 1996).

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