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# 4. Optimization with control of robustness

For clarity, let us now assume that is a one-to-one map. The method presented in Sect. 3 then yields a solution  of Eq. (38), and thereby the solution of the problem: . 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 , directly, as the solution of the normal Eq. (4):

where

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 is assumed to be a one-to-one map, is strictly positive. In the general case where the  generating  E do not form an orthogonal set, the reader must keep in mind the fact that the action of  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  in  E , the iterates converge to  in at most m  iterations, getting closer to  at each iteration. In this algorithm, is the "direction of research" in , whereas  is the corresponding "parameter of exact line search"; is the residue of the normal Eq. (39) for :

As , we have , hence:

Denoting by an estimate of , we therefore have:

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

ALGORITHM| 3:

#&# Step 0: &Set (for example) and n=0 ; &choose a natural starting point in E ; &compute ; &set . Step 1: &Compute &, 5pt&, 5pt&, 5pt&, 5pt&; 6pt&if , termination. 6pt &Compute &, 5pt&. 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 ,

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),  minimizes  on .

Provided that  n  is sufficiently large, the least-squares solutions in  E and  are very close to one another. At the end of the reconstruction process,  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  onto the corresponding eigenspaces. We are thus led to consider the operator

where is the projection (operator) onto .

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

and

The eigenvalues of  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 ; b) improved clean map  provided by the regularized version of CLEAN (). 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 onto the eigenspaces corresponding to  and  are different from zero, a condition which is always numerically satisfied in practice,  and  respectively tend as  n tends to  m (see Fig. 3 (click here) of Lannes et al. 1996).

In our reconstruction processes, the eigenvalues of are computed at each iteration. (The cost for this is negligible compared to that of the action of  B .) As soon as  is less than say ,

are very good approximations to  and , 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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