14.4.7 Uniqueness and robustness

When the problem is well-posed, A_E is a one-to-one map ( $\ker A_E = \{0\}$) from E onto F; the solution is then unique: there exists only one vector $\phi\in E$ such that  $A_E\phi = \Psi_F$. This vector, $\Phi _E$, is said to be the least-squares solution of the equation $A_E\phi$ ``=''  $\Psi_{\! d}$.

In this case, let  $\delta\Psi_F$ be a variation of $\Psi _F$ in F, and  $\delta\Phi_E$ be the corresponding variation of $\Phi _E$ in E(see Fig. 14.7). It is easy to show that the robustness of the reconstruction process is governed by the inequality:

$$\frac{\Vert\delta\Phi_E\Vert_o}{\Vert\Phi_E\Vert_o} \leq \kappa_E \frac{\Vert\delta\Psi_F\Vert_d}{\Vert\Psi_F\Vert_d}.$$ (14.17)

The error amplifier factor $\kappa _E$ is the condition number of A_E:

$$\kappa_E = \frac{\sqrt{\lambda'}}{\sqrt{\lambda}};$$ (14.18)

here $\lambda$ and $\lambda'$ respectively denote the smallest and the largest eigenvalues of  $A^*_E A^{\vphantom{*}}_E$. The closer to 1 is the condition number, the easier and the more robust is the reconstruction process (see Fig. 14.8 and 14.9).

  

Figure: Uniqueness of the solution and robustness of the reconstruction process. Operator A is an operator from the object space H_o into the data space K_d. The object representation space E is a particular subspace of H_o. The image of E by A, the range of A_E, is denoted by F. In this representation, $\Psi _F$ is the projection of the data vector  $\Psi_{\! d}$ onto F. The inverse problem must be stated so that A_E is a one-to-one map from E onto F, the condition number $\kappa _E$ having a reasonable value.

The part played by inequality 14.17 in the development of the corresponding error analysis shows that a good reconstruction procedure must also provide, in particular, the condition number $\kappa _E$. This is the case of the current implementation of WIPE which uses the conjugate gradient method for solving the normal equation 14.16.

To conduct the final error analysis, one is led to consider the eigenvalue decomposition of  $A^*_E A^{\vphantom{*}}_E$. This is done, once again, with the aid of the conjugate gradient method associated with the QR algorithm. At the cost of some memory overhead (that of the M successive residues), the latter also yields approximations of the eigenvalues $\lambda_k$of  $A^*_E A^{\vphantom{*}}_E$. It is thus possible to obtain the scalar components of the associated eigenmodes $\Phi_k$ in the interpolation basis of H_o. The purpose of this analysis is to check whether some of them (in particular those corresponding to the smallest eigenvalues) are excited or not in $\Phi _E$. If so, the corresponding details may be artefacts of the reconstruction.

  

Figure: Reconstructed neat map $\Phi _E$ (left hand) and eigenmode $\Phi_1$ (right hand) corresponding to the smallest eigenvalue  $\lambda_1=0.165$ of $A^*_E A^{\vphantom{*}}_E$. The conditions of the simulations are those of Fig. 14.4 and 14.5: in particular, the diameter of  $\mathcal{H}_s$ is equal to  $40\,\delta u$. The final condition number is  $\kappa_E=2.46$ (the eigenvalues of  $A^*_E A^{\vphantom{*}}_E$ are plotted on the bar code below). This eigenmode is not excited in $\Phi _E$: the separation angle $\theta_1$ between $\Phi _E$ and $\Phi_1$ is greater than $89^\circ$. In other situations, when the final condition number is greater, this mode may be at the origin of some artefacts in the neat map (see Fig. 14.9).

The reconstructed map is then decomposed in the form:

$$\Phi_E = \sum_{k=1}^M w_k \Phi_k, \quad w_k = ( \Phi_k \mid \Phi_E ).$$ (14.19)

The separation angle $\theta_k$ between $\Phi _E$and $\Phi_k$ is explicitly given by the relationship:

$$\cos\theta_k = \frac{w_k}{\sqrt{\sum_{k=1}^M w_k^2}} \quad (0 \le \theta_k \le \pi / 2).$$ (14.20)

The closer to $\pi/2$ is $\theta_k$, the less excited is the corresponding eigenmode $\Phi_k$ in the reconstructed neat map $\Phi _E$.

To illustrate in a concrete manner the interest of equations 14.19 and 14.20, let us consider the simulations presented in Fig. 14.4 and 14.9. Whatever the value of the final condition number is, the error analysis allows the astronomer to check if there exists a certain similitude between some details in the neat map and some features of the critical eigenmodes. This information is very attractive, in particular when the resolution of the reconstruction process is greater than a reasonable value (the larger is the aperture to be synthesized  $\mathcal{H}_s$, the smaller is the full width at half-maximum of $\Theta _s$). In such situations of ``super resolution,'' the error analysis will suggest the astronomer to redefined the problem at a lower level of resolution, or to keep in mind that some details in the reconstructed neat map may be artefacts of the reconstruction process.

  

Figure: Reconstructed neat map $\Phi _E$ (left hand) and related critical eigenmode $\Phi_1$ (right hand). The latter corresponds to the smallest eigenvalue $\lambda_1=0.057$ of $A^*_E A^{\vphantom{*}}_E$. The conditions of the simulations are those of Fig. 14.5, but here the diameter of  $\mathcal{H}_s$ is taken equal to  $48\,\delta u$: the final condition number is  $\kappa_E=4.19$ (the eigenvalues of  $A^*_E A^{\vphantom{*}}_E$ are plotted on the bar code below). The critical eigenmode $\Phi_1$ is at the origin of the oscillations along the main structuring entity of $\Phi _E$. This mode is slightly excited (the separation angle $\theta_1$between $\Phi _E$ and $\Phi_1$ is less than $86^\circ$), thus the corresponding details may be artefacts. In this case of ``super-resolution'' the error analysis provided by WIPE suggests that the procedure should be restarted at a lower level of resolution (see Fig. 14.8), so that the final solution be more stable and reliable.