Simulated Results with the HST

In our numerical experiments we have mainly considered two test objects provided by R. Hanisch and characterized by different types of brightness distribution; one of the objects simulates an elliptical galaxy and the other one a star cluster with size 512512 and 256256 pixels, respectively. Simulated blurred images are obtained from the object convolved with a computed PSF, with the addition of Poissonian noise and readout noise (Hanisch 1993). The discrepancy between the reconstructed solution and the known solution may be measured by the RMS norm of their difference divided by the norm of (here )

Another possible way to evaluate algorithm performance is the so called Mean Square Error Improvement Factor (MSEIF) which reads (Busko 1993)

in the sequel we use both. By varying the regularization parameter, it is hoped to determine a minimum of (Eq. 13) or a maximum of (Eq. 14). An example is given in Fig. 1, where this has been done for the case of the elliptical galaxy.

We now give our results in the comparison of the various methods and we consider first the case of the star cluster.

It is worthwhile to introduce here the condition number of the matrix , which gives an useful estimate of the error propagation:

where is the error on the blurred image (Poisson + read-out) and is the consequent error on the restored image (this simple two-norm condition number doesn't take into account the error on the PSF). The condition number (Eq. 15) can be simply evaluated as follows:

where For the PSF of the star cluster we have

that sounds exceedingly low for people working in the field of inverse and ill-posed problems; consequently one expects the discrepancy (13) to be rather small at the minimum. The best performances in the restoration of the star cluster image are provided by the R-L algorithm and the LWPOS one; they give accuracies of 6%and 5%, respectively (see Table 1). This is clearly an excellent mathematical result but not exceedingly satisfactory on the practical side as many false stars appear when the cluster's core is examined in details; this is shown in Fig. 2. This is partially due to the fact that the solution itself is represented by discrete points, whereas the image and reconstructed solution are blurred representations thereof. In fact the choice of such solutions are particularly demanding for the restoration routines, since they are delta-function-like objects with sharp cut-offs.

As a matter of fact, the LWPOS method seems to perform better. This is highlighted from the analysis of the computational efforts (see last column of Table 1): we recall that LWPOS just needs two FFTs per iteration while R-L needs four. Other methods examined seem to be very fast at the expenses of a rather poor accuracy (all the RMSs are greater than ).

Let us turn our attention to the case of the elliptical galaxy. Once more LWPOS gets the best performance among the regularization methods but this time the R-L algorithm seems to work better, both in terms of convergence rate and accuracy (see Fig. 1 and Table ).

For the elliptical galaxy we have

a value not too different from (17). This means that the lower accuracy, with respect to the star cluster, my be imputed to a lower signal-to-noise ratio

rather than to a worse conditioning of the problem. Indeed we found, in the two cases, that Furthermore it is not clear how important the main features of these images (brightness, shape, etc.) are in restoring our noisy data. It seems evident that for this test object the R-L algorithm provides the best restoration. Fig. 3 shows the comparison between R-L and LWPOS results.