Numerical Results

In our numerical experiments we mainly considered two sets of simulated data provided by R. Hanisch, namely the Star Cluster and the Elliptical Galaxy 2 (Hanisch 1993).

Fig. 1 shows the relative error (the inverse of the signal-to-noise ratio) achieved using the Tikhonov method in the restoration of the above mentioned images.

These two cases show that much better accuracy is achieved for the Star Cluster than for the Elliptical Galaxy image. We mainly impute this fact to the strong difference existing between the signal-to-noise ratio associated with the Star Cluster and with the Elliptical Galaxy. Indeed their ratio is approximately .

We recall that any linear system , where is the inverse of the signal-to-noise ratio of the noisy data and is the corresponding ratio for the solution, provides the following inequality (Stoer and Bulirsch 1980):

Therefore, as the condition numbers of the Fourier transforms of the two PSFs provided by R. Hanisch have approximately the same value, we must expect a better signal-to-noise ratio in the restoration of Star Cluster images. For this reason we feel that, for solving Eq. (1), the use of a regularization method is recommended in the case of the Star Cluster and it is mandatory for the Elliptical Galaxy. Furthermore we believe that, as a consequence of its worse signal-to-noise ratio, the latter case provides a more critical but significant test to assess the effectiveness of the numerical methods used for the restoration.

Fig. 2 shows the comparison among the true image (top left), the simulated data (top right) and the Tikhonov regularized solution (bottom left).

The image at the bottom right of Fig. 2 will be called into play later on. The most evident effect is that the regularization method is only partially effective in cleaning up the image background contaminated from the simulated noise (compare the true image with the simulated data). The problem is not of particular relevance in this case but it could be if we treat images including low-intensity sources distributed in the background. It can be noticed that while the inside structure of the Galaxy is satisfactorily well restored in the center, where the image function takes its highest values, the restoration is more critical near to the edge, where the Tikhonov method is not fully effective in discriminating between the low values of the true image and the noisy background.

As an example of numerical computation based on relations (11)-(12) we used J413, a real image of Jupiter observed on 4 March 1992 with the WFPC. This is an extended image (without sharp cut-offs) which is essentially band limited. Tikhonov regularization coupled with interpolation by sinc functions allows to restore images on an additional ``shifted'' grid by the following relation: The overall computational complexity is , which must be compared with for the Tikhonov restoration. The use of effective BLAS 3 routines for matrix-matrix product (ESSL 2, 1992) makes acceptable the CPU-time increase. As a result, Tikhonov regularization takes about sec on the initial mesh and sec on an extended mesh (IBM RISC 6000/560 with ESSL 2). Fig. 3 shows two Tikhonov restorations of J413 on the above mentioned grids.

For validating the method described in § 3 we use again the already shown Elliptical Galaxy image. We partially restored the simulated image using WAVE II and got a whole set of smoothed images corresponding to different scale edges. Taking the smoothed image characterized by the finest scale and treating it by the Tikhonov method, we obtained a significant improvement of the accuracy, as shown in Fig. 4.

It can be appreciated that the new minimum of the accuracy takes the value instead provided by the Tikhonov method. It is worthwhile to note that the improvement is appreciable for low values of the regularization parameter. The restored image obtained by this method is shown in Fig. 2 (bottom right).