Results and Discussions

In this section we describe different restoration results with FOC and simulated data. Before describing the restoration details we discuss the restoration method.

Maximum Entropy Image Restoration

We used the maximum entropy method (MEM) of Hollis, Dorband, Yusef-Zadeh (1992) who restored images that were plagued by the FOC detector saturation which corrupted the data in certain regions. Use of a mask was required to ensure that the saturated data were ignored in the resultant restoration which these authors obtained with an observed PSF. We have determined that the size and shape of the mask affects the detailed structures of the MEM restoration for a given number of iterations, but not the global structure. Moreover, we found the detailed structures are independent of the mask geometry when allowed to iterate long enough.

Deconvolution: Observed Versus Analytic PSFs

Hollis and colleagues obtained two public domain FOC F/96 images of the binary system R Aquarii and an observed PSF both in the light of twice ionized oxygen which emits a spectral line at 5007 Å. These images were plagued by spherical aberration and, in the case of R Aquarii, also by detector saturation. The MEM technique described above was used to restore one R Aquarii image with the observed PSF to the original design resolution of the HST/FOC combination. It was precisely because the observed FOC PSF was larger than the saturated region of the R Aquarii image that the restoration was successful (see Hollis, Dorband, and Yusef-Zadeh 1992). However, as noted in the Introduction, restoration of two extended FOC images taken at two different times does not lead to similar small-scale details when using the observed PSF. We used the two FOC images and the observed PSF used by Hollis, Dorband, and Yusef-Zadeh (1992) and repeated the MEM deconvolution. Restored images after 1000 iterations are shown in Fig. 3. These images are not identical in details,

suggesting that that the observed PSF does not correspond well enough to the images for deconvolution purposes. Hence, it is desirable to be able to synthesize two different analytical PSFs to remedy the differences between the two extended images of interest.

In our effort to compute and use an analytical PSF in the restoration process, we noted that the structures prominent in the center core of the observed PSF are better modeled by the and the Zernike ``clover'' coefficients which correspond to the and polynomial terms. Restoration results of the FOC R Aquarii images using an analytical PSF, computed using our estimated annular Zernike polynomial coefficients and , are shown in Fig. 4. It can be noted that this single

analytical PSF leads to fairly consistent enhanced details compared to the restoration using the observed PSF. This may be due to the noise-free analytical PSF. However, some of the faint structures shown in the restoration using the observed and the analytical PSf are computational artifacts. This suggests that neither the observed PSF nor the analytical PSF matches the scene to be restored and, moreover, two different analytical PSFs will be eventually required to restore two images separated in time. The following simulation study demonstrates that these artifacts are a measure of the accuracy with which the PSFs are modeled.

Effect of PSF Mismatch on the Deconvolution

A test image was generated by convolving five spatially distributed delta functions with the ``correct'' PSF which was calculated by means of Zernike coefficients. Using the MEM algorithm, the test image was subsequently deconvolved with both the correct PSF and a ``mismatched'' PSF which was calculated using the same Zernike coefficients as in the correct PSF except that , the focus term, was changed by 0.1 waves rms. When logarithmically stretched, both image restorations gave a subtle mottled pattern with extremely weak linear artifacts that seem to bridge the gaps between delta functions, but the mismatched PSF artifacts were more exaggerated. Increasing the number of iterations for each type restoration reduces the prominence of the artifacts in the correct PSF case. Moreover, the mismatched PSF restores more slowly than the PSF which was used to generate the image in the first place. Hence, for a given number of iterations, the correct PSF will restore points to larger numerical values with fewer artifacts than the mismatched PSF. This property of ``maximized signal and minimized artifacts'' could be used in an empirical fashion to decide between the results given by two different PSF restorations for a fixed number of iterations when using this algorithm. Unfortunately, this method would be very time consuming, and thus, a convergence criterion based on some statistical property of the converging image, or some property grounded in the physics, or some smoothness property is sorely needed. Fig. 5 shows a 1000 iteration restoration of the

test image with a mismatched PSF along side of a 1000 iteration restoration of FOC data of the R Aquarii Jet using a calculated PSF whose Zernike coefficients ( were published in the STScI Newsletter of March 1992 (hereafter the ``ST ScI PSF''). Comparison of these highly similar results suggest that both the test image and the actual HST data restorations are plagued by mismatched PSFs.

Our objective is to be able to calculate the correct PSF for use in subsequent deconvolution of an extended image. Thus, it is of interest to compare the results of a 1000 iteration restorations of FOC data of the R Aquarii Jet using the ST ScI PSF and an observed PSF, and these two results are shown in Fig. 6 as a three-dimensional plot. The results are very

similar, and from a science information point of view, are indistinguishable. Considering the fact that the observed PSF data were taken 5 days later than the R Aquarii Jet data by the FOC and HST secondary mirror movement is suspected, it is not unexpected that this observed PSF would be mismatched to R Aquarii Jet data for restoration purposes. However, in a relative sense, note that the signal-to-noise on the image produced with the observed PSF is less than the image produced by the ST ScI PSF. Here the noisy areas lay outside of the areas dominated by regions of intense signal. This suggests that the observed PSF is better matched to the data than the calculated ST ScI PSF.

In conclusion, we still need to restore two scenes of the same object separated in time to validate the contention that we should be able to do better with a calculated PSF than an observed PSF. Once we are able to do this, then, in principle, we can sample a calculated PSF at any desired resolution and generate imagery that exceeds the design resolution of the HST/FOC combination (see Dorband and Hollis 1992). Our quest in this effort is still on-going.

Acknowledgments

We gratefully acknowledge the help of Mr. Rick Lyon of RADEX and Mr. Anthony Gruszak of HDOS.