Test Results

Since the most demanding test of a deconvolution technique is to apply it to an image which contains many objects, we have generated an artificial image of NGC 6293, a globular star cluster for which we have HST Planetary Camera (PC) data. The PC is a mosaic of 4 800800 Texas Instruments CCD detectors (numbers 5-8). The filters used for our observations were F555W and F785LP which are similar to the and filters of the ground-based Johnson photometric system. Our artificial image simulates a 600 second exposure from PC 7 with the F555W filter. Busko (1993) presents results from this algorithm for the simulated cluster image available from ST ScI. We present restoration results for real HST data in a separate paper (Fullton 1993).

To generate the artificial image, we used the color-magnitude diagram (CMD) of NGC 6293 as determined from ground-based photometry (Janes 1991). We used the stellar coordinates as measured by Janes and Heasley in their images (Janes, private communication), to generate a 300300 pixel (about 1313 arc second) image in which the measured magnitudes, , were converted into photon counts and the stellar coordinates were used as the location of point sources with these fluxes. It was necessary to correct for the difference in pixel scale between the ground-based CCD images and those obtained with HST.

Because the photometry of Janes and Heasley is not complete at fainter magnitudes, i.e., , it was necessary to correct the observed data to account for those stars which were missed either because they were too faint to be detected or because of crowding effects. In order to do this, we used the luminosity function of the globular cluster M92 (Stetson and Harris 1988) which tells us how many stars we expect to see as a function of magnitude. M92 was chosen because its luminosity function has been well-determined and because its CMD morphology is similar to that of NGC 6293, indicating that the two clusters are likely similar in age and chemical composition. The number of artificial stars generated in each 0.25 magnitude bin from to was determined from the luminosity function and corrected for the observed area of NGC 6293 by a multiplicative factor determined by requiring that the luminosity function reproduce the observed counts from Janes and Heasley brighter than . The magnitude and position of each star were selected randomly using an algorithm for generating uniformly-distributed random numbers based on Convex's pseudo-random number generator RAN. The generated magnitudes were constrained to fall within the given magnitude bin for each star. The final frame contained 425 artificial stars which were added to a uniformly bright image whose intensity was chosen to be representative of the sky background in actual HST images of the cluster.

The image was blurred by convolution with an estimated HST PSF computed using the Tiny Tim software (Krist 1992). Additive Poisson noise was simulated in the blurred image by adding to each pixel a random value drawn from a Poisson distribution with a mean equal to the square root of the intensity at each pixel. The resulting image is shown in Fig. 3. No attempt was made to simulate

cosmic ray hits or saturation effects which are often present in real HST data. These effects create portions of an image that are not the result of convolution of a light source with the PSF, so such areas will need to be removed or ignored when using the iterative/recursive deblurring algorithm on real HST images.

The simulated image was deconvolved using the iterative/recursive method coded in C++ on a Convex C-240. The restoration reported here used 5 recursion levels with 3 iterations per level. Thus, iterations of the BID algorithm at the lowest recursion level were performed. Restoration of the 300300 image took approximately 38 CPU minutes, but this computation involved forward and inverse discrete Fourier transforms in each iteration. This time can be drastically shortened by performing the entire computation in the frequency domain, but it is then not possible to observe intermediate results during the restoration.

The restored image is shown in Fig. 4. The Fourier

transform-induced ringing may appear exaggerated due to contrast stretching for this display. Fig. 5 shows intensity plots through

the same row in the blurred and deconvolved images. The star located at column 163 in Fig. 5 has a simulated magnitude of 23.44. The star at pixel 194 has magnitude 21.14. The plot from the deconvolved image demonstrates the low amplitude of the ringing compared to the flux in the central peak of the star. This ringing does not violate flux conservation and does not interfere with intensity measurements of the stars in the restored images. The ringing may interfere with detection of stars at the faintest limits. As a star's magnitude decreases well below the sky brightness, it becomes increasingly difficult to distinguish the star from the star combined with the sky background. However, detection in the deconvolved image will be easier since the light from the star is not blurred over such a large area.

To test the suitability of this deconvolution technique for astronomical purposes, we have measured the brightnesses of all the stars in the deconvolved output image. For each star in the input star list, the sum of the pixel intensities in circular apertures of various radii centered on the stars' input coordinates was computed. The brightness of the uniform sky background was estimated by taking the mean of the median pixel values in several ``star-free'' regions in the deconvolved image. This sky value was subtracted from each pixel within the stellar apertures to correct for the flux contributed by the sky.

Fig. 6 is a plot of input magnitude (the magnitude used to create

the artificial image) vs. the magnitude measured in an aperture of radius 3 pixels centered on each star in the output image after correcting for the sky background. A 45 degree line indicates perfect agreement between input and output magnitudes. A constant additive aperture correction was applied to the magnitudes since an aperture of only 3 pixels does not properly account for all the stellar flux due to the ringing apparent in the image. The aperture correction was determined by plotting the measured magnitudes of 10 bright, relatively uncrowded stars as a function of aperture radius. It was estimated from these growth curves that at an aperture of radius 6 pixels the ringing had subsided sufficiently that an accurate measurement of the stellar flux could be made. Thus the average difference between the magnitude in the 6 pixel aperture and the 3 pixel aperture ( mags) was defined to be the aperture correction and was added to the magnitudes of all the stars shown in Fig. 6. This correction corresponds to less than 1%of the stellar flux. Note that in the smaller aperture, all the stars were measured too bright. Thus, while one might expect the smaller aperture to yield a fainter estimate of the stellar brightness, this is not necessarily the case in the restored image. Due to the requirement of flux conservation, the central pixels are enhanced in brightness to compensate for the ringing, which may include negative flux values. However, as shown in Fig. 6, the aperture correction is constant with magnitude. Thus the percentage overestimate of brightness is the same for all the stars. Six stars with input coordinates within 1 pixel from another star were omitted from Fig. 6. Twenty-nine other stars had coordinates so near the edges of the frame that the 3 pixel radius aperture extended outside the boundaries, so these stars were also omitted.

There are two important observations to be made concerning Fig. 6. First, it is obvious that the relationship between input and output magnitudes is linear. This is to be expected since there are no nonlinear operations in the iterative/recursive algorithm. Second, the agreement between input and output magnitudes is extremely good, with most stars falling on the line of slope unity. Stars whose nearest neighbor was less than 2 pixels away are plotted as squares, those whose nearest neighbor was greater than 2 but less than or equal to 5 pixels away are plotted as crosses, and those whose nearest neighbor was over 5 pixels away are plotted as dashes. Almost all of the stars whose output magnitudes deviate substantially from the input magnitudes are stars with nearby neighbors, some of whose light is being included in the 3 pixel aperture used. This problem would be worse if a larger aperture had been used to encompass all the light from the star. All the affected stars are measured systematically too bright because of this added light from nearby stars. It is important to note that this is a result of our use of aperture photometry in crowded conditions and is not an artifact of the deconvolution.