Comparisons of Implementations

In this section, we describe the results of experiments that we have performed to examine the effect on restored images of the Poisson and saddlepoint approximations to the function .

Experiment 1: Restoration of Star Cluster Data Using 50 EM Iterations

We performed restorations for the simulated star cluster (file ``truth.fit'') provided by the Space Telescope Science Institute. The star cluster, shown in the left panel of Fig. 1, consists of a field of unresolved stars at random locations and having randomly selected brightnesses. We simulated an HST image of the star cluster using a space-invariant PSF (file ``mpsf12.fit" provided by ST ScI) representative of the HSTs PSF, Poisson-distributed photo-counts, and a read-out noise level of electrons r.m.s. This is shown in the right panel of Fig. 1. In these and the following images, gray scale values are shown in a logarithmic scale so that low level noise and distortions can be seen.

The restorations obtained from the simulated data by performing 50 EM iterations (Eq. 3) based on the Poisson (Eq. 11) and saddlepoint (Eq. 23) approximations are shown in Fig. 2.

It is difficult to compare these two restorations directly, so we have summarized one feature of them in Fig. 3 where shown are the ratios of estimated to true brightness versus true brightness for the restorations in Fig. 2. These scatter diagrams were obtained by placing a point at location (tb, eb/tb) for each star in the star cluster for restorations based on the Poisson (left) and saddlepoint (right) approximations. The solid lines were obtained by averaging the logarithmic values in the scatter diagrams with a rectangular window of width 0.1 moved in steps of 0.05.

The two solid lines in Fig. 3 are redrawn together in Fig. 4 so they can be compared more readily. It is evident from this graph that, for 50 EM iterations, the saddlepoint approximation tends to yield more accurate brightness estimates, especially for fainter stars. The difference does not appear to be enormous, but it is present, with the saddlepoint approximation yielding brightness estimates closer to truth than with the Poisson approximation. Recognizing that these are logarithmic plots, there is seen to be an improvement in the accuracy of the brightness estimate by a factor of about 1.5 for fainter objects, which can be significant when accurate photometry is sought.

Experiment 2: Averages for 25 Repeated Star Cluster Simulations, 50 Iterations Each

For this experiment, we simulated 25 independent realizations of HST images that would be acquired by the HST for the star cluster field. These images were produced in the same manner as the image in the right-hand panel of Fig. 1. We then restored each of the 25 images using 50 EM iterations for both the Poisson and saddlepoint approximations. Shown in Fig. 5 is the average ratio of estimated brightness to brightness versus brightness for the star cluster data; this graph was obtained by averaging the 25 versions of Fig. 4 obtained in the multiple simulations.

Shown in Fig. 6 is the ratio of the mean error to brightness versus brightness for the two approximations. These scatter plots were obtained by averaging the error (estimated brightness minus true brightness) across the 25 restorations, dividing by the true brightness, and creating a point at (tb, error/tb) for each star in the star cluster. Shown in Fig. 7 is a corresponding scatter plot showing the ratio of standard deviation to brightness versus brightness. The solid lines in Figs. 6 and 7 were obtained by averaging the values in the scatter diagrams with a rectangular window of width 0.2 moved in steps of 0.01. These are replotted in Fig. 8 to facilitate comparisons.

Experiment 3: Restoration of Star Cluster Data Using 500 EM Iterations

Shown in Fig. 9 are the results for the same conditions as for Fig. 3 but with 500 EM iterations having been performed. When these extended numbers of iterations are performed, we find less scatter for fainter stars compared to restorations performed with 50 EM iterations, and there is less bias for all brightness levels. We also find at 500 EM iterations that the Poisson and saddlepoint approximations yield virtually identical results so far as these scatter diagrams are concerned. However, these scatter diagrams only reflect the restored images at the true star locations. Restored images of the star cluster for 500 iterations are in Fig. 10, which is the result of performing 500 EM iterations on one simulated HST star cluster image. These images are shown on a logarithmic scale, so low level noise is also evident, but this is noise is small, typically at a level of about 50 compared to stars that are at a level several orders of magnitude larger (about ). The low level noise is reduced substantially in Fig. 11, which is the average of 25 restorations of HST star cluster images, each being the result of performing 500 EM iterations. The average results shown for the Poisson and saddlepoint approximations are virtually identical at 500 EM iterations and low level noise is greatly reduced.