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The Restoration of HST Images and Spectra II
Space Telescope Science Institute, 1994
R. J. Hanisch and R. L. White, eds.
Evaluation of Image Restoration Algorithms Applied to HST Images
I. C. Busko
Space Telescope Science Institute, Baltimore, MD, and
Astrophysics Division, INPE, S. J. dos Campos, Brazil.
Abstract. This work reports results on intercomparison of image restoration algorithms,
when used in the speciîc context of stellar îelds imaged by the HST WFPC. Properties as
îdelity to the original image and photometric linearity, as well as computation performance,
were evaluated.
1. Introduction
In the Image Restoration Project carried out at STScI, the restoration quality and scientiîc usefulness
of several restoration algorithms were evaluated in a comparative way by numerical experiments in
which a #perfect# (or #truth#) image was artiîcially degraded, next restored by a given technique,
and înally the output was compared with the original. Previous work on this line in the astronomical
literature (e.g. Cohen 1991, Thomson et al. 1992) mostly restricted their efforts to the MEMSYS
implementation of the Maximum Entropy method. With the availability of other image restoration
techniques in public domain format (as in the Space Telescope Science Data Analysis System,
STSDAS), as well as a suite of simulated data sets available on the STScI Electronic Information
Service (STEIS), more systematic and complete studies were possible.
2. Methodology
This study focussed on pure stellar images, since they are probably the kind of object most often
imaged by HST. Also, they present the most difîcult challenge to any restoration algorithm, since
the intensity gradient and dynamic range in such an image are very large when compared to usual
#terrestrial# scenes, to which restoration algorithms are usually designed.
The main data set used in this study is a simulated WFPC I observation of a #star cluster#,
similar to the ones available in STEIS. The simulated image was built on a 8 \Theta 8 oversampled data
grid, later blockíaveraged to the actual instrument resolution. The PSF was computed by TinyTIM
(Krist 1991) at Ö = 5500 #
A, at center of CCD #1. Finally, the appropriate noise model for WF
CCDs, including Poisson and readout (Gaussian) components, was added to the image.
Algorithms studied include current STSDAS implementationsof the RichardsoníLucy iteration
(Richardson 1972, Lucy 1974), Maximum Entropy (Wu 1993), the Wiener îlter (Andrews & Hunt
1977), oeíCLEAN (Keel 1991) and a standard Iterative Least Squares algorithm (e.g. Katsaggelos
1991). Also, an independent implementation of the Iterative/Recursive least squares algorithm
(Coggins et al. 1993, Fullton et al. 1993) was tested.
Criteria to evaluate restoration quality include both generic and astronomicalíspeciîc ones.
An often used goodnessíofíît criterion in image restoration work is a measure of the #distance#
between the restored image Ó
f(i; j) and the #truth# image f(i; j) (the one without any degradation)
j 2 = 10 log
P
i;j (g(i; j) \Gamma f(i; j)) 2
P
i;j ( Ó
f(i; j) \Gamma f(i; j)) 2
279

280 Busko
where g(i; j) is the observation at pixel i; j. The measure is expressed in dB, and increases as long
as the restored image becomes #closer# to the truth image. We also used an absoluteívalue distance
j 1 = 10 log
P
i;j jg(i; j) \Gamma f(i; j)j
P
i;j j Ó
f(i; j) \Gamma f(i; j)j
which correlates better than j 2 with visual quality evaluation. Both j 1 and j 2 are independent of
image content.
Astronomical criteria include photometric linearity, precision, and sky background statistics.
Photometric evaluation was performed on star images using standard aperture techniques.
Algorithm sensitivity to PSF errors was evaluated by restoring the test image using both the
original PSF, as well as another TinyTIM PSF, computed for a bluer star and situated ¦ 200 pixels
away from the the CCD center.
3. Results
Table 1 summarizes some distance measure results. These measures were computed both for the
full image and for a 50 pixel sq. region centered on the star cluster, dominated by crowded star
images. Thus, they are sensitive to different image properties. The full image measures asses the
global goodnessíofíît, but in the particular image under study they are strongly sensitive to the sky
background ît. The crowded region measures, on the other hand, are more sensitive to how well
star images were îtted. The highest peak value is a measure on how close the brightest star's peak
approached the truth value, and so it is also a goodnessíofíît measure for star images.
Algorithms are arranged in Table 1 in increasing order of execution speed. All algorithms
were iterated up to the point where at least one of j 1 or j 2 just started to decrease. Most often
this happened for the full image's j 1 , which is sensitive to spurious noise peaks introduced in the
sky background. The exception is the Iterative/Recursive algorithm, which was run arbitrarily for 5
levels of recursion and 3 iterations at each level. oeíCLEAN was iterated to the 2:6oe level.
Visual inspection alone showed that the three algorithms: RíL, MEM0, and ILS+P are capable
of very similar results. This is conîrmed by the similar j 1 values for them. The solution obtained
by the ILS+P algorithm, being a leastísquares one, had the highest j 2 , as expected. RíL produced
a somewhat better solution than MEM0, but at a considerable speed penalty. oeíCLEAN produced
not so good j 1 results as the methods above, mostly because of the large PSF residuals left on the
CLEAN+residual map. The RILS results show that inclusion of a spatiallyíadaptive smoothness
constraint in the leastísquares iteration actually decreased restoration quality.
Results also suggest that the ILS+P iteration is the most robust against PSF mismatches,
followed by RíL and MEM0, which showed very similar degradations. oeíCLEAN was the most
sensitive to PSF errors.
Figure 1 depict typical results from photometry with a small aperture. Results were also
obtained for a subset of isolated (nonícrowded) star images, using larger apertures up to OE = 8
pixel. This largeíaperture data show that systematic zeroípoint shifts are a common feature of
restored stellar images. Only the RíL iteration showed negligible zeroípoint shifts; all other
algorithms showed ¦ 0:05 mag shifts to either side of the zeroíresidual locus.
Restoration with PSF error led to an increased scatter and more confusion. Besides, largeí
aperture measurements showed measurable (¦ \Gamma0:02) slopes in RíL and MEM0 residuals. ILS+P
showed only a slight increase in zeroípoint shift.
Comparison of results in Figure 1 with the ones in Figure 2 points to the existence of a tradeíoff
between a particular algorithm's linearity, and its ability to control highífrequency noise buildup.
Extreme cases are the Iterative Inverse and IR algorithms, which are very linear but do not control
sky background noise, and RILS+P , which produces very smooth backgrounds but at the cost of
losing ¦ 0:5 mag of light at the faint levels in this particular test image. RíL, MEM0 and ILS+P
seem all to offer a reasonable compromise among these extremes.

Evaluation of Algorithms 281
Figure 1. Aperture photometry in restored and unrestored images. A 2.2 pixel diameter
aperture was used to measure each one of the 500 stars in the simulation input list. Abcissa
is magnitude in truth image, ordinate is residual in the sense restored image minus truth.
A negative residual indicates excess light in the aperture, probably due to crowding.
A positive residual means some light is missing from the aperture, either because it
was not recovered by the restoration algorithm, or because the restored star proîle does
not ît inside the aperture. Sky background is constant and set to zero in the aperture
measurements.

282 Busko
Figure 2. Sky background intensity distribution (log scale) in a 35 sq. pixel area devoid
of any stars.

Evaluation of Algorithms 283
Figure 3. Effectiveness in minimizing neighbor star confusion. Residuals (in mag
scale) are from star's intensity in central pixel only. They are plotted as a function of
star's distance to cluster center. Unrestored image shows the largest degree of confusion;
both iterative inverse and IR restorations the smallest. An algorithm's ability in reducing
confusion is directly related to its resolution enhancement power.

284 Busko
Table 1. Goodnessíofíît measures. Meaning of symbols is as follows: RíL:
RichardsoníLucy; MEM0: Maximum Entropy; ILS: Iterative Least Squares (no regí
ularization); RILS: ILS with Miller regularization and spatial adaptivity; IR: Iteraí
tive/Recursive; P : projection (positivity).
PSF Algorithm Number of CPU time a Full image Crowded region Highest peak b
errors iterations (minutes) j 2 j 1 j 2 j 1 (%)
NO RíL 3000 132 6.90 5.08 10.61 5.83 52
MEM0 300 19 5.98 4.87 9.88 5.68 46
oeíCLEAN c 40000 13 9.32 3.89 10.16 5.01 87
ILS + P 300 12 9.48 5.43 11.82 5.88 69
RILS+P d 150 12 9.03 4.88 11.94 5.70 65
RILS e 100 3 8.44 0.82 10.31 3.18 78
Iterative inverse 80 1 7.68 í2.90 9.90 2.61 74
IR 3 X 5 f 7.41 í3.91 8.27 1.79 89
Wiener 6 sec. 1.30 í0.52 1.39 0.74 29
YES RíL 1500 65 5.92 4.49 9.64 5.18 46
MEM0 300 19 4.81 4.21 8.76 4.97 39
oeíCLEAN 30300 10 7.94 2.84 8.31 3.73 90
ILS + P 200 8 8.79 5.03 11.06 5.34 66
a SPARC10, 256 sq. image.
b Ratio of brightest star's highest peak in restored and truth images.
c Loop gain = 0.05.
d ff = 0:2, ï = 0:5.
e ff = 0:03, ï = 0:5.
f Algorithm wasn't run in a SPARC10, but code analysis suggests that it is roughly twice as fast as the iterative inverse.
Preliminary results from an ongoing study of photometric errors from aperture photometry in
restored images suggests that restoration by RíL, MEM0 and ILS+P does not degrade the random
error level in largeíaperture photometry of isolated stars. There is marginal evidence that restoration
may even decrease the random error in faint stars. This point needs more study, and of course a
detailed comparison with sophisticated PSFíîtting techniques (Stetson 1993) is fundamental to
answer the question: should stellar photometry be performed in restored or unrestored images ?
References
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Evaluation of Algorithms 285
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