EXPERIMENTS IN BLIND DECONVOLUTION OF
AEOS ADAPTIVE OPTICS IMAGES
Michael E. Brown 1
Sarah Hörst
Antonin H. Bouchez1
Division of Geological and Planetary Sciences
California Institute of Technology
Pasadena, CA 91125
1.
INTRODUCTION
Adaptive optics (AO) systems are capable of partially removing the blurring of astronomical images caused by
turbulence in the earth’s atmosphere, allowing large ground-based telescopes to potentially achieve angular
resolutions comparable to their diffraction limit. The AO correction is only partial, however, and a significant
fraction of the light from a point source remains in a seeing-limited halo considerably larger than the diffraction
limit (Fig. 1). The presence of this halo reduces the contrast and masks features in AO images. Deconvolution of the
image by the correct point spread function (PSF) is crucial to recovering the maximum information in AO images.
Unfortunately, accurate deconvolution is difficult for AO systems owing to the time-variability of the PSF. As seen
in Fig. 2 the core of the PSF can range from nearly diffraction limited to more than twice that size on times scales of
only seconds. While deconvolution can be attempted by measuring an average PSF in conditions similar to those of
the target observation, this average will be degraded from the best instantaneous PSF and might be substantially
different from the PSF of the target.
Fig. 1. Comparison of the PSF measured from the Advanced Electro Optical
System (AEOS) 3.67-meter telescope and AO/Vis imaging system of a V=5 star
in a single 12ms exposure with the theoretical diffraction limited PSF.
Fig 2. The measured full width half max (FWHM) from a sequence of 50 12ms
observations of a V=5 star at AEOS.
Blind deconvolution attempts to circumvent the problem of unknown PSFs by simultaneously solving for both the
underlying astronomical object and the PSF [1]. Clearly, this problem is severely underconstrained and comparable
to trying to determine 2 unknowns from a single observation. The problem can be made less underconstrained by
using multiple observations of the same target; one is now attempting to solve for N PSFs and 1 underlying object
using N observations. A unique solution is still not possible, but applying additional constraints to the problem
allows an optimal solution to be found. Potential constraints that can be applied include positivity, where all pixels
of the PSF and reconstructed image must be positive, and bandwidth limitation, where no power is allowed in the
1
Visiting Astronomer, Maui Space Surveillance System, which is operated by Detachment 15 of the US Air Force
Research Laboratory.
PSF at spatial frequencies greater than the diffraction limit. In addition, any other a priori information about the
object or the PSF can be incorporated into the constraint [2].
While tests of blind deconvolution comparing actual and reconstructed objects have been performed with various
simulated data sets and have been generally successful [2], few tests have been performed on actual astronomical
data owing to the simple fact that it is impossible in general to know a priori what an astronomical source is
supposed to look like. This difficulty can be circumvented by observing solar system objects which have been
visited by spacecraft. Of all solar system bodies, the Galilean satellites are unique in their suitability for such tests:
they are bright enough yet small enough to be observable with AO systems, they are large enough to have multiple
resolution elements across their surfaces, they have invariant surfaces at these scales (except for Io, whose
volcanism can cause large-scale resurfacing on timescales of years), and they have been extensively imaged by the
Galileo spacecraft. We have therefore chosen to test blind deconvolution of AO images by direct comparison of
Galileo spacecraft images of Galilean satellites with ground-based AO observations.
2.
DATA
The highest resolution full disk images of the Galilean satellites have been obtained at visible wavelengths with the
CCD camera on the Galileo spacecraft. AEOS is the largest telescope with an AO system allowing observations at
visible wavelengths, therefore observations from this telescope will provide the best tests of blind deconvolution of
Galilean satellite images.
We obtained images of the Galilean satellites of Jupiter with the AO/Vis system on AEOS on several nights in
November and December 2000. Two sets of images of Ganymede, the largest satellite, were obtained at orientations
very similar to existing Galileo images. We therefore predominantly use those data sets for the deconvolution
experiments.
Fig. 3 compares Galileo Ganymede images obtained at a wavelength of 725nm with AEOS images obtained with the
700-800 nm filter on 6 Nov and 9 Dec 2000. Surface features are clearly apparent on the AEOS images, and these
all correspond to features seen in the much higher resolution Galileo data.
3.
DECONVOLUTION
We use an implementation of blind deconvolution called IDAC (iterative deconvolution algorithm in C) [2] which
is available at babcock.ucsd.edu/cfao_ucsd/idac/idac_package/idac_index.html. Given initial guesses of the object
and the PSF along with different possible constraints, IDAC iteratively works to a solution which minimizes the
error between the object convolved with the PSFs and the observed data and also best satisfies the additional
constraints. The experiments that we have performed using IDAC have generally consisted of attempts to modify the
starting guess for the object and the PSFs or modify the additional constraints and determine how these
modifications affect the deconvolution. As the criteria for success of the deconvolution, we use the FWHM of the
reconstructed image (determined by convolving the Galileo image with a gaussian and matching the surface
appearance), the reliability of apparent surface features, the intensity contrast between known light and dark features
on the surface, and the appearance of deconvolution artifacts.
Fig. 3. Comparison of high-resolution Galileo images and AEOS images of
Ganymede. Note the ~40 degree rotation between the bottom two images.
3.1
Maximum likelihood deconvolution of average images
On November 6, 2000, we obtained images of Ganymede and a calibrator star well-matched to Ganymede’s V=5
brightness within 5 minutes of each other and within 2 degrees on the sky. This situation was ideal for attempting a
simple straight deconvolution rather than blind deconvolution, as the measured PSF from the calibrator star has the
most chance of closely resembling the Ganymede PSF. We performed a standard Lucy-Richardson deconvolution of
the average Ganymede image with the average PSF. The final deconvolved image, shown in Fig. 4, has a FWHM of
0.08 arcseconds, almost 50% larger than the expected diffraction limited FWHM of 0.055 arcseconds. As Fig. 2
suggests, however, many of the individual images likely have better FWHM than the average. We have therefore
also taken the best Ganymede image (defined as having the most power in the power spectrum close to the cutoff
frequency) and deconvolved it with the best PSF. The result is also shown in Fig. 4. The FWHM of this
deconvolution 0.06 arcseconds, which is comparable to the diffraction limit. We also show in Fig. 4 a comparison of
the intensity of one column of the Galileo, raw, and deconvolved images. As can be seen, the deconvolved images
nicely reproduce the intensity contrast between the bright and dark regions of the disk which are otherwise washed
out in the average image. Also seen however, is the extra intensity at the very edge of the disk profile. This extra
intensity can be seen as a ring of higher intensity at the very edge of Ganymede’s disk in the deconvolved images.
Such ringing is a common characteristic of deconvolution of extended objects with hard edges and is caused by the
finite sampling and bandwidth of the PSF. (In fact, the ringing appears if a uniform disk is convolved with the
AEOS PSF and the Lucy-Richardson deconvolved even in the absence of any noise sources.) Other than the ringing,
however, every single feature observed in the deconvolved image is present in the Galileo image of Ganymede.
Fig. 4. Comparison of smoothed Galileo image with Lucy-Richardson
deconvolved AEOS images. The first image shows the image from Galileo
smeared to a resolution with FWHM of 0.055 arcseconds. The second image is
the deconvolution of the average AEOS image with the average AEOS PSF. The
third image is the deconvolution of the best AEOS image with the best PSF. The
plot shows the intensity along the marked column; the thickest line is from the
smeared Galileo image, the middle line is from the average deconvolution, and
the thin line is from the best deconvolution.
The results from this deconvolution suggest that excellent results can be obtained from simply measuring multiple
target images and PSF images close in angular distance and close in time and using the best of each, alleviating the
need for blind deconvolution. In many cases, however, such a technique is not practical and either an appropriate
calibration PSF star cannot be measured or the signal-to-noise in short exposures is insufficient. To simulate such a
case, we have attempted similar Lucy-Richardson deconvolutions of the Dec 9th data using the PSF measured from
Nov 6th. In this case, the poor match of the data and PSF results in a low quality deconvolution with a FWHM of
about 0.09 arcseconds, almost twice the diffraction limit of the telescope. A data set such as this, for which no
appropriate PSF data is available, is a perfect test case for blind deconvolution.
3.2
Completely blind deconvolution
For the ideal blind deconvolution, one should be able to reconstruct the object from the data with no knowledge
whatsoever of the PSF except for the bandwidth limit. Early tests of blind deconvolution advocated using white
noise as the initial estimate to prevent any biasing of the final reconstructed object and found that while the
reconstructed object and PSFs varied with different white noise starts, high quality results were often achieved.
We attempted such completely blind deconvolution with white noise object and PSF starts and a full bandwidth
limit, but results were spectacularly bad. The PSF remained broad and never converged to a narrow core with broad
wings. The resulting deconvolutions were completely dominated by multiple broad ringing with no actual surface
detail visible. The final object was a significantly worse representation of Ganymede than the simple raw data.
Multiple attempts were made with different white noise starting points, but the results were always similar.
We suspect that the lack of a success with completely blind deconvolution comes from a combination of the PSF of
the system and the object observed. Previous white noise tests used data sets composed primarily of combinations of
point sources and PSFs that were appropriate for earlier speckle-type data, where the PSF was composed of several
discrete speckles of otherwise near diffraction-limited images. Reconstructing an extended source with a complex
broad PSF is a significantly more difficult task , and one for which this type of completely blind deconvolution
appears unsuited.
As a second type of completely blind deconvolution, we tried a diffraction-limited Airy disk as our beginning PSF
3.3
Simple myopic deconvolution
Deconvolution with some prior knowledge of the PSF or of the object can no longer be strictly considered blind, but
merely myopic. One implementation of myopic deconvolution is to use some measured PSF, even if it is not the
appropriate PSF, as the starting point, and to use the average of the data as the starting point for the object. In our
case, we use the average of the November PSFs with the December Ganymede data.
Fig. 5. Deconvolution using the average of the Nov PSFs as the starting PSF and
the average of the December data as the starting object guess. The left image
shows the Galileo image smeared to a resolution of 0.10 arcseconds. The middle
image shows the deconvolution result, and the right panel shows the intensity in
the area marked with white lines for the smeared Galileo image (thick line) and
the deconvolved image (thin line). The resulting image has almost a factor of 2
lower contrast than the true object and a slight ring appears at the edge. All
features seen on the reconstructed image appear in the Galileo image.
The deconvolution, seen in Fig. 5, is moderately good. The resolution, at ~0.10 arcseconds, is well above the
diffraction limit, a slight ring appears at the edge, and the intensity contrast is almost a factor of two worse than it
should be. Part of the reason for the poor contrast can be seen in the image: much of the light from the object still
lies in a halo well outside where we know the object should be. The reason for the existence of this halo is that the
starting guess for the object – the average data – contains a large generally invariant halo around it and the algorithm
has no way of knowing if the changes need to happen to the PSF or to the object, so changes are made to both.
Several experiments were tried for removing the halo. The first two relied on the a priori knowledge that the object
is a disk with a certain size, so we regard these trials as less satisfactory though we note that in many cases for
extended objects this knowledge does, in fact, exist. First, we attempted to remove the halo by using the averaged
data as the initial object guess but truncating the data at the known edge of Ganymede. Results from this trial were
poor, as might be expected from such an artificial starting guess. Second, we forced halo removal by adding to the
deconvolution the constraint that the final object can only be non-zero inside the region of the known size of
Ganymede (we will refer to this constraint as “support”). Results from this trial were also surprisingly poor and
turned out worse than the unsupported case above. Reasons for this failure are still under investigation.
The simplest successful method for removing most of the halo turned out to be using the Lucy-Richardson
deconvolution result as the initial object guess. While the mismatch between the November PSF and the December
data prevented this deconvolution from producing good results, starting with this as the object guess and the average
PSF as the PSF guess lead to interesting results, see in Fig. 6. The resulting image has no halo, but extremely bright
wings. The resolution of the final image is 0.08 arcseconds, the best achieve on this dataset so far, but the contrast is
extremely poor
Fig. 6. Deconvolution using the Lucy-Richardson deconvolved image as the
starting object guess and the average PSF for the PSF. The halo prevalent in
earlier trials is now gone and the resolution is good, but ringing is significantly
worse and contrast is poor.
3.4
More complex myopic deconvolution
One of the problems with all of the above trials is imperfect registration of the data images; while the images were
cross-correlated and shifted to be registered within one pixel, even sub-pixel differences become apparent when
examining residuals. Ideally, blind deconvolution should easily take care of a simple PSF translation, unfortunately,
however, in a pixel-by-pixel implementation such as IDAC, a PSF shift is actually a large series of pixel value
changes rather than a single parameter. One method of addressing this difficulty would be to allow a PSF shift to be
another parameter of the deconvolution. We have chosen instead to automatically match the PSF position to each
individual data image with the following technique: We begin by using, as above, the Lucy-Richardson deconvolved
image as the starting guess, but now, rather than using the average PSF as the starting PSF guess for each data
image, we instead Lucy-Richardson deconvolve the starting guess by each individual data image to obtain a starting
PSF guess for each image. We now are starting with a case in which each PSF is appropriately centered and begins
with many of the features that the real PSF must have. The result is shown in Fig. 7. This trial best reproduced the
contrast in the center of the disk and lacks a halo, but ringing is strong and resolution is only a moderate ~0.10
arcseconds.
Fig. 7. Deconvolution using the Lucy-Richardson deconvolution result as the
starting object guess and deconvolving this object with each individual data
image to obtain individual starting PSF guesses. The contrast in this trial is the
best achieved, but ringing is strong and resolution is only moderate.
As a variation of this method and a test for when no PSF information is available at all, we perform a trial where we
use the average data as the starting object and determine individual PSF by deconvolving each data image with a
uniform disk of the expected size of Ganymede. The PSFs created in this manner will lack the high spatial frequency
information available from features on the disk, but should faithfully reproduce the general shape of the wings.
Fig. 8 shows the results of this trial. As expected, the resolution of the final image is the lowest of any of the
attempted trials, at about 0.20 arcseconds. Surprisingly, however, the contrast is also quite poor. This method
appears incapable of providing any type of acceptable results.
\
Fig. 8 Deconvolution using PSFs created by deconvolving uniform disks with
the data images. The resolution and contrast are both poor.
3.5
Unconstrained
As an additional experiment, performed trials where we removed the bandwidth limit constraint, so the PSFs are
allowed to have arbitrarily high power at the highest spatial frequencies. We had hoped that removal of this
constraint would have some effect on lowering or removing the ringing structure, which is caused by a finite
bandwidth. Fig. 9 shows results of a trial identical to that shown in Fig. 7 except without bandwidth constraints.
The final resolution achieved is the highest of any of the trials, at under 0.08 arcseconds. The ringing is only
moderate, but a large halo appears and the contrast is bad. More significantly, for the first time, features appear in
the deconvolved image which do not appear at all in the Galileo image. It appears, not surprisingly, that bandwidth
limitation is an important feature for maintaining reliability of features at the highest spatial frequencies.
Fig. 9. Deconvolution identical to that in Fig. 6 except without bandwidth limit
constraints. The resolution is the highest achieved, but the halo and contrast are
poor.
4.
CONCLUSIONS
These experiments have shown that the results from blind deconvolution of astronomical data depend strongly on
initial guesses of the object and PSF. Different choices for these guesses lead to different tradeoffs between
resolution, contrast, and deconvolution artifacts. Careful consideration should be given to the science result desired
before deciding on the deconvolution starting points.
Resolution: If high-resolution of the interior of the object is important and absolute intensity levels are not needed,
one of the best and simplest methods is deconvolution using the average data as the object guess and the average
PSF as the PSF guess. The contrast is poor, but all features seen are real.
Contrast: The only method we found of preserving correct relative intensity levels was using the Lucy-Richardson
deconvolution result as the starting object guess and deconvolving each of the data images with this result to obtain
starting PSFs. This trial faithfully reproduced the correct contrast and had moderate resolution. As with most trials,
rings at the edge of the object were apparent, so only intensities in the interior of the object can be trusted.
5.
REFERENCES
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2
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