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Subject: SPACE Digest V12 #34

SPACE Digest                                      Volume 12 : Issue 34

Today's Topics:
      Re: Image Restoration (was Hubble Trouble) [LONG]

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Date: 9 Jul 90 00:29:45 GMT
From: uvaarpa!murdoch!fits![e-mail]  (Don Wells)
Subject: Re: Image Restoration (was Hubble Trouble) [LONG]


  (Notations [xxyy], e.g. [PSB88], refer to references given at the end.)
        (There is also a glossary at the end.)

A week ago, in article <[e-mail]> [e-mail]
(Steve Willner, OIR) wrote:

sw> Here is some information on the HST status and some comments of mine.
sw> The information came from a talk given here on 6/29.  The speaker got
sw> his information directly from a briefing to the Science Working Group...
sw> The (Wide Field/Planetary Camera and Faint Object Camera) will be
sw> the worst affected.  At best, image restoration might be able to derive
sw> full-resolution images from the 10% of the light still within the
sw> central spot, but at least a factor of 10 (2.5 magnitudes) in
sw> sensitivity will thereby be lost.  Ultraviolet imaging will be possible
sw> at spatial resolution comparable to ground-based telescopes in visible
sw> light.

The last two sentences sound to me like a reasonable projection of
what will happen, with the exception that the factor of 10 loss of
sensitivity estimate might prove to be unduly pessimistic, especially
in the ultraviolet where night sky background may be lower (remember,
the real problem is not so much the broadening of the PSF as it is the
excess sky noise).  There is reason to think that the full spectrum of
spatial frequencies provided by the 2.4 meter aperture (the "baseline"
in interferometer terminology) are present in the PSF, so that high
quality (i.e., "beautiful!") restorations are likely to be possible
(see discussion below).

sw> ---------What follows is from me, not from the talk-------------------
sw> The problem of image restoration is rather different than the radio
sw> case.  An optical detector measures the wave "intensity," not the
sw> "amplitude."  The difference is that phase information is lost.  Some
sw> degree of reconstruction should certainly be possible, but it is not
sw> clear how much.

The waves interfere to form diffraction fringes. The positions of
these fringes in the image matrix carry the information about relative
source positions. The orientation of the sources is not ambiguous.
Therefore, I do not agree with the assertion that the phase
information needed to restore the scene has been lost. The key
question is whether fringes are seen. If they are (see below), then
the information about the sky is encoded in the position and amplitude
of these fringes, and within the range of spatial frequencies which
have been sampled we are limited only by the S/N of the data. The only
real difference for restoration purposes is that radio interferometers
do not usually measure the zero baseline and so they have a PSF with
integral of zero. Because of this fact (and also the zeroes elsewhere
in the UV-plane due to unsampled baselines), linear restoration is not
used in radio synthesis imaging; all radio restoration is nonlinear
[NN86,PSB88]. I do not expect that this difference in the form of the
PSF will prevent nonlinear techniques from restoring the HST data.

sw> Since any reconstruction costs signal-to-noise, the
sw> faintest objects will not be reconstructable.

Nonlinear restoration does not degrade S/N (see discussions in [F72],
[FW78], [W80], [NN86]). A better gross generalization would be "the
MEM is maximally non-committal regarding unmeasured data and...
produces images that are as featureless as possible" (i.e., as
noise-free as possible while consistent with the measured data; quote
is from Sect.1.2 of [NN86]). It is true that S/N is degraded with
*linear* algorithms, but I do not expect that linear restoration will
be used for HST data once observers see how much better the nonlinear
algorithms perform.

sw> However, there are lots
sw> of bright objects where HST resolution will be valuable...

Yes. For example, I expect than a well-exposed UV image of the jet in
M87 taken with the FOC (and maybe even with the PC) will enable a fine
restoration to be produced, perhaps to nearly the diffraction limit.
It would be very interesting to compare such an image to the VLA image
which has a PSF of slightly less than 0.1 arcsec.

sw> Steve Willner            Phone [phone]         Bitnet:   willner@cfa
sw> Cambridge, MA 02138 USA                 Internet: [e-mail]


    =-=-=-=-= The HST PSF does show fringes! =-=-=-=-=

In a followup a week ago <Sat, 30 Jun 90 05:33:47 GMT,
[e-mail]>, I said that: "I
would be curious to know if the FOC sees diffraction patterns in the
out-of-focus images, especially when using a narrow band filter (e.g.,
H-alpha).  If it does, then those irregular "rings" would be carrying
the *full* spatial frequency range of the 2.4meter aperture!"  A day
or so later a friend who reads sci.astro and who had heard a technical
presentation by a member of one of the IDTs told me via E-mail that:
"Lovely diffraction patterns are seen and they fit them to a simple
model incorporating pure spherical aberration, obstructions due to the
telescope mount and so on." Then, later in the week I mentioned this
statement in the presence of another person who is associated with
another IDT, and he confirmed it. He said that the fit of the
diffraction fringe pattern to textbook theory [BW59, Ch.9] is so exact
that computed and observed images are practically indistinguishable.
So, when you hear that the wavefront is wrong by half a wave, what it
probably means is that that number is the amount that makes the
diffraction model fit the data.  It sounds like this is a *perfect*
mirror, perfectly wrong, and the magnitude of the wavefront error is
now exactly known. I.e., the necessary information to design proper
correcting optics for the replacement instruments is available.

I have not seen any actual imagery from HST, and I have not computed
any model images from theory, but I will speculate a bit anyway.
Normally a perfect mirror would bring the wavefront from all parts of
the aperture to about the same point, where interference fringes would
be formed and add up to form the Airy disk surrounded by rings. But
there is no focal point at which this can occur in the HST as built;
by choosing a focal point you choose which ring of the aperture will
be able to interfere. My intuitive guess is that it would be best to
choose to focus a ring near the outer edge of the aperture (the
"marginal focus") in order to maximize the area coming to a focus and
the longest baseline (but that may be naive; better to do the
calculations to confirm whether the marginal focus is the best
choice).

I expect that the outer ring can form a sharp core with width similar
to the Airy disk (see Fig.9.4(a) on p.476 of [BW59]). Apparently about
10% of the flux is in this core and nearby rings.  Note that all
baselines from zero up to the diameter of the HST are making fringes
in this core region because of all the chord baselines connecting
elements around the circumference of the ring aperture.  Therefore,
probably this will be a fine PSF for restoration work, and I expect to
see pretty (and scientifically useful) images from any sources for
which sufficient photons can be accumulated, either because they are
bright or because sufficient exposure time can be obtained. If the
background noise level is low enough (e.g., in the UV) even the
out_of_focus halo may be able to contribute to the restoration and
thereby recover some of the lost S/N. The best, most easily restored,
fringes will be seen with narrowband filters, but more photons will be
available as filters are broadened. This tradeoff of S/N vs. fringe
visibility as a function of optical bandwidth will probably turn out
to be a very interesting question.

    =-=-=-=-= Exhortation =-=-=-=-=

Image restoration has not yet become credible in optical astronomy, in
spite of the beliefs and hopes of all the people who have worked on it
over the past 20 years. This is because there has never before been a
direct imaging problem in optical astronomy in which scientific
results depended *critically* on restoration technology. Imaging on
the (crippled) HST really does need nonlinear deconvolution. Now is
the moment for the restoration specialists to exert maximum effort on
this problem!

    =-=-=-=-= Selected references =-=-=-=-=

[BW59] M.Born & E.Wolf, "Principles of Optics", Pergamon Press 1959
    (1st ed), LOC=QC355.B63.

See Ch.9 "The Diffraction Theory of Abberations", Sect.9.4 "The
Diffraction Pattern Associated with a Single Abberation", and
Sect.9.4.1 "Primary Spherical Abberation". Note Fig.9.4 "Images in the
marginal focal plane... in the presence of primary spherical
abberation". The examples are for filled apertures; central
obscuration cases like HST follow from the Babinet principle. See also
Fig.9.15 "Normalized frequency response curves for incoherent
illumination, at selected focal settings of a system suffering from a
small amount of primary spherical abberation...".

[F72] B.R.Frieden, 1972, J. Opt. Soc. Am. 62, 511-18.

This is the original paper on MEM restoration of photon-noise-limited,
diffraction-limited optical signals. Frieden showed that information
even beyond the Rayleigh limit is recoverable in favorable cases. His
graduate student's unpublished thesis (R.Herschel, "Unified Approach
to Restoring Degraded Images in the Presence of Noise", Univ.  Arizona
1971, also Opt. Sciences Ctr. Tech. Rep. 72) was produced in the same
period and contains a description of what was probably the first
operational 2-D nonlinear image restoration algorithm.

[FW78] Frieden,B.R., Wells, D.C., 1978, J. Opt. Soc. Am. 68, 93-103,
    "Restoring with Maximum Entropy. III. Poisson Sources and
    Backgrounds".

Much of the discussion in this paper is relevant to the HST
diffraction-limited case with its photon (Poisson) noise, but the use
of a separable Gaussian kernel is not (it is the right tactic for
seeing-limited imagery). This is the original reference for the
concept of subtracting the background image (separate handling of high
and low spatial frequencies) while accounting for the signal-dependent
noise variation.

[W80] Wells, D.C., 1980, Proc. Soc. Photo-Opt. Instrum. Eng. 264,
    148-156, "Nonlinear Image Restoration: What We Have Learned".

This is a review paper, with references for the first ten years of
non-linear deconvolution research, especially oriented toward the
optical case.  Last week I went back into my files and re-read this
paper, and found that I still agree with everything I said ten years
ago. I was also surprised to find that I had even mentioned the Space
Telescope in one of the sections!  SPIE volume 264 is "Applications of
Digital Image Processing to Astronomy", the proceedings of a
conference held at JPL/Caltech in August 1980.

[NN86] R.Narayan and R.Nityananda, "Maximum Entropy Image Restoration
    in Astronomy", Ann. Rev. Astron. Astrophys. 1986, 24, 127-70.

This is a superb, comprehensive review of the MEM field, with special
attention to the successes of deconvolution in radio astronomy, and
with good comparisons of MEM to other techniques like CLEAN. This ARAA
chapter is *required reading* for any astronomer who is seriously
interested in nonlinear deconvolution. I agree with the final
sentences: "Ultimately, in our view, any image consistent with the
data and free from obvious artifacts must be taken seriously. The goal
of restoration techniques is to produce such images. When the results
are relatively independent of the method, we gain confidence in them.
However, if we obtain significantly different restorations, the places
where they agree should tell us what we can really believe, and the
differences should indicate what extra data are needed."  Any radio
astronomer in the 80's knows just what they mean! Their Fig.5 (p.145)
shows restorations of test data by several variations on MEM and by
CLEAN.

[PSB88] "Synthesis Imaging In Radio Astronomy", ed. R.Perley, F.Schwab
    & A.Bridle, LOC=QB479.2.S96, ISBN=0-937707-23-6, San Francisco:
    Astronomical Society of the Pacific, 1988, 509 pages.

This [text]book is an editted version of notes from the Third NRAO
Synthesis Imaging Summer School, held at Socorro, NM, June 1988, for
which the staff of NRAO prepared a series of lectures for serious
students of synthesis imaging and image processing. Both the theory
and the practice of synthesis imaging, including MEM and CLEAN, are
covered.

[GGHP90] P.Gorham, A.Ghez, C.Haniff and T.Prince, 1990, Astron. J.
    100, 294, "Recovery of Diffraction-Limited Object Autocorrelations
    from Astronomical Speckle Interferograms using the CLEAN Algorithm".

I chose this reference, in the July 1990 (current) issue of AJ, to
illustrate that use of nonlinear deconvolution is not confined to
radio astronomy. The authors, all from Caltech, are reducing data from
the Palomar 5 meter. The abstract says: "We present a new technique
for processing speckle interferometric data which uses the CLEAN
algorithm, originally developed for the removal of the effects of
incomplete spatial frequency coverage in aperture synthesis radio maps
...  because of the immmunity of CLEAN to gaps in the spatial
frequency coverage of the power spectrum, deconvolution is robust
under conditions where regions of low signal-to-noise ratio in the raw
speckle data effectively introduce such gaps..." Later in the paper we
read "Ebstein (1987) has developed an iterative Fourier deconvolution
technique which uses a series of projections onto convex sets to
successively constrain an initial estimate of the object spectrum,
with requirements such as positivity and symmetry in the object power
spectrum and positivity of the corresponding autocorrelation function
(ACF). Here we present an alternative technique which employs CLEAN
algorithm to perform the deconvolution entirely in the domain of the
specklegram average ACF ... the technique may be applied to recover a
diffraction-limited autocorrelation of the observed astronomical
object, even if the SNR of the data is of order unity or less
throughout large portions of the raw object power spectrum..."  Note
that the specklegrams are optical *photon_noise_limited* images.  The
details of the speckle problem are different from those of the HST
imagery, but the basic problem of inverting an integral equation is
the same. The key to getting good results in all such problems is to
introduce non-linear constraints (usually positivity) into the
algorithms.

    =-=-=-=-= Glossary of Selected Terms =-=-=-=-=

Restoration    }    Synonyms. "restoration" is used by image processing
Reconstruction    }    people. "inversion" (of integral equations) is often
Deconvolution    }    used by mathematicians. I myself often prefer to use
Inversion    }    "deconvolution", but use "restoration" interchangeably
            with it.

PSF        }    Synonyms. PSF = Point_Spread_Function. Radio
kernel        }    astronomers often refer to the "beam". Mathematicians
beam        }    often prefer "kernel". Many signal processing people
impulse_response}    like "impulse response". Most image processing people
            prefer "PSF", and so do I, but I use all four terms
            interchangeably.

UV-plane    = 2-D Fourier Transform of image plane. Coordinates are
        spatial frequency, e.g. cycles per arcsecond, and values
        in cells of matrix are complex, amplitude and phase. Radio
        interferometers measure "complex [fringe] visibilities"; these
        are "gridded" into the UV matrix and then an inverse Fourier
        Transform produces the image. The UV-plane is the arena in
        which much of the discussion of image restoration takes place.
        Note!! Do not confuse this use of "UV" with UV as an
        abbreviation for "ultraviolet". "UV-plane" is a traditional
        notation which happens to use the same letters for variable
        names.

IDT        = HST Instrument Definition Team (aka Guaranteed Time
        Observers [GTOs]). Each of the five instruments of the HST
        was proposed by a team who have had responsibility for its
        implementation over the past decade. In return team members
        will get guaranteed time during the first months of HST
        operation. There is also the Astrometry IDT, who will use
        the FGS [Fine Guidance Sensors] for their science.

FOC        = HST Faint Object Camera. The FOC has a long focal length,
        large image scale, in order to fully sample the diffraction
        pattern of the HST.

PC        = Planetary Camera portion of the WF/PC HST instrument.
        The PC has a longer focal length (f/48?) than the Wide Field
        camera; it is approximately critically sampled for the
        visible and near-IR bands.

sampling    = as in "critically sampled", "under-sampled", "fully sample
        the diffraction pattern". This refers to the famous Sampling
        Theorem, which says (roughly) that any bandlimited function
        is fully specified by samples taken at a frequency at least
        twice as high as the bandlimit. In the case of the HST the
        bandlimit is set by the main mirror diameter, 2.4 meters; any
        focal plane of the HST in which wavefronts from the outer
        ring of the aperture interfere will contain interference
        fringes with angular spatial frequency (in radians) of about
        lambda in meters divided by the diameter in meters [2.4m].
        Critical sampling needs angular pixel frequency twice this
        interference fringe frequency limit. Failure to meet this
        criterion results in the aliasing of high spatial frequency
        information back to lower frequencies; this mixing of signals
        is *very* difficult to untangle once it has been allowed to
        occur.

baseline    = as used in radio interferometry, either (1) a pair of
        antennas, (2) the geometric line connecting such a pair,
        (3) the projection on the sky of the connecting line, or
        (4) the point in the UV-plane whose spatial frequency
        coordinates correspond to definition (3). NRAO's VLA, the
        largest radio interferometer, with 27 antennas, produces
        fringe visibilities (samples in the UV-plane) for 351
        baselines (27*26/2). Earlier, when discussing the HST imaging,
        I used the term "baseline" in several places as though the
        HST aperture could be considered to be made up of a mosaic
        of antennas whose wavefronts interfere to form "baselines"
        in the radio sense.

            =-=-=-=-=

Donald C. Wells, Associate Scientist | NSFnet: [e-mail] [192.33.115.2]
National Radio Astronomy Observatory | SPAN:   NRAO::DWELLS    [6654::]
Edgemont Road                        | BITnet: DWELLS@NRAO
Charlottesville, VA 22903-2475 USA   | UUCP:   ...!uunet!nrao.edu!dwells
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End of SPACE Digest V12 #34
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