Amplitude Recovery.
The problem of amplitude distortion was found by Labeyrie (1970) to be
solved by calibration of the amplitudes by those of a point source.
The point source (an unresolved star) produced a measure of the
speckle transfer function of the atmosphere which then allowed
deconvolution of the transfer function from the
amplitudes of the object of interest. Thus
speckle interferometry (SI) was developed, where the filled-aperture
optical telescope was viewed as a continuous interferometer
arrayed with ~10 cm subaperatures, corresponding to the coherence
size of the atmosphere; and sequences of ~10 ms snapshots were used to
`freeze-frame' the atmospheric turbulence pattern. Although successful in
recovering diffraction-limited structure information from ground-based
optical and infrared observations, SI does not directly produce images,
because the phase recovery problem is not solved by simple deconvolution.
Solution to the Phase Problem. The groundwork for solving the phase recovery problem was laid by the first measurements of closure phase at optical wavelengths, using discrete element optical interferometers formed by pupil masks over the apertures of 4-5 m class optical telescopes. This work was done by the Cambridge and Caltech groups in the mid-to-late-1980s (Baldwin et al. 1986; Readhead et al. 1988). Closure phase measurements rely on the principle that stochastic phase errors associated with the individual elements (antennas or subapertures) in an interferometer will cancel if the phase is summed around a closed loop of interferometer baselines (the simplest being a triangle). Thus the source structure phase may be recovered (via a set of linear combinations) by constructing a set of such "closure triangles" from the baselines of the array, even when the raw visibility data appears to have entirely corrupted phase values. Although it was noted in the late sixties by Rogstad (1968) that this principle could be applied to optical wavelengths as well as the radio wavelengths where it was first used, it was not until much later that experiments to directly verify this were made.
In the late 1970s and early 1980s, G. Weigelt and his
collaborators developed an independent approach to the use of closure phase
in optical astronomy, without realizing that closure phase was the active
principle involved. Through the use of third moments of the Fourier
transform of SI snapshot frames, they were able to recover the structure
phase for number of different objects up to the diffraction limit (~ 0.1")
of the 1-1.5 m class telescopes they were using (Weigelt and Wirnitzer 1983;
Lohmann et al 1983; Hofmann and Weigelt 1986).
Bispectral Analysis. The phase recovery part of this
technique of complete imaging in SI was termed bispectral analysis by
analogy to the much simpler techniques of Fourier spectral analysis. The
complex bispectrum function used in this analysis was later shown to contain
the exact optical equivalent of radio closure phase in its phase (Roddier
1986; Cornwell 1987). Thus the apparently dissimilar efforts to achieve
ground-based diffraction-limited optical imaging with both filled aperture
telescopes and discrete element interferometers are unified in their
exploitation of the closure phase principle to solve the phase corruption
problem.
Although the Hubble Space Telescope (HST) will provide high resolution
images that are unaffected by the atmospheric limitations of ground-based
telescopes, the optical resolution of ground-based telescopes such as the
Keck 10 m will still be significantly higher for brighter objects where
interferometric imaging can be utilized. The HST also does not observe in
the infrared, where significant work has now just begun using the new array
detectors. The K band (2.2 1m) is particularly interesting for speckle
imaging, since the atmospheric degradation is much less severe than in the
optical, but still requires speckle techniques. Thus ground-based
observations in both optical and infrared can make significant contributions
in a number of areas:
Computational Load.
To illustrate the computational load demanded by bispectral analysis,
we describe some of the parameters of our optical and infrared analysis.
The bispectrum function may be expressed as
For our optical and infrared Palomar data, the resulting complete bispectra
had typically 10**5 to 10**6 values, each requiring ~ 50 floating-point
operations
to calculate. Accumulation of the bispectrum for a set of speckle frames
required storage of the complex bispectrum values (8 bytes per value), the u,v
coordinates (4 bytes per value), and the variances of the real and imaginary
parts of the bispectrum (8 bytes per value). A data set could contain as
many as 10**6 frames in the optical and 5 x 10**4 in the infrared,
and the bispectrum
and its variance is calculated and accumulated for the Fourier transform of
each frame. Thus in an extreme case, 1 flop could be required, along with
storage of 20 Mbyte. Memory is thus typically not a problem at present, but
for the largest data sets, even the NCUBE (running at about 50 Mflops for
our bispectrum code) requires days for complete data reduction. Because of
this, our efforts in exploring the entire bispectrum have concentrated on
infrared data, where the problem is more tractable.
Similar diffraction-limited reconstructions are now being achieved with
our infrared data (Ghez et al. 1990).
In Figure 3 we show two examples of
such reconstructions for binary stars. In Fig. 3(a) we show a 2.2 micron
recovered image of the unusual system T Tauri, which contains the prototype
of the pre-main-sequence stellar class which bears its name, along with the
infrared companion shown. This system also has a reported optical companion
(Nisenson et al. 1985) which is distinct from the other two
components. Fig. 3(b) shows a previously unresolved system, HR 8028, which
has a companion unseen in optical speckle work. The dynamic range of both of
these images is ~50:1, giving a contrast of more than 4 mag.
The work to date can be characterized as primarily developmental since
the yield in scientific results has been somewhat secondary to the work on
understanding the technique and optimizing the algorithms. This emphasis has
however been a fruitful one: we feel that our present algorithms for speckle
imaging are state-of-the-art, and the time allocation committee for the 200
inch telescope has reinforced this belief by granting nearly all of the
requested observation time (20 nights) for infrared and optical speckle
imaging for 1990. Improved Amplitude Recovery Techniques for Speckle Interferometry.
In addition to development of new algorithms for phase recovery in SI data,
a spinoff occurred from our SI work which has led to a significant
improvement in the techniques for accurate diffraction-limited amplitude
recovery, even at surprisingly low signal-to-noise ratios in the raw
data. Because much of the optical speckle observations made with the 200
inch telescope in our observing program were in what is known as the
photon-limited regime, where the number of detected photons per speckle
falls below unity, we found that existing techniques of amplitude
calibration were inadequate to achieve the diffraction-limit of the 200
inch. The solution was found through a novel adaptation of the CLEAN
algorithm, commonly used in radio synthesis mapping, to deconvolve the noisy
speckle data (Gorham et al 1990). The algorithm is found to be very robust,
because CLEAN effectively interpolates over the regions of low
signal-to-noise in the Fourier plane in much the same way that it
interpolates over the unmeasured visibility regions in the sparsely sampled
regions of aperture synthesis visibility data.
Figure 4 shows a typical result using
this algorithm. Fig. 4(a) shows the
raw power spectrum, with the diffraction-limited information appearing only
sparsely at the high spatial frequencies due to the low signal-to-noise. In
Fig. 4(b) the result of the application of our algorithm is shown as an
autocorrelation function of the binary system (a Fourier transform of the
reconstructed power spectrum), showing that the diffraction-limited
information has been recovered. Standard techniques of SI are completely
unable to deal with data like this. The availability of the NCUBE in this
case allowed us to quickly explore the parameter space necessary to optimize
this technique.
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Return to Caltech Computational Astronomy
2. Scientific Objectives
The scientific objectives of applying interferometric techniques to
filled-aperture telescopes at optical and infrared wavelengths are based on
the desire for complete structure information in astronomical images, up to
the diffraction limit of the telescope. The two techniques that appear the
most promising at present are: direct solutions to the phase recovery
problem, combined with speckle interferometry; and active optics systems,
which attempt to correct the phase distortion of the wavefront before
detection. We are involved only with the first technique here; in section
3.2 we outline briefly what we see as the future relationship between the
two.
There is then clear scientific justification for such programs; in the
following section we present some of the details of why they are
interesting from the standpoint of computational astronomy.
3. Computational Aspects of Bispectral Imaging
Because the bispectrum function is a quantity which is third order
in the Fourier transform of the intensity, and thus sixth order in the
electric field of the source, it is extremely
demanding of computational resources. The efforts we report
here actually led recently to the first functioning code
(Gorham 1988; Gorham et al 1989) which could
utilize all of the available parameter space afforded in the bispectrum for
typical astronomical data.
This has led to significant improvements in the signal-to-noise ratio
of recovered images at infrared wavelengths (Ghez et al 1990). Previous
efforts concentrated on using very limited portions of the available
closure phase information because of the computational demands. However
it has been our intent at Caltech to achieve image recovery which pushes the
limits of ground-based instruments, and to this end we wished to achieve
the full imaging resolution of the Palomar 200 inch telescope, ~ 30
mas at 630 nm (our typical optical observation wavelength),
or ~100 mas at 2.2 microns in the infrared. The availability of
a concurrent supercomputer (the NCUBE) at Caltech has been essential to
the development of these imaging techniques, because of the freedom it has
provided from computational constraints that have previously hindered
these developments.
B(u,v) = J(u)J(v)J(-u-v) (1)
where u and v are vector positions in the Fourier
plane, corresponding
to an interferometer baseline, and 1 is the complex value of the image
Fourier transform at 1. For a typical optical speckle snapshot, the
~1 arcsec seeing disk size together with the requirement for
Nyquist sampling of the diffraction-limited spatial frequencies implies that
at least ~60 pixels are needed across the image for the 200 inch telescope
at 630 nm, and ~20 pixels at 2.2 microns. In practice, the field of view
must be about twice as large as the largest structure being imaged, to
avoid the problem of edge structure "folding back" into the image.
Sampling at frequencies well above the Nyquist limit is also recommended to
avoid attenuating the higher frequencies. We have typically used at least
128**2 pixels for the optical and 64**2 in the infrared. Thus
J(u) has an equal number of complex values, and combinatorics on
equation (1) shows that the bispectrum of an M**2 pixel image
has a 4-dimensional volume bounded by
(4)
VB <= (M**2 (M**2 - 1)) / 2! (2)
Taking into account the symmetry properties of the bispectrum,
requirements that the bispectrum values correspond to a physical
closure triangle in the telescope aperture, and possible oversampling,
the actual number of bispectrum values used in computation can be reduced to
(4)
VB' ~ M'**4 / 40 (3)
where M' = M/s and s is the oversampling factor (Gorham et al
1989).
4. Results
Closure-Phase Imaging. The program in optical and infrared speckle
imaging and aperture mask interferometry has been completely successful to
date. We have achieved our goals of diffraction-limited imaging in each of
the attempted areas: optical non-redundant mask interferometry (Nakajima et
al 1989); optical speckle imaging (Gorham et al 1989); infrared
non-redundant mask interferometry (Weir et al 1990); and infrared speckle
imaging (Ghez et al. 1990). Figures 1 - 2 show a sample of results from the
optical speckle imaging work (from Gorham et al 1989):
Figs. 1(a),(b), and
Fig. 2(a) are diffraction-limited
reconstructions of a number of binary stars
from optical data, showing signal-to-noise ratios comparable to those of
snapshot VLA radio synthesis images.
In Fig. 2(b) we show the first (to our
knowledge) reconstruction of an optical diffraction-limited image from
closure phase information only, using the same data which gave the
reconstruction in Fig. 2(a). Although the dynamic range is much lower than
in Fig. 2(a), the structure information is clearly present in the closure
phase data alone.
5. References
6. Publications by the Caltech Computational Astronomy group:
Closure Phase Imaging and Speckle Interferometry in
Optical and Infrared Astronomy
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