Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.adass.org/adass/proceedings/adass94/adorfhm1.ps Äàòà èçìåíåíèÿ: Tue Jun 13 20:41:55 1995 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 01:51:06 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: universe

Astronomical Data Analysis Software and Systems IV
ASP Conference Series, Vol. 77, 1995
R. A. Shaw, H. E. Payne, and J. J. E. Hayes, eds.
On the Combination of Undersampled Multiframes
H.­M. Adorf
Space Telescope--European Coordinating Facility, European Southern
Observatory, Karl­Schwarzschild­Str. 2, D­85748 Garching b. M¨unchen
Abstract. Some cameras with digital detectors, such as the Wide Field
and Planetary Camera 2 onboard the Hubble Space Telescope, undersam­
ple the joint point spread function of telescope and camera optics. In
order to overcome undersampling, two or more frames of the same field
may be obtained which are shifted with respect to each other by frac­
tions of a pixel. During data analysis such multiframes must be ``regis­
tered'' and ``combined''. This contribution investigates a novel method
for merging undersampled multiframes based on projections onto convex
sets (POCS). The method does not require a point­spread function; it
can cope with missing data, and may incorporate prior knowledge such
as non­negativity and band­limit constraints.
1. Introduction
At present the Wide Field and Planetary Camera 2 (WFPC­2) is the primary
science instrument on­board HST . Correcting HST 's spherical aberration, it
displays a vastly improved image quality compared to WF/PC­1. However,
it's CCDs coarse sampling cannot exploit HST 's high spatial resolving power.
Obviously science programs such as photometry of dense star fields and the
morphological classification of distant galaxies would benefit from a method
allowing a proper combination of two or more ``dithered'' WFPC­2 frames, i.e.,
frames that are shifted with respect to each other by fractions of a pixel.
Resolution improvement by combining (potentially undersampled) multi­
frames has in the recent past been investigated by a number of researchers
outside astronomy (for references see Adorf 1994). Within astronomy a com­
bination method has been proposed by Lucy (1993 and references therein) and
further developed by Hook & Lucy (1993 and references therein). Being a gener­
alization of the well­known Richardson­Lucy restoration method, this algorithm
requires knowledge of the point spread function (PSF).
An alternative combination algorithm is presented here which is based on
the concept of projections onto convex sets (POCS, Sezan, & Stark 1983; Youla
1987 and references therein). The proposed algorithm is similar in spirit to the
POCS­method for reconstructing irregularly sampled data series (Adorf 1993).
The algorithm does not presume knowledge of the PSF; it can accommodate
missing data, as well as bandpass and non­negativity constraints. When a non­
negativity constraint is effective, the algorithm becomes non­linear.
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2. The POCS­Based Combination Algorithm
Assuming that the relative shifts between the undersampled frames are known,
the POCS­algorithm (for two undersampled frames) can be stated as follows:
1. guess initial high­resolution (hi­res) estimate
2. fractionally shift hi­res frame to register it with low­resolution (lo­res)
frame #1 and replace pixel values in hi­res frame by those from lo­res
frame #1
3. fractionally shift hi­res frame to register it with lo­res frame #2 and replace
pixel values in hi­res frame by those from lo­res frame #2
4. apply bandpass constraint
5. optionally apply non­negativity constraint
6. if not converged, iterate starting from step 2
This reconstruction algorithm belongs to the class of POCS­algorithms since
each of the steps 2 to 5 is a projection onto a convex set in the linear space of
all images. The iteration provably converges if there is a common point in the
intersection of all convex sets. The POCS­algorithm has been implemented in
the Interactive Data Language (IDL) image processing package.
3. Registration
Using fine lock on guide stars, HST usually achieves a pointing precision of about
7 milliarcsec RMS, or better. However, for non­contiguous observations its abso­
lute pointing accuracy on the detector (i.e., the difference between commanded
and observed pointing) has been worse. Therefore, in practice, the registration
parameters have to be estimated from the data---a non­trivial problem in the
presence of undersampling.
Traditionally, registration parameters have been derived from position mea­
surements of stars contained in the field. In the context of this work, the reg­
istration parameters were estimated using a cross­correlation technique: the
asymmetric shape of the correlation function in the vicinity of its peak was ex­
ploited for locating the peak to subpixel accuracy. The correlation approach
has the merit of not requiring the presence of point sources in the field. While
this technique has passed a few simple tests, further investigations are necessary
to prove that it is a viable general method for registering ``dithered'' WFPC­2
frames.
A remaining problem concerns the WFPC­2 CCD­chips, which have a non­
regular pixel grid. Single rows with narrower pixels are interlaced with the
normal rows. This pattern repeats itself. Thus there is no set of registration
parameters globally valid across the field of view.

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Figure 1. A ``peculiar'' galaxy in the distant, rich cluster
CL0939+4713. Top left: part of a single coarsely sampled ``subframe
#1'' at a scale of 0.1 arcsec/pixel; top right: part of the displaced
subframe #2, shifted by \Deltax = \Gamma0:13 and \Deltay = 0:31 WFC pixels;
bottom left: high­resolution output of the POCS­reconstruction al­
gorithm; bottom right: same as before, but upsampled by a factor
of four using sinc­interpolation. The combined high resolution frame
reveals that the ``peculiar'' galaxy presumably consists of a pair of in­
teracting galaxies.
4. Application to WFPC­2 Data
The POCS combination algorithm was applied to public HST Early Release
Observations of the distant, rich cluster of galaxies CL0939+4713 (HST program
number 5190). The data set has been analyzed by Dressler et al. (1994), who
comment: ``As remarkable as these images of systems at high redshift are, they
point up the need for : : : higher resolution: : : Substepping : : : would dramatically
increase the detail in these very high redshift systems : : :''
The CL0939+4713 data set consists of 10 WFPC­2 exposures divided into
two sets of 5 consecutive exposures. Each set was combined into a single frame
using the STSDAS task ``crrej'', which simultaneously rejects cosmic ray hits.
Subsequent work was carried out on chip #2 frames. The two undersampled
frames in the working set still contained a large number of so­called ``warm/hot
pixels'' at fixed CCD positions which were detected using a simple threshold

4
algorithm. Data values at the location of warm/hot pixels were marked as
``missing''.
From the resulting two clean 800 \Theta 800 pixel frames, two 128 \Theta 128 pixel
subframes centered on a peculiar galaxy were extracted (Figure 1a, b), and
registered via cross­correlation. Fractional offsets of \Deltax = \Gamma0:13 and \Deltay =
+0:31 WFC pixels (i.e., \Deltax = \Gamma13 and \Deltay = +31 milliarcsec, respectively)
were found.
The POCS combination algorithm was applied to these subframes in a
staged approach, going through a low­resolution and a medium­resolution phase
to the final high resolution. Altogether 30 iterations were executed. The re­
sulting high­resolution frame (Figure 1c, d) nicely interpolates the missing data,
and reveals fine spatial structure difficult, if not impossible, to gather from the
two individual undersampled frames. The contrast in the high resolution image
is also increased compared to the coarsely sampled frames.
5. Summary
A novel, iterative POCS (projections onto convex sets) based algorithm has
been presented for combining two or more fractionally shifted and potentially
undersampled observations of the same scene into a high­resolution image. The
method does not require knowledge of the point spread function, and can ac­
commodate missing data as well as bandpass non­negativity constraints. The
algorithm can also be used for solely increasing the signal­to­noise ratio without
resolution enhancement. The POCS­algorithm has been successfully used to
generate high resolution images from HST WFPC­2 observations of the distant,
faint galaxy cluster CL0939+4713, revealing morphological detail which cannot
be easily discerned in the individual coarsely sampled observations.
References
Adorf, H.­M. 1993, in Proc. 5th ESO/ST­ECF Data Analysis Workshop, eds.
P. GrosbÜl et al. (Garching, ESO), p. 191
Adorf, H.­M. 1994, ST­ECF Newsl., 22, in press
Dressler, A., Oemler, J. Augustus, Sparks, W. B., & Lucas, R. A. 1995, preprint
Hook, R. N., Lucy, L. B. 1993, ST­ECF Newsl., 19, 6
Lucy, L. 1993, in Proc. Science with the Hubble Space Telescope, Chia Laguna,
Sardinia, eds. P. Benvenuti and E. Schreier (Baltimore, Space Telescope
Science Institute), p. 207
Sezan, M. I., & Stark, H. 1983, Appl. Opt., 22, 2781
Youla, D. C. 1987, in Image Recovery---Theory and Application, ed. H. Stark,
(Orlando, Academic Press), p. 29