Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.adass.org/adass/proceedings/adass94/hookr.ps Äàòà èçìåíåíèÿ: Tue Jun 13 20:47:45 1995 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 00:34:53 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: star

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.
Star Finding and PSF Determination using Image
Restoration
R. N. Hook and L. B. Lucy 1
Space Telescope--European Coordinating Facility, European Southern
Observatory, Karl­Schwarzschild­Str. 2, D­85748 Garching bei
M¨unchen, Germany
Abstract. Crowded­field stellar photometry consists of three main
phases: locating the point sources, determining the point spread func­
tion (PSF) and measuring the point source brightnesses. In earlier work
(Lucy 1994; Hook & Lucy 1994) we have described a two­channel restora­
tion method which provides photometric fidelity and addresses the last
of these items in a restoration context. Here we describe two further en­
hancements. First, an experimental method for locating stellar images
by enhancing the sharp cores of stars using a multi­channel entropy min­
imization technique is described. In addition an extended version of the
two­channel method is given which allows PSFs to be extracted from des­
ignated point­sources in images during restoration. A default PSF may
be used to regularize the result. A simple but accurate model for the
form of ground­based PSFs (Saglia et al. 1993) has been implemented
and seems a suitable choice of default PSF for ground­based images. Ex­
amples using a typical ground­based CCD image of a star cluster are
given.
These methods are available in preliminary implementations running
within both the IRAF and MIDAS data analysis packages.
1. Introduction
In earlier papers (Lucy 1994; Hook & Lucy 1994) we described a two­channel
restoration method in which one channel contains point­sources and the other
contains a smooth background to represent the sky or an extended object. The
second channel is regularized by the addition of an entropy term to the expression
being maximized, but the first is treated as a simple likelihood maximization.
This method has many advantages including the suppression of the artifacts
often seen around bright stars in conventional single channel restorations and
photometric fidelity without the bias found in most restoration techniques (e.g.,
Cohen 1991).
The technique of multiple channels with different regularization is here ex­
tended in two directions of relevance to crowded field photometry. First an
experimental three­channel method is described in which one channel has a neg­
1 Affiliated with the Astrophysics Division, Space Science Department, European Space Agency
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ative coefficient for the entropy term. This has the opposite effect to the normal
smoothing which follows from positive entropy coefficients and enhances point
sources, ultimately to the point where they become ffi­functions (single pixels
or sub­pixels). Secondly, ways of extracting a PSF from an image containing
designated point­sources are described as well as different ways in which such a
PSF may be regularized. An implementation of a recently suggested theoretical
form for the PSF of ground­based telescopes is available and is suitable for use
as a default PSF.
2. Star Finding
Maximum likelihood restorations often develop noise ``spikes'' when large num­
bers of iterations are used and regularization methods, often based on maxi­
mizing an entropy expression, are used to impose smoothness. However, point
sources in images are really ffi­functions on the sky and hence are the opposite
of smooth. This suggests a method for finding point sources using an entropy
minimization rather than the normal maximization used to impose smoothness.
In this case an objective function of the following form is maximized:
Q =
X
i
~
\Phi i
ln\Phi i
+ ffS + fiT ; (1)
where ~
\Phi is the observed intensity distribution, \Phi is the current estimate and
the summation is performed over all pixels in the image. The first term is the
likelihood, and S and T are entropy­type expressions. The second term has
positive ff and acts as a standard regularization to enforce smoothness on the
background channel. The third term has fi ! 0 and leads to a minimization
of the entropy of a ``points'' channel. Both entropy expressions are evaluated
relative to floating priors, the second term relative to a highly smoothed version
of the total estimated intensity in the image and the third relative to a slightly
smoothed version of the current estimate for the points channel. It is necessary
to choose the parameters ff and fi, as well as the degree of smoothing applied
when creating the floating defaults, so that even faint points are successfully
found but noise clumps are not detected. An experimental implementation has
been coded as a program called stars. Unlike normal star finding methods which
seek local maxima this method is global and works on the entire image.
Figure 1 (left) shows a typical deep CCD image of a star cluster. This
frame is part of an I \Gammaband image of the cluster M71 taken by F. G. Jensen
(Aarhus) using the Nordic Optical Telescope, and used with his permission.
Figure 1 (center) shows the points (low entropy) channel produced by applying
this method to Figure 1 (left), and Figure 1 (right) shows the smooth (high
entropy) background channel obtained simultaneously. All the stars visible to
the eye in the input have been found and there are few spurious detections. The
background smooth image shows small artifacts caused by the displacements of
the stars from the centers of pixels. This map of the star positions may then
be used directly as input to the PLUCY two­channel code to obtain unbiased
magnitudes for the designated point­sources.

3
Figure 1. An example of star finding: see text for details.
3. PSF Determination
Successful image restoration and photometry both require a good knowledge
of the PSF. Many restoration methods can be generalized to allow simultane­
ous ``blind iterative restoration'' in which the PSF is obtained simultaneously
with the restored image. Such methods are generally thought to be unreliable
and are little used. However, when the additional information of designated
point­sources is added much greater robustness and reliability is achieved. Such
simultaneous PSF determination can easily be added to the PLUCY two­channel
code and a preliminary example of its use on HST data is given in Hook & Lucy
(1994).
Such results tend to retain noise features from the data frames, particularly
around bright stars, and are clearly not optimal. It would be advantageous to
use extra information about the PSF expected and also to provide regularization
to produce a resultant PSF which is smooth. The code has now been updated
to include such regularization and found to be effective. We now need a suitable
choice of form for the regularizing, default PSF.
Several models for ground­based PSFs have been proposed and used. These
are typically simple analytic functions (such as the Moffat function or multiple
Gaussians) which may be conveniently fitted to stars but do not have any phys­
ical basis. However, recently Saglia et al. (1993) have investigated another form
for such PSFs which can be derived from the theory of atmospheric seeing and
they show that this form fits observed PSFs as well as, if not better than, the
more traditional ad hoc forms. This new form has only two parameters (one of
which is simply the size of the star images as defined by the FWHM) but has the
minor disadvantage of not being analytic, being instead the Fourier transform
of a simple exponential function. This form for the PSF has been implemented
as an IRAF compatible task called ``seeing'' and seems an excellent choice for a
default PSF.
Figure 2 shows the steps in the derivation of a good PSF from the same
M71 image used above. First the FWHM of bright (but unsaturated) stars in
the frame is measured. This value is then used to produce a default PSF of the
Saglia et al. (1993) form (upper­right). This in turn is used as the default PSF
with the PLUCY code to finally give the required best estimate for the PSF
(lower­left). A simple circular disc is used as the first approximation for the

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Figure 2. PSF Determination---see text for details.
PSF (upper­left). A clear elongation of the images has been well modelled and
the result is smooth and free of noise features. The PSF found by DAOPHOT,
using the same input data, is given at the lower­right for comparison.
4. Conclusions
We have extended earlier work on multi­channel, regularized image restoration
to produce experimental codes which allow both the mapping of point­source
positions in an image and the estimation of the PSF during a subsequent photo­
metric image restoration. A recently suggested form for ground­based PSFs has
been implemented and found to be a suitable default. Tests have successfully
been made using a typical deep, ground­based CCD frame. All codes have been
implemented using the F77/VOS interface to IRAF and are available on request.
References
Cohen, J. G. 1991, AJ, 101, 734
Hook, R. N. & Lucy, L. B. 1994, in The Restoration of HST Images and Spectra
II, ed. R. J. Hanisch & R. L. White (Baltimore, Space Telescope Science
Institute), p. 86
Lucy, L. B. 1994, in The Restoration of HST Images and Spectra II, ed. R. J.
Hanisch & R. L. White (Baltimore, Space Telescope Science Institute),
p. 79
Saglia, R. P., et al. 1993, MNRAS, 264, 961