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Astronomical Data Analysis Software and Systems VII
ASP Conference Series, Vol. 145, 1998
R. Albrecht, R. N. Hook and H. A. Bushouse, e
Ö Copyright 1998 Astronomical Society of the Pacific. All rights reserved.
ds.
A Posteriori Guidance for Astronomical Images
P. Melon, M. Guillaume and Ph. Refregier
Laboratoire Signal et Image, Ecole Nationale de Physique de Marseille,
Domaine Universitaire de Saint Jerome, 13397 Marseille Cedex 20,
France
A. Llebaria, R. Cautain and L. Leporati
Laboratoire d'Astronomie Spatiale du CNRS. Marseille. BP 8, Traverse
du Siphon, 133376 Marseille Cedex 12, France
Abstract.
Astronomical ultraviolet images were obtained from a balloon­borne
telescope equipped with a photon­counting detector. The final images
are built from the list of temporal photoevent addresses produced by the
detector. We take advantage of the high temporal resolution available,
to significantly reduce, during the image reconstruction process, the blur
induced by the residual movements of the guidance system.
1. Context of image acquisition
For many years the FOCA experiment (Milliard et al. 1991) has been the main
instrument for a balloon­borne middle­UV imaging programme in astronomy.
The present version of the experiment consists of an UV camera of 40 cm di­
ameter equipped with a 2D photon­counting device. The image field of view is
about 1 # of diameter with an angular resolution of 12 arcsec.
The camera is flown in a stratospheric gondola actively stabilized. The
guidance error signal is taken from an independent star tracker centered on a
guide star (using the visible part of its spectrum). An active system compensates
for most of the astronomical field rotation around the central guide star. The
residual movements are in the range 2-10 arcsec. They consist mainly of a slow
X­Y drift and of small, random and faster oscillations in X­Y and in angular
rotation. Significant power is still present at frequencies above 1 Hz.
In the focal plane the photo­counting device gives the 2D position of each
detected photo­event. Position coordinates are digitized and they define a 1024â
1024 array of pixels each 3.4 arcsec wide.
Telemetry sends to the ground equipment a sequence of position coordi­
nates in the order of time arrival for each event. They are grouped in sets of
photoevents detected in the same short interval of time; each group is called a
``frame''. An interval of 20 ms per frame is then standard. The full sequence
of frames consists of the ordered (in time) set of groups of events. When pro­
cessed, successive frames are grouped in blocks of the same size (between 1 and
10 frames per block). The block size must be chosen to be small enough to
71

72 Melon, Guillaume, Refregier, Llebaria, Cautain and Leporati
sample fast movements and big enough to include enough events, between 25
and 100 typically. This point is discussed later in this paper. The typical flow
of data is about of 2000 events per sec.
The image is built from individual events in a 1024â1024 array by increasing
by one the pixel value whose address is defined in the event. If the final image
is built without any correction of residual movements of the guidance system a
significative blur will be added.
2. Algorithm
Thereafter we consider that an elementary, i.e., a high temporal resolution image,
is built from all events included in a block of frames by a simple pile­up of
photoevents at their original X­Y address.
We consider the sequence of such elementary images at high temporal res­
olution. We make the following assumptions about these images:
1) The noise present in the images is photon shot noise and is described by
Poisson Law.
2) The scene does not change during the acquisition.
3) The movement is negligible during the block duration.
4) The images are periodic.
5) Rotations are negligible, so we consider only the translations between the
di#erent images.
6) We do not have any a priori knowledge either about the translations or the
imaged astronomical field.
Let s p (i) denote the intensity of the p th observed image s p at pixel i where
i # [1, N ], and N is the number of pixels of the image (we use one­dimensional
notations for simplicity). Let r(i) denote the intensity of the imaged astronomi­
cal field at the same pixel i. It is the perfect, non­noisy and non­blurred image of
the observed portion of the sky. The observed image s p is translated by j p pixels
from the reference r. Without any a priori knowledge of the movement, we de­
termine the likelihood of the hypothesis that the translation between the images
s p and r is j p pixels. In a previous article (Guillaume et al. 1997), the value of
r(i) is considered known, and it has been proved that the optimal estimation of
j p is obtained by maximizing the intercorrelation between the observed image
and the logarithm of the reference image r(i):
j ML
p = arg max
jp
# N
# i=1
s p (i + j p ) ln[r(i)] # (1)
In the present paper, the true value of the reference image r(i) is considered to
be not available. In this case, we consider the maximum likelihood estimation of
the reference r ML (i) = # P
p=1 s p (i + j p ), and then the set of relative translations
•
J = (j 1 , j 2 , ......j P ) of the P observed images s p . Assuming the statistical in­
dependence between the images, it can be shown that the maximum likelihood
estimate •
J ML of •
J is found as:
•
J ML = arg min
•
J
# # - # j
r ML (j) lnr ML (j) # # = arg min
•
J
S( •
J) (2)

A Posteriori Guidance for Astronomical Images 73
The determination of •
J ML is performed by an iterative algorithm which is very
close to a steepest descent procedure:
. Choose the size m of the search window and the number of iterations
. For each iteration k:
-- For each image p of the sequence:
# Calculate the variation #S( •
J k ) of S( •
J k ) for all the mâm neigh­
bours of j k
p
# Choose the value j k+1
p for which #S( •
J k ) is negative and mini­
mum
We take advantage of the low photon level ( only few pixels have non­zero value)
and we develop a fast algorithm inspired by the Nieto­Llebaria algorithm (Nieto,
Llebaria & di Serego, 1987) by calculating #S( •
J k ) on tables of photo­events
addresses rather than on images. For example, for 2000 images with a hundred
photons per image, the computation time can be reduced from 40 hours for the
direct calculation to 5 minutes for the fast algorithm on a Sun Sparc station 10.
We note that:
1) In practice, for all the performed simulations, the convergence for S( •
J) has
always been attained with less than 10 iterations (i.e., 10 presentations of all
the images).
2) The size m of the search window can be adapted to the amplitude of the
translations in order to avoid local minima of S( •
J).
3. Results
This algorithm has been tried on simulated fields as well as real images from
balloon flights. We present here the results produced in an image centered on
the M3 globular cluster. The experimental event series includes 24000 frames
1024â1024 pixels size and a mean of 25 photons/frame.
In the algorithm, the only free parameter is the number of frames per block.
Previous simulations (Melon 1997) showed (see Table 1) that for each block
the probability of exact recentering drops as the number of photons per block
decreases. As rule of thumb: the threshold is between 50 and 100 photons per
block depending on the spatial arrangement and flux distribution of stars, and
background intensity.
Nb of blocks Photons/block Frames/block Percentage
5000 110 4 75
10000 45 2 45
20000 22 1 20
Table 1. Probability of exact correction of a block

74 Melon, Guillaume, Refregier, Llebaria, Cautain and Leporati
Figure 1. Left: Histograms of FWHM for case 1) & 2). Right: His­
tograms of FWHM for case 1) & 3)
In our case a block of 2 frames, that is, a 40 ms interval was enough to
assure a good bandwidth to sample the movement and to get a correct recon­
struction. To characterize the e#ects of the algorithm we measured for a limited
set of stars (N # 50) their FWHM (full wide to half maximum). We build the
FWHM distribution for: 1) Stars from an uncorrected image, 2) Stars from a
corrected image using a reference (see algorithm 1), 3) Stars from a corrected
image without reference (see algorithm 2).
Figure 1 shows the histograms of these distributions. On the left side we
compare 1) with 2) and in the right side we compare 1) with 3). As can be seen,
a decrease of 20 µm over 120 µm is clearly visible for both corrected images.
From that and from other work, not shown here, we can conclude that the
new algorithm improves resolution as e#ciently as the algorithm with reference
image. The new algorithm does not show any ringing side­e#ect and has the
major advantage of not needing a reference image
The lack of ringing side e#ects and the ``self­su#cient'' use of event series
with any loss in accuracy are encouraging results for a large and deep study of
this new method.
Acknowledgments. We are grateful to M. Laget and B. Milliard for their
valuable help and fruitful discussion of this paper.
References
Milliard, B., Donas, J., & Laget, M., 1991, Adv. Space Res.
J.L. Nieto, A. Llebaria, & S. di Serego Aligheri, 1987, Astron. and Astroph.
178, 301
M. Guillaume, Th. Amouroux, Ph. Refregier, B. Milliard & A. Llebaria, 1997,
Opt. Lett. 22, 322
P.Melon, 1997, Rapport de DEA, Ecole Nationale de Physique, Marseille