G. Cowan
                                                     7 December 2001


            Using jet-finding algorithms to find stars
            ------------------------------------------

The problem of identifying stars observed in a CCD pixel array by
means of a computer algorithm is very similar to that of
reconstructing jets from high-energy particle collisions by the energy
that they deposit in a calorimeter.  For jets, several procedures have
been developed, such as the cone-based algorithm used in hadron
collider experiments (see e.g.  J. Huth et al. 1990; M. Seymour,
1997).

It should be reasonably straightforward to adapt the cone algorithm to
find stars in a CCD array.  The algorithm requires two parameters to
be specified in advance: a threshold number of counts n0 and a radius
R.  Translated into CCD terminology, the steps are:

1) Every pixel with contents n above the threshold n0 is considered 
   a `seed', with position (x_seed, y_seed) taken as the centre of the 
   pixel.

2)  A provisional "star" is defined by summing all pixels within  
    radius R of the seed.

3) The position of the star is the centre-of-gravity of the 
   pixels within the radius R of the seed, i.e.,

      x_star = sum_{i,j} n_{ij} x_{ij} / sum_{i,j} n_{ij} ,
      y_star = sum_{i,j} n_{ij} y_{ij} / sum_{i,j} n_{ij} ,

   where n_{ij} is the number of counts in pixel (i,j), x_{ij} and
   y_{ij} are the x and y positions of the centre of pixel (i,j), and 
   the sums run over all pixels within radius R of the seed.

4) The position (x_star, y_star) will not in general be the same
   as the position of the seed.  Redraw the circle of radius R around
   the new (x_star, y_star) and repeat step 3.  Continue until the
   positions converge to stable values.

5) There is now a list of provisional stars. Some of these are duplicates, 
   which are discarded.

6) Some stars will be overlapping.  For every star, compute the fraction
   of counts shared by other stars.  Any star that has more 50% of its 
   counts in common with another is merged with it.  The position 
   (x_star, y_star) is recalculated according to the formulae above using 
   the merged set of pixels.

7) Any star that has less than 50% of its counts in common with a 
   brighter star is split from it.  Each pixel is counted only as part of 
   the star to which it is nearest.

It would be interesting to try this procedure on star clusters or
close binaries and to adjust n0 and R to provide the best separation
(with some appropriate definition of `best').

For the best possible measurements of stellar positions it may be
useful to correct for a small bias that results from use of the
weighted averages for x_star and y_star (Akopjanov 1977).  The nature
of this bias is to pull the estimated coordinates of a star towards
the centre of the nearest pixel.  This is easy to see if all the
energy is deposited into a single pixel; the estimated position is
then always in the pixel's exact centre.  A Monte Carlo calculation
could be carried out to estimate the bias and a correction based on
this could then be applied.  Similar procedures have been carried out
for determining the positions of clusters of energy deposited in the
BaBar electromagnetic calorimeter (Anashkin, 1997; Marsiske 1998).


----------------------

References:

John Huth et al., TOWARD A STANDARDIZATION OF JET DEFINITIONS,
FERMILAB-CONF-90-249-E, Dec 1990. 6pp. Presented at Summer Study on
High Energy Physics, Research Directions for the Decade, Snowmass, CO,
Jun 25 - Jul 13, 1990.  Published in Snowmass Summer Study
1990:134-136.

Michael Seymour, JET PHENOMENOLOGY, Proceedings of Les Rencontres de
la Valee d'Aoste, Results and Perspectives in Particle Physics, La
Thuile, Italy, March 2-8, 1997.

G. A. Akopjanov et al., NIM 140 (1977) 441-445.

E. Anashkin, SYSTEMATIC SHIFT CORRECTIONS FOR BABAR EMC BARREL,
BaBar internal note #400, 6 November 1997.

D. Bukin and H. Marsiske, STUDY TO IMPROVE THE SPATIAL RESOLUTION
OF PHOTONS IN THE BABAR ELECTROMAGNETIC CALORIMEER BY TUNING THE 
WEIGHTING METHOD, BaBar internal note #412, 4 March 1998.