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M. W. Regan, R. A. Gruendl
Department of Astronomy, University of Maryland, College Park, MD 20742
minimization technique. This
technique uses the offset between all overlapping pairs of frames to
determine the best background offset for each frame. This technique was
developed for use with infrared arrays; it has been used with the existing
Simultaneous Quad Infrared Imaging Device (SQIID) software within IRAF,
but applies to any image that is created from several individual frames.
We also describe several techniques that allow objective criteria to be
used to identify individual frames to be excluded from a mosaic. We present
examples of our technique using two computer programs and compare the
results with simpler methods. This technique is extremely powerful for
examining low surface brightness objects.
Due to the high background in the near-infrared, it is necessary to split an observation into several small individual integrations. These individual frames must then be reformed into a composite image. Since the sky background can change on minute time-scales, the individual frames will have different background intensities. Usually a sky frame is formed and subtracted from all frames, but since the intensity of the sky can change between the object frame and the sky frame, each frame has a different zero-point.
The normal method used to create a mosaic is to determine the background offset in a region of overlap between two frames, and bootstrap, adding one frame at a time. The problem with this method is that an error in one of the relative offset determinations can propagate, leading to background intensities that do not match in regions where one frame overlaps with more than one other frame. A better way to match the backgrounds between individual frames is to use all the information that exists in the data. In general, each frame overlaps with several other frames, and a relative offset between each pair of overlapping frames can be determined.
In this paper we discuss a method for determining the correct background offset for each frame by generating a least-squares solution that minimizes the errors in the offsets for all frames.
If i and j are the indices of two frames that overlap, then the model equation is
where 
is the measured difference in background intensities and
and 
are the true background offsets of frames i and j,
respectively.
We wish to minimize the deviation between the observed difference and that
expected. The quantity to be minimized in the least-squares solution is
where 
is set to one if frames i and j overlap, and is set to
zero otherwise. Assuming that every frame overlaps with at least one other
frame, there are m unknowns. The m-1 normal equations for the m
parameters can be written as
Since there are only m-1 equations for the m unknowns, one of the offsets
must be fixed before the equations can be solved. By setting 
to zero,
the set of equations becomes
solvable. The final set of offsets has an arbitrary zero-point, which
must be determined by other means. The best way to determine the correct
zero-point is to include a set of overlapping frames that extend far enough
from an object of interest to be sure that there is no low surface
brightness emission. This region will determine the overall zero-point.
The algorithm is currently implemented in two computer programs:
(1) a FORTRAN routine that reads all the individual frames into
memory, determines the background offset difference between each pair of
overlapping frames, and writes this to a file, and (2) a C
routine that reads the file containing the background offset differences
and performs the 
minimization. This routine also reads in a file
that lists the frames to be excluded from the fit. Typically these are
frames with some known defect (see § 4). This program creates output that
can be used by the SQIID routine nircombine to create a mosaic.
Determining which frames to exclude from the final mosaic is critical, since one bad frame can create background offset errors that propagate throughout the mosaic.
The most common method---trial and error---is very subjective and can
take a lot of time.
A side benefit of our technique is that we determine the residual for each
of the frames, which can be used as an objective basis for rejecting frames
whose residual is greater than 3
above the average residual.
We used an iterative process to remove bad frames. On each iteration
the frame with the highest residual above 3
is removed and a new
solution is calculated. This process is repeated until no frames have
residuals above 3
.
While it is clear that this process is applicable when sky emission
variations occur, as in the near-infrared, a similar technique can be
applied to determine and correct for the effects of variable atmospheric
extinction. Variable extinction can have two causes: a poorly determined
airmass correction, or the presence of high clouds. We have developed a
similar set of routines that perform a 
minimization of the gain
variation between frames. The goal of this routine is to look for frames
that show a very low gain relative to the other frames in the mosaic.
If there is enough redundant coverage, the most conservative thing to
do is to remove those frames that show excessive extinction.
If there is not enough redundant coverage, then one corrects
the frames by multiplying them by the inverse of the amount of extinction.
Table: Resultant Fit and Subsequent Improvement of Offsets
Figure:
H-band mosaic of the cluster IC 348 taken with SQIID on the Kitt Peak
1.3 m telescope. Frame A ( left) shows a mosaic made with the
bootstrap method of 23 frames laid out in a grid. There are no two frames
that share more than 
1.' 5 (75 pixels) overlap. It is
apparent that errors in the calculated background offset cause significant
mismatches from frame to frame. Frame B ( right) shows the
same mosaic with the 
minimization run and the rejection
criteria applied. Low surface brightness emission features can be
identified with some degree of confidence now that the variations
have been reduced.
Original PostScript figure (4208 kB)
As an example of how to use this 
minimization technique, and how to
reject bad frames, Figure 1a shows a SQIID image of IC 348 made with the
standard bootstrap approach.
The background offsets mismatch at several places in the figure. The
minimization technique produces the initial solution shown in
Table 1. It is clear from Table 1 that frame 7 is defective; we removed
it from the mosaic and re-determined the background offsets. These results
are shown in the last two columns of Table 1. It is apparent that the
residuals for all the frames with poor fits have been reduced. The final
mosaic (Figure 1b) is much improved and low-level emission features near
the zero-point level are now visible and believable.
A copy of the source code that implements this algorithm is available by request from the authors.
We would like to thank Elizabeth Lada for letting us use her images of IC 348. We would also like to thank Stuart Vogel for his comments and direction throughout this work.
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