To construct a properly sampled image for our Cycle 1 data (The HST
primary mirror has a diffraction-limited resolution of 57.7 mas at
550 nm and 37.8 mas at 720nm) we chose to ``dither'', i.e., observe the object
at four slightly shifted positions (offset in and
by
different pointing combinations of 0.022 mas as shown in Fig. 2.
In fact, because of scheduling constraints, the observations were
pointed in the way shown in Fig. 2b with offsets of several pixels
from the
initial pointing position and differences in the spacecraft roll
angle ranging between 1.°1 and 24.°2 (Fig. 2b).
A rotated image is the same as a non-rotated image convolved with
a PSF of opposite rotation. That is, if is an image at roll angle
and
is its exposure time, then:
where and P are the image and PSF at 90°, the subscript 90
outside the parentheses
means the image has been rotated back to 90°, and
is the PSF at 90°rotated through the negative roll angle,
.
Thus,
becomes a new, composite PSF for the composite image which is the sum of the individual images shifted to a common center and rotated back to a common angle.
Note that since the rotation operation can be expressed in terms of
two orthogonal functions (sine and cosine), as long as two conditions
hold: a) the undersampling is less than a factor of 2, and b) for each frame,
if is the pixel size, and both the offsets from the common
center,
and
, include a pixel fraction,
, such that
then all of the pixels in the registered images will have sufficient offset spacings to sample a different part of the image. Both these conditions hold for the Cycle 1 data discussed here.
The technique described above was applied to our Cycle 1 data which was resampled on a double grid, shifted to a common center, rotated using a sinc interpolation, and then co-added. The composite PSF was generated in the same way using a set of observed PC6 PSFs (at 502nm). These were slightly shifted from the PC6 position of the galaxy nucleus but, again, this (along with the slight mis-match in effective wavelength of the PSFs) is a secondary effect.
One immediate advantage of this serendipitously employed technique is shown in Fig. 3 where the patterns of ``hot pixels'' in the co-added frames have been emphasized. Not only is it easy to spot these artifacts and thus remove them, it is also clear that over a period of 3 to 4 weeks (mid-September to October 8, 1991) some of them have decayed in intensity. This suggests that at least some of the ``hot pixels'' are transient, possibly induced by energetic cosmic ray events.
Another advantage of the technique (which is more intuitive) is that the
composite PSF, , is much more radially symmetric, with the ``tendrils''
deemphasized (and thus, their propensity to induce artificial, radially
extensive features minimized).