In the standard case of known PSF (Code I), knowledge of the PSF is
often derived from the very image to be restored. This is of necessity
true for ground-based images where the seeing-dominated and thus
time-variable PSF must be determined from suitably isolated stars in
the field. For such cases, a natural generalization of Code I is a
code that incorporates these ``PSF stars'' into the list of designated
point sources and treats the PSF as an unknown function to be
determined simultaneously with and
.
Clearly, this generalization effectively mandates the assumption of a
spatially-invariant PSF.
The objective function whose maximization defines the above restoration problem is
The new symbols introduced here are the unknown PSF and a
second Lagrangian multiplier
to impose normalization of the PSF.
Because the PSF is now assumed to be translationally invariant, the
predicted intensity distribution in the image plane is
As with Code I, an algorithm for maximizing is derived following
the operational procedure of § 4. Details are omitted.
Code II is not without pitfalls. The first is that has multiple
maxima of approximately equal height. In fact, if
point sources
are designated, there are
such maxima, with each of the
spurious maxima corresponding to the entire image (
)
being attributed to the PSF-broadening of just one of the point
sources. These spurious solutions (
) thus have
the form:
all
,
for
,
, and
.
Fortunately, these unwelcome solutions are readily avoided with
sensible initialization (see § 5) and would in any case be readily
recognized as spurious.
A more serious problem arises when all the designated point sources are superposed on distributed emission. The resulting PSF and the allocation of emission between point sources and background are then determined by and sensitive to the regularization procedure. Limited experiments suggest that sensible results in this circumstance require strong regularization with a rather broad resolution kernel in Eq. (3).
Code II might seem to be an example of blind deconvolution (Nisenson et al. 1990) since both the restored image and the PSF are derived simultaneously from a single observed image. However, in contrast to blind deconvolution, the designated point sources here play an essential role in making the PSF determinate. Accordingly, as noted above, this code should be regarded as consolidating conventional image-processing procedures into a single, automatic code.