Some simulations have been carried out to investigate how much pixelation the conventional RL-method with sub-sampling option (where the data-to-prediction ratio is upsampled by pixel replication) imprints onto the restoration of an undersampled distorted image frame.
Two grids were used, a "fine" grid with pixels
and a "coarse" grid with
pixels, i.e., half the sampling
rate of the fine grid. As an image model a single sine wave was used
with a wavelength of 6 pixels on the fine grid, which is still
sufficiently sampled on the coarse grid (2/3 of the coarse grid's
Nyquist frequency). A value of 1 was added to create a non-negative
image.
Four different PSFs were used: a delta function (PSF0) and 3 diffraction limited PSFs with "aperture" radii of 31, 15, and 7 pixels (PSF1 to 3), respectively. PSF1 is approximately two-fold (four-fold) undersampled on the fine (coarse) grid. PSF2 is sufficiently sampled on the fine grid, but about two-fold undersampled on the coarse grid. PSF3 is sufficiently sampled on both grids.
The model was convolved with each of the PSFs, then
block-averaged and down-sampled (decimated) to
pixels, generating four simulated data frames (REST0
to 3). No noise was added. The ST-ECF's IRAF code for the RL-method
with sub-sampling option was used to restore the four simulated
"data" frames.
As expected, the restoration REST0 (after some 20 accelerated iterations) clearly showed the coarse-grid pixel pattern: pairs of two pixels on the fine grid had exactly the same value. The restoration REST3 on the other hand showed very little pixelation, if any. The intermediate restorations REST2 and REST1 showed a degree of remnant pixelation on the fine grid which increased with the degree of undersampling.
Acknowledgments
Thanks are due to Bob Hanisch, STScI, for suggesting the SV-PSF restoration problem and to Richard Hook, ST-ECF, who helped to improve the written version of this contribution. I am particularly indepted to Rick White, STScI, for enlightening discussions and several valuable suggestions.