Ghost images and feasibility of reconstructions with the Richardson-Lucy algorithm

Jorge Llacer, Jorge Nunez, Timothy J. Schulz, Donald L. Snyder
1994 Image Reconstruction and Restoration  
DISCLAIMER This docummt was prepared as an account of work sponsored by the United Statcs Government While this document is believed to contain correct information, neither the United States Government nor any agency thereof. nor The Regents of the University of California, nor any of heir employees, malccs any w-ty, exprrss or implied, or assumes any legal responsibility for the -cy. completeness. or usefulness of any information. apparatus, product, or proccss disclosed, or npnsents that its
more » ... sc would not infringe privately owned rights. Reference berein to any specific c o d a l product, process, or service by its trade name, trademark, manufacturer, or orhawise, does not oecessarily constitute or imply its endommcnt, recommendation, or favoring by the United States Govanment or any agency thuwf. or The Regents of the University of California. The views and opinions of authors expressed herein do not d l y statc or reflect those of the United States Gov-nt or any agency thereof, or The Regents of che University of California Lawrence Berkeley Laboratory is an qual oppormnity employer. ABSTRACT This paper is the result of a question that was raised at the recent workshop on "The Restoration of HST Images and Spectra II", that took place at the Space Telescope Science Institute in November 1993, for which there was no forthcoming answer at that time. The question was: What is the null space (ghost images) of the Richardson-Lucy (RL) algorithm?. Another question that came up for which there is a straight-forward answer was: What does the MLE algorithm really do?. In this paper we attempt to answer both questions. This paper will begin with a brief description of the null space of an imaging system, with particular emphasis on the Hubble telescope. The imaging conditions under which there is a possibly damaging null space will be described in terms of linear methods of reconstruction. For the uncorrected Hubble telescope, it is shown that for a PSF computed by TINYTIM on a 512 x 512 dimension, there is no null space. We introduce the concept of a "nearly null" space, with an unsharp distinction between the "measurement" and the "null" components of an image and generate a reduced resolution Hubble Point Spread Function (PSF) that has that nearly null space. We then study the propagation characteristics of null images in the Maximum Likelihood Estimator W E ) , or Richardson-Lucy algorithm, and the nature of its possible effects, but we find in computer simulations that the algorithm is very robust to those effects: if they exist,the effects are local and tend to disappear with increasing iteration numbers. We then demonstrate how a PSF that has small components in frequency domain results in noise magnification, just as one would expect in linear reconstruction. The answer to the second question is given in terms of the residuals of a reconstruction and the concept of feasibility.
doi:10.1117/12.188042 fatcat:vr4puhf6xrawlbpt5237pc466i