  Restoration of digital images from their degraded measurements has always
been a problem of great theoretical and practical importance in numerous
applications of imaging sciences. A specific solution to the problem of image
restoration is generally determined by the nature of degradation phenomenon as
well as by the statistical properties of measurement noises. The present study
is concerned with the case in which the images of interest are corrupted by
convolutional blurs and Poisson noises. To deal with such problems, there
exists a range of solution methods which are based on the principles
originating from the fixed-point algorithm of Richardson and Lucy (RL). In this
paper, we provide conceptual and experimental proof that such methods tend to
converge to sparse solutions, which makes them applicable only to those images
which can be represented by a relatively small number of non-zero samples in
the spatial domain. Unfortunately, the set of such images is relatively small,
which restricts the applicability of RL-type methods. On the other hand,
virtually all practical images admit sparse representations in the domain of a
properly designed linear transform. To take advantage of this fact, it is
therefore tempting to modify the RL algorithm so as to make it recover
representation coefficients, rather than the values of their associated image.
Such modification is introduced in this paper. Apart from the generality of its
assumptions, the proposed method is also superior to many established
reconstruction approaches in terms of estimation accuracy and computational
complexity. This and other conclusions of this study are validated through a
series of numerical experiments.
