Least-squares methods with Poissonian noise: Analysis and comparison with the Richardson-Lucy algorithm

It is well-known that the noise associated with the collection of an astronomical image by a CCD camera is largely Poissonian. One would expect, therefore, that computational approaches that incorporate this a priori information will be more effective than those that do not. The Richardson-Lucy (RL) algorithm, for example, can be viewed as a maximum-likelihood (ML) method for image deblurring when the data noise is assumed to be Poissonian. Least-squares (LS) approaches, on the other hand, are based on the assumption that the noise is Gaussian with fixed variance across pixels, which is rarely accurate. Given this, it is surprising that in many cases results obtained using LS techniques are relatively insensitive to whether the noise is Poissonian or Gaussian. Furthermore, in the presence of Poisson noise, results obtained using LS techniques are often comparable with those obtained by the RL algorithm. We seek an explanation of these phenomena via an examination of the regularization properties of particular LS algorithms. In addition, a careful analysis of the RL algorithm yields an explanation as to why it is more effective than LS approaches for star-like objects, and why it provides similar reconstructions for extended objects. Finally a comparative convergence analysis of the two algorithms is carried out, with a section devoted to the convergence properties of the RL algorithm. Numerical results are presented throughout the paper. The subject treated in this paper is not purely academic. In comparison with many ML algorithms, the LS algorithms are much easier to use and to implement, are computationally more efficient, and are more flexible regarding the incorporation of constraints on the solution. Consequently, if little to no improvement is gained in the use of an ML approach over an LS algorithm, the latter will often be the preferred approach.