Introduction

Consider an image characterized by its intensity distribution , corresponding to the observation of an object through an optical system. If the imaging system is linear and shift-invariant, the relation between the object and the image in the same coordinate frame is a convolution:

is the point spread function (PSF) of the imaging system, and is additive noise. We want to determine knowing and . This inverse problem has led to a large amount of work, the main difficulties being the existence of: (i) a cut-off frequency of the PSF, and (ii) an intensity noise (see for example Cornwell 1988). Eq. 1 is always an ill-posed problem. This means that there is no unique least-squares solution of minimal norm , and a regularization is necessary.

The best restoration algorithms are generally iterative (Katsaggelos 1991). Van Cittert (1931) proposed the following iteration:

where is a convergence parameter generally taken as 1. In this equation, the object distribution is modified by adding a term proportional to the residual. Another iterative algorithm is provided by the minimization of the norm (Landweber 1951) and leads to:

where . The Richardson-Lucy method (Lucy 1974, Richardson 1972) uses an iterative approach to compute a maximum likelihood estimate:

and

where is the transpose of the PSF.