Richardson–Lucy deconvolution

Not to be confused with Modified Richardson iteration.

The Richardson–Lucy algorithm, also known as Lucy–Richardson deconvolution, is an iterative procedure for recovering a latent image that has been blurred by a known point spread function.

Pixels in the observed image can be represented in terms of the point spread function and the latent image as

$d_{i} = \sum_{j} p_{ij} u_{j}\,$

where p_ij is the point spread function (the fraction of light coming from true location j that is observed at position i), u_j is the pixel value at location j in the latent image, and d_i is the observed value at pixel location i. The statistics are performed under the assumption that u_j are Poisson distributed, which is appropriate for photon noise in the data.

The basic idea is to calculate the most likely u_j given the observed d_i and known p_ij. This leads to an equation for u_j which can be solved iteratively according to

$u_{j}^{(t+1)} = u_j^{(t)} \sum_{i} \frac{d_{i}}{c_{i}}p_{ij}$

where

$c_{i} = \sum_{j} p_{ij} u_{j}^{(t)}.$

It has been shown empirically that if this iteration converges, it converges to the maximum likelihood solution for u_j.

In problems where the point spread function p_ij is dependent on one or more unknown parameters, the Richardson–Lucy algorithm cannot be used. A later and more general class of algorithms, the expectation-maximization algorithms, have been applied to this type of problem with great success