Richardson-Lucy deconvolution

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The Richardson-Lucy algorithm, also known as Richardson-Lucy 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

c_i =	∑	p_iju_j
	j

where $p i j$ 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 $c i$ 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 $c i$ and known $p i j$ . This leads to an equation for $u j$ which can be solved iteratively according to

$\bold{u}_{j}^{(t+1)} = \bold{u}_j^{(t)} \sum_{i} \frac{c_{i}}{\bold{c}_{i}}p_{ij}$

where

$\bold{c}_{i} = \sum_{j} \bold{u}_{j}^{(t)}p_{ij}$ .

It has been shown empirically ^[1] that if this iteration converges, it converges to the maximum likelihood solution for $u j$ .

In problems where the point spread function $p i j$ 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 can however be applied to this type of problem.

[edit] References

W.H. Richardson, 1972, Bayesian-Based Iterative Method of Image Restoration, J. Opt. Soc. Am. 62 (1), pp. 55-
A.P. Dempster, N.M. Laird, D.B. Rubin, 1977, Maximum likelihood from incomplete data via the EM algorithm, J. Royal Stat. Soc. Ser. B, 39 (1), pp. 1-38