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.

In the presence of noise, 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, $u j$ is the pixel value at location $j$ in the latent image, and $c i$ is the observed value at pixel location $i$ .

The basic idea is to calculate values of $u j$ iteratively according to

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

where

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

The Richardson-Lucy algorithm was the precursor to the widely used Expectation-maximization algorithm.

[edit] References

Richardson, W. H. 1972, J.Opt.Soc.Am., 62, 55
AP Dempster, NM Laird, DB Rubin, Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society Ser. B, 1977