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
- <math> c_{i} = \sum_{j} p_{ij} u_{j} </math>
where <math>p_{ij}</math> is the point spread function, <math>u_{j}</math> is the pixel value at location <math>j</math> in the latent image, and <math>c_{i}</math> is the observed value at pixel location <math>i</math>.
The basic idea is to calculate values of <math>u_{j}</math> iteratively according to
- <math>\bold{u}_{j}^{(t+1)} = \bold{u}_j^{(t)} \sum_{i} \frac{c_{i}}{\bold{c}_{i}}p_{ij}</math>
where
- <math> \bold{c}_{i} = \sum_{j} \bold{u}_{j}^{(t)}p_{ij}</math>
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
