Richardson–Lucy deconvolution

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Richardson–Lucy deconvolution

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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.^[1]^[2]

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 $u_j$ .^[3]

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, theexpectation-maximization algorithms,^[4] have been applied to this type of problem with great success

[edit]References

^ Richardson, William Hadley (1972). "Bayesian-Based Iterative Method of Image Restoration". JOSA 62 (1): 55–59. doi:10.1364/JOSA.62.000055.
^ Lucy, L. B. (1974). "An iterative technique for the rectification of observed distributions". Astronomical Journal 79 (6): 745–754. doi:10.1086/111605.
^ Shepp, L. A.; Vardi, Y. (1982), "Maximum Likelihood Reconstruction for Emission Tomography", IEEE Transactions on Medical Imaging 1: 113, doi:10.1109/TMI.1982.4307558
^ 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

来源：http://en.wikipedia.org/wiki/Richardson%E2%80%93Lucy_deconvolution