Richardson-Lucy deconvolution

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.cite journal
author = Richardson, William Hadley
title = Bayesian-Based Iterative Method of Image Restoration
year = 1972
journal = JOSA
volume = 62
issue = 1
pages = 55–59
url = http://www.opticsinfobase.org/abstract.cfm?id=54565] cite journal
author = Lucy, L. B.
year = 1974
title = An iterative technique for the rectification of observed distributions
journal = Astronomical Journal
volume = 79
issue = 6
pages = 745–754
doi = 10.1086/111605]

Pixels in the observed image can be represented in terms of the point spread function and the latent image as :$c\_\{i\}\; =\; sum\_\{j\}\; p\_\{ij\}\; u\_\{j\},$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 $c\_\{i\}$ 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 $p\_\{ij\}$. This leads to an equation for $u\_j$ which can be solved iteratively according to :where:.

It has been shown empirically that if this iteration converges, it converges to the maximum likelihood solution for $u\_j$.citation
author = Shepp, L. A.
coauthors = Vardi, Y.
year = 1982
title = Maximum Likelihood Reconstruction for Emission Tomography
journal = IEEE Transactions on Medical Imaging
volume = 1
pages = 113
doi = 10.1109/TMI.1982.4307558]

In problems where the point spread function $p\_\{ij\}$ is dependent on one or more unknown parameters, the Richardson-Lucy algorithm can not be used. A later and more general class of algorithms, the expectation-maximization algorithms, [A.P. Dempster, N.M. Laird, D.B. Rubin, 1977, " [http://groups.csail.mit.edu/drl/journal_club/papers/DempsterEMAlgorithm77.pdf Maximum likelihood from incomplete data via the EM algorithm] ", J. Royal Stat. Soc. Ser. B, 39 (1), pp. 1-38] have been applied to this type of problem with great success.cite journal
author = Shan, Qi; Jia, Jiaya; Agarwala, Aseem
year = 2008
month = August
title = High-quality motion deblurring from a single image
journal = ACM Transactions on Graphics
volume = 27
issue = 3
pages = 1
doi = 10.1145/1360612.1360672
url = http://www.cse.cuhk.edu.hk/~leojia/projects/motion_deblurring/index.html]

References