# Richardson–Lucy deconvolution

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The Richardson–Lucy algorithm, also known as Lucy–Richardson deconvolution, is an iterative procedure for recovering an underlying image that has been blurred by a known point spread function. It was named after William Richardson and Leon Lucy, who described it independently.[1][2]

## Description

When an image is produced using an optical system and detected using photographic film or a charge coupled device for instance, it is inevitably blurred, with an ideal point source not appearing as a point but being spread out into what is known as the point spread function. Extended sources can be decomposed into the sum of many individual point sources, thus the observed image can be represented in terms of a transition matrix p operating on an underlying image:

${\displaystyle d_{i}=\sum _{j}p_{i,j}u_{j}\,}$

where ${\displaystyle u_{j}}$ is the intensity of the underlying image at pixel ${\displaystyle j}$ and ${\displaystyle d_{i}}$ is the detected intensity at pixel ${\displaystyle i}$. In general, a matrix whose elements are ${\displaystyle p_{i,j}}$ describes the portion of light from source pixel j that is detected in pixel i. In most good optical systems (or in general, linear systems that are described as shift invariant) the transfer function p can be expressed simply in terms of the spatial offset between the source pixel j and the observation pixel i:

${\displaystyle p_{i,j}=P(i-j)}$

where P(Δi) is called a point spread function. In that case the above equation becomes a convolution. This has been written for one spatial dimension, but of course most imaging systems are two dimensional, with the source, detected image, and point spread function all having two indices. So a two dimensional detected image is a convolution of the underlying image with a two dimensional point spread function P(Δx,Δy) plus added detection noise.

In order to estimate ${\displaystyle u_{j}}$ given the observed ${\displaystyle d_{i}}$ and a known P(Δix,sub>y) we employ the following iterative procedure in which the estimate of ${\displaystyle u_{j}}$ which we call ${\displaystyle {\hat {u}}_{j}^{(t)}}$ for iteration number t is updated as follows:

${\displaystyle {\hat {u}}_{j}^{(t+1)}={\hat {u}}_{j}^{(t)}\sum _{i}{\frac {d_{i}}{c_{i}}}p_{ij}}$

where

${\displaystyle c_{i}=\sum _{j}p_{ij}u_{j}^{(t)}.}$

It has been shown empirically that if this iteration converges, it converges to the maximum likelihood solution for ${\displaystyle u_{j}}$.[3]

Writing this more generally for two (or more) dimensions in terms of convolution with a point spread function P:

${\displaystyle {\hat {u}}^{(t+1)}={\hat {u}}^{(t)}\cdot \left({\frac {d}{{\hat {u}}^{(t)}\otimes P}}\otimes P^{*}\right)}$

where the division and multiplication are element wise, and ${\displaystyle P^{*}}$ is the flipped point spread function.

In problems where the point spread function ${\displaystyle p_{ij}}$ is not known a priori, a modification of the Richardson–Lucy algorithm has been proposed, in order to accomplish blind deconvolution.[4]

## References

1. ^ Richardson, William Hadley (1972). "Bayesian-Based Iterative Method of Image Restoration". JOSA. 62 (1): 55–59. doi:10.1364/JOSA.62.000055.
2. ^ Lucy, L. B. (1974). "An iterative technique for the rectification of observed distributions". Astronomical Journal. 79 (6): 745–754. Bibcode:1974AJ.....79..745L. doi:10.1086/111605.
3. ^ Shepp, L. A.; Vardi, Y. (1982), "Maximum Likelihood Reconstruction for Emission Tomography", IEEE Transactions on Medical Imaging, 1: 113–22, doi:10.1109/TMI.1982.4307558, PMID 18238264
4. ^ Fish D. A.,; Brinicombe A. M.; Pike E. R.; Walker J. G. (1995), "Blind deconvolution by means of the Richardson–Lucy algorithm" (PDF), Journal of the Optical Society of America A, 12 (1): 58–65, Bibcode:1995JOSAA..12...58F, doi:10.1364/JOSAA.12.000058