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. It was named after William Richardson and Leon Lucy, who described it independently.[1][2]

Description[edit]

When an image is recorded on a detector such as photographic film or a charge coupled device, it is generally slightly blurred, with an ideal point source not appearing as a point but being spread out, into what is known as the point spread function. Non-point sources are effectively the sum of many individual point sources, and pixels in an observed image can be represented in terms of the point spread function and the latent image as

where is the point spread function (the fraction of light coming from true location that is observed at position ), is the pixel value at location in the latent image, and is the observed value at pixel location . The statistics are performed under the assumption that are Poisson distributed, which is appropriate for photon noise in the data.

The basic idea is to calculate the most likely given the observed and known . This leads to an equation for 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 .[3]

This can also be written more generally (for more dimensions) in terms of convolution,[4]

where the division and multiplication are element wise, and is the flipped point spread function, such that

In problems where the point spread function is dependent on one or more unknown parameters, the Richardson–Lucy algorithm cannot be used.

References[edit]

  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