Re: Richardson-Lucy deconvolution ?

On Feb 25, 10:32 am, aruzinsky <aruzin...@xxxxxxxxxxxxxxxxxxxx> wrote:
On Feb 25, 9:00 am, jomarbue...@xxxxxxxxxxx wrote:



On Feb 25, 4:34 am, Fokko Beekhof <Fokko.Beek...@xxxxxxxxxxxx> wrote:

Hello,

I'm trying to understand the Richardson-Lucy deconvolution algorithm,
primarily from the description here:http://en.wikipedia.org/wiki/Richardson-Lucy_deconvolution

In this notation, u_i is a pixel in the latent image(unknown), c_i the
corresponding pixel in the observed image, and p_{i, j} is the
point-spread-function, i.e. the fraction of the pixel at j observed at i.

This is an iterative scheme, so an estimate for u is updated in every
iteration.

There are some things not entirely clear to me:
1) What should be the initial value for the latent image U ?
2) How does one detect convergence ?

Any help would be much appreciated!

F. Beekhof

Hi Fokko,

There are two widely used ways to initialize the RL iteration. One is
simply to set the latent image to a constant -ususally 1.0. The other
is to use the recorded image as the initial guess. The latter choice,
however, has the drawback that the recorded image has noise and thus
the initial estimate also has noise. This is a problem because
deconvolution is ill-conditioned and thus noise grows with iteration.

Regarding the convergence, you could check some norm of the difference
between two consecutive iterates and iterate until the norm becomes
smaller than a small number of your choice. However, most people let
the deconvolution iterate until results "look satisfactory". One of
the problems of the RL iteration is that it has a rather slow
convergence rate. The algorithm improves the estimated image at a rate
that keeps decreasing. For example, the improvements achieved in the
first 200 iterations will be similar to those obtained in the next
2000 iterations (for an assessment of the rate of convergence see
Conchello, J.-A., Super-resolution and convergence properties of the
expectation maximization for maximum-likelihood deconvolution of
incoherent images. Journal of the Optical Society of America-A, 1998.
15(10): p. 2609-2619. For a faster method see Markham, J. and J.-A.
Conchello, Fast maximum-likelihood image restoration algorithms for
three-dimensional fluorescence microscopy. Journal of the Optical
Society of America-A, 2001. 18(5): p. 1062-1071.)

HTH

Jomar- Hide quoted text -

- Show quoted text -

Without assuming unknown boundary conditions, is the RL solution
unique? If not, why doesn't every treatise on RL explicitly state
that near the top?

If one can assume 1) that the blurring function p sub i,j is an
analytic function, non-negative, and band limited 2) that the latent
image is non-negative and has finite support,and 3) that the algorithm
accounts for the fact that the recorded image is only a portion of the
blurred image, then, in theory, the solution is unique. However, this
also requires that the recorded image is captured without noise and
with infinite resolution. In practice the solution is "reasonably
unique" if the iteration is stopped before the noise becomes
significant. For an analysis on the uniqueness of the solution see
Sementilli, P.J., B.R. Hunt, and M.S. Nadar, Analysis if the limit to
superresolution in incoherent imaging. Journal of the Optical Society
of America-A, 1994. 10(11): p. 2265-2276. To see how to deal with the
fact that the recorded image is only a portion of the blurred image
see Holmes, T.J., Expectation-maximization restoration of band
limited, truncated point-process intensities with application in
microscopy. Journal of the Optical Society of America-A, 1989. 6(7):
p. 1006-1014.

Hope this helps

Jomar

.

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