# Richardson-Lucy iterative deconvolution is producing erroneous deconvoluted images

I am working on producing deconvoluted image from degraded image where the process of degradation is known to be linear and space invariant. I have used Richardson-Lucy iterative deconvolution method for that. The formula used is given here under.$${\hat{f}}^{k+1}=\left ( {h^{*}}\otimes \frac{g}{h\otimes {\hat{f}}^{k}} \right ). {\hat{f}}^{k}$$ where g is the degraded image, h is the point spread function (PSF) and f's are deconvoluted images. I have tried two processes of degradations (both linear and space invariant). In both cases the RL deconvolution is not working, producing erroneuos images. Although same RL deconvolution script is used in other cases with success. On top of that when I tried Weiner filtering with apodization then in both cases I was able to produce physically meaningful image.

Original activity

Richardson-Lucy deconvoluted image iterates (1,3,6,9,12,15,18,21,24,27th iterate images)

Richardson-Lucy deconvoluted image iterates (1,3,6,9,12,15,18,21,24,27th iterate images)

Weiner filtering with apodization (for Degradation Type-1)

Original activity image + degraded image + deconvoluted image

The fact that Weiner filtering works put us in a big question of why RL algorithm is not working specially in the scenario where same RL algorithm script is used for successful image production in a case where degradation is less (not shown here). Point to be noticed that in RL algorithm staring from first iteration image it gets worse than degraded image. All the intensity gets accumulated in center region. Why is it happening?

I will be higly obliged if someone provides me hint about where things are wrong. Thanks in Advance.

• I will really appreciate any response. – aranyak Oct 21 '20 at 15:52
• It looks to me like you are getting a point spread function out of your Richardson-Lucy function. What library are you using, or did you implement it yourself? – JRE Oct 21 '20 at 20:59
• Thank you for the comment. Yes I have written a MATLAB script implementing the RL equation mentioned above and used the input data (PSF and degraded image) obtained from MC simulation. Can you provide a more eloborate hint with some reading materials. Thank you :) – aranyak Oct 22 '20 at 0:00
• I will really appreciate any response. – aranyak Oct 23 '20 at 5:50
• Is it possible that you have the input images swapped going into the Ruchardson-Lucy algorithm? It looks to me like it is deconvolving the PSF with the image. – JRE Oct 23 '20 at 6:12