Deconvolution

Deconvolution! It holds out the promise of turning noisy, out-of-focus images into tableaux of sparkling delight! And saving my PhD!

• As any fule kno, the classic algorithm is Richardson-Lucy; you supply an image and a point spread function (you could estimate that from the parameters of your microscope - but how? - or measure it by imaging point sources, ie fluorescent subresolution beads), and it gives you back a reconstructed image after a bunch of iterations.
• As with any iterative algorith, RL needs a stopping condition. The traditional one is a chi-squared test
• If you believe what Matthias Pruksch and Frank Fleischmann have to say about Positive Iterative Deconvolution in Comparison to Richardson-Lucy Like Algorithms, then RL may be better for point sources (eg stars), but 'positive iterative deconvolution' works better on detailed things (eg asteroids, cells).
• Then we've got the Expectation-maximization algorithm, which is a development of RL. I get the impression that EM is joint king of the hill with maximum entropy.
• The Wiener filter is, according to Mathworld, "an optimal filter used for the removal of noise from a signal which is corrupted by the measuring process itself", which sounds good, but unlikely. A more detailed description sheds more light; it's clever, elegant, but not enormously effective.
• M. D. Cahill's Unshake is nifty, although probably not useful to me.
• On the question of unshaking, super-nifty results from Fergus, Singh, Hertzmann, Roweis, and Freeman.. Lyndsey sent this to me, so unsurprisingly, it's Bayesian.
• Richard L. White wrote about Image Restoration Using the Damped Richardson-Lucy Method a while ago; this is "a modification of the Richardson-Lucy iteration that reduces noise amplification in restored images".
• The maximum entropy method is a sophisticated form of deconvolution used in radioastronomy.
• Dr Peter Steinbach has very kindly written An Introduction to the Maximum Entropy Method which makes everything clear - this looks very useful! In particular, the algorithm of Cornwell and Evans, with tweaks, is the one we want.
• Molecular Expressions has a fairly high-level article on Algorithms for Deconvolution Microscopy.
• RestoreTools sounds interesting, but has fallen off the internet at the moment (20060216194830).
• Scientific Volume Imaging's SVWiki has useful, practical stuff (based on their Huygens software, but still of some general interest).
• Apple's vImage library has deconvolution functions.
• A radioastronomy-oriented deconvolution tutorial.
• A review on Iterative methods for image deblurring - J. Biemond, R. L. Lagendijk, R. M. Mersereau (May 1990); Proc IEEE 78(5):856
• Mathworks' tutorial on Deblurring Images Using the Blind Deconvolution Algorithm, ie RL; has some useful practical stuff.

Still haven't worked out if Tschumperlé-Deriche vector-valued regularization PDEs (or, probably, its scalar, ie monochrome, relatives) are any use.

PLAN OF ACTION: get or build implementations of Richardson-Lucy, positive iterative, expectation maximisation and maximum entropy filters, and run them over some sample data.

Back on the trail of MEM ...

• Maximum a posteriori estimation with Good's roughness for three-dimensional optical-sectioning microscopy - S. Joshi, M. I. Miller (1993); J Opt Soc Am A 10:1078
• Image restoration based on Good's roughness penalty with application to fluorescence microscopy - P. J. Verveer, T. M. Jovin (1998); J Opt Soc Am A 15:1077
• Artifacts in computational optical-sectioning microscopy - J. G. McNally, C. Preza, J.-A. Conchello, L. J. Thomas Jr (1994); J Opt Soc Am A 11:1056
• A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy - P. J. Verveer, M. J. Gemkow, T. M. Jovin (Jan 1999); J Microsc 193(1):50 [doi:10.1046/j.1365-2818.1999.00421.x]
• Radio Astronomy: Lecture #7; the first section deals with MEM and explains it a bit more simply (but still with maths)

PP-TSVD looks like magic.