Efficient Maximum A Posteriori Deconvolution of Sparse Structures

A new algorithm for maximum a posteriori deconvolution, Iterated Window Maximization (IWM), was recently proposed by Kaaresen. The algorithm was derived for a particular sparse spike train model, and was in a later simulation study found to perform better than a number of established alternatives. The purpose of the present paper is to show how the same simple and efficient technique can be generalized to a number of other situations. In seismic and ultrasonic applications, deconvolution is often made difficult by colored noise, non-stationary noise, or by a position variant convolution filter. It is shown here how IWM can be adapted to all of these situations. Then, simple formulas are derived for multi-channel and spatial deconvolution. As an illustration of the latter, deconvolution results from synthetic and real astronomical star field images are presented. Finally, it is shown that IWM can also be used to deconvolve step functions. This extension is exemplified on synthetic and real seismic well log data. It is argued that IWM can also be used for other problems than those studied here, as long as the function to be reconstructed is in some sense sparse. KeywordsSparse deconvolution, iterated window maximization, colored noise, non-stationary noise, position variant psf., multichannel deconvolution, spatial deconvolution, deblurring, star fields, Hubble Space Telescope, well-log processing. The author is with the Department of Mathematics, University of Oslo, P.B. 1053 Blindern, N-0316 Oslo, Norway. E-mail: [email protected]. The author is supported by grants from the Research Council of Norway. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version will be superseded.