Notes on Deconvolution

The MR/1 Approach to Deconvolution

• Detection of signal, and hence of noise, provides a very effective way to regularize the solution.
• There is no need to guess or estimate any regularizing parameter.
• In fact, convergence is usually very rapid (a few iterations only).
• In such regularization through noise removal (and the use of variance stabilization if called for), the image so treated is the discrepancy image between given input image and the approximation to this at any iteration (the "residual image"). So variance stabilization does not alter the basic nature of the inverse problem.
• The future: software for maximum entropy deconvolution is nearing completion and will be part of the MR/2 package, due for release around mid-1999. Multiscale entropy and noise modeling is used in this method.
• For various examples, evaluation, and to read further:
  • Chapter 3 of JL Starck, F Murtagh and A Bijaoui, Image and Data Analysis: the Multiscale Approach", Cambridge University Press, 1998.
  • A Bijaoui, JL Starck and F Murtagh, "Restauration des images multi-échelles par l'algorithme à trous", Traitement du Signal, 11, 229-243, 1994.
  • JL Starck and A Bijaoui, "Filtering and deconvolution by the wavelet transform", Signal Processing, 35, 195-211, 1994.
  • JL Starck and F Murtagh, "Image restoration with noise suppression using the wavelet transform", Astronomy and Astrophysics, 288, 343-348, 1994.
  • F Murtagh, JL Starck and A Bijaoui, "Image restoration with noise suppression using a multiresolution support", Astronomy and Astrophysics Supplement Series, 112, 179-189, 1995.
  • JL Starck and E Pantin, "Multiscale maximum entropy image restoration", Vistas in Astronomy, 40, 1-6, 1996.