Total variationpenalized Poisson likelihood estimation for illposed problems
 Johnathan M. Bardsley,
 Aaron Luttman
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The noise contained in data measured by imaging instruments is often primarily of Poisson type. This motivates, in many cases, the use of the Poisson negativelog likelihood function in place of the ubiquitous least squares data fidelity when solving image deblurring problems. We assume that the underlying blurring operator is compact, so that, as in the least squares case, the resulting minimization problem is illposed and must be regularized. In this paper, we focus on total variation regularization and show that the problem of computing the minimizer of the resulting total variationpenalized Poisson likelihood functional is wellposed. We then prove that, as the errors in the data and in the blurring operator tend to zero, the resulting minimizers converge to the minimizer of the exact likelihood function. Finally, the practical effectiveness of the approach is demonstrated on synthetically generated data, and a nonnegatively constrained, projected quasiNewton method is introduced.
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 Title
 Total variationpenalized Poisson likelihood estimation for illposed problems
 Journal

Advances in Computational Mathematics
Volume 31, Issue 13 , pp 3559
 Cover Date
 20091001
 DOI
 10.1007/s1044400890818
 Print ISSN
 10197168
 Online ISSN
 15729044
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Total variation regularization
 Illposed problems
 Maximum likelihood estimation
 Image deblurring
 Nonnegatively constrained minimization
 02.30.Zz
 02.50.r
 07.05.Pj
 Authors

 Johnathan M. Bardsley ^{(1)}
 Aaron Luttman ^{(2)}
 Author Affiliations

 1. Department of Mathematical Sciences, University of Montana, Missoula, MT, USA
 2. Division of Mathematics and Computer Science, Clarkson University, Potsdam, New York, USA