The University of Montana
Department of Mathematical Sciences

Technical report #8/2006

Total Variation-Penalized Poisson Likelihood Estimation for Ill-Posed Problems

J. Bardsley
Department of Mathematical Sciences
The University of Montana (USA)

Aaron Luttmann
Division of Science and Mathematics
Bethany Lutheran College (USA)

Abstract

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 likelihood functional 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 ill-posed 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 variation-penalized Poisson likelihood functional is well-posed. 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 quasi-Newton method is introduced.

Keywords: total variation regularization, ill-posed problems, maximum likelihood estimation, image deblurring, nonnegatively constrained minimization

PACS numbers: 02.30.Zz, 02.50.-r, 07.05.Pj

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