An analysis of regularization by diffusion for ill-posed Poisson likelihood estimations

Johnathan M. Bardsley, N'Djekornom Laobeul

Research output: Contribution to journalArticlepeer-review

Abstract

The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative-log Poisson likelihood function in place of the least-squares fit-to-data function. However, if the underlying mathematical model is assumed to have the form z= Au+γ, where z is the data and A is a compact operator and γ is the background light intensity, minimizing the negative-log Poisson likelihood function is an ill-posed problem, and hence some form of regularization is required. In previous work, the authors have performed theoretical analyses of two approaches for regularization in this setting: standard Tikhonov and total variation regularization. In this article, we consider a class of regularization functionals defined by differential operators of diffusion type, and our main results constitute a theoretical justification of this approach. However, in order to demonstrate that the approach is effective in practice, we follow our theoretical analysis with a numerical experiment.

Original languageEnglish
Pages (from-to)537-550
Number of pages14
JournalInverse Problems in Science and Engineering
Volume17
Issue number4
DOIs
StatePublished - Jun 2009

Keywords

  • Compact operator equations
  • Ill-posed problems
  • Optimization
  • Poisson likelihood estimation

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