This paper deals with reconstructions of density images from data with Poisson statistics, with particular focus on cartoon reconstructions. We propose a total variation (TV) based regularization technique adapted to the case of Poisson data and use a forward-backward (FB) splitting strategy to create a numerical scheme that alternates a standard Expectation-Maximization (EM) reconstruction step with a TV correction step, which is realized by a modified version of the Rudin-Osher-Fatemi (ROF) model. We illustrate our technique at a synthetic object simulating optical nanoscopy measurements.Inspired by the high redundancy of information in a typical biological image, we give an outlook for an extension of the FB-EM-TV method to a nonlocal TV (NL-TV) regularization. NL-TV uses a nonlocal graph regularization, which allows to take advantage of the high degree of redundancy of natural images. Furthermore, we discuss the main challenges of the nonlocal regularization to create efficient numerical solvers which occurs automatically due to the high structure of complexity of the nonlocal regularization graph.

@InProceedings{SB10,
  author       = {Sawatzky, A. and Burger, M.},
  title        = {Edge-Preserving Regularization for the Deconvolution of Biological Images in Nanoscopy},
  booktitle    = {Proceedings of the 8th International Conference of Numerical Analysis and Applied Mathematics 2010},
  series       = {Conference Proceedings},
  volume       = {1281},
  pages        = {1983-1986},
  month        = {september},
  year         = {2010},
  editor       = {(Eds.) T.E. Simos, G. Psihoyios and Ch. Tsitouras},
  publisher    = {AIP},
  doi          = {10.1063/1.3498325},
  url          = \{/2010/SB10},
}