Journal of Mathematical Imaging and Vision

, Volume 46, Issue 3, pp 370-387

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Statistical Multiresolution Estimation for Variational Imaging: With an Application in Poisson-Biophotonics

  • Klaus FrickAffiliated withInstitute for Mathematical Stochastics, University of Göttingen Email author 
  • , Philipp MarnitzAffiliated withInstitute for Mathematical Stochastics, University of Göttingen
  • , Axel MunkAffiliated withInstitute for Mathematical Stochastics, University of GöttingenMax Planck Institute for Biophysical Chemistry


In this paper we present a spatially-adaptive method for image reconstruction that is based on the concept of statistical multiresolution estimation as introduced in Frick et al. (Electron. J. Stat. 6:231–268, 2012). It constitutes a variational regularization technique that uses an -type distance measure as data-fidelity combined with a convex cost functional. The resulting convex optimization problem is approached by a combination of an inexact alternating direction method of multipliers and Dykstra’s projection algorithm. We describe a novel method for balancing data-fit and regularity that is fully automatic and allows for a sound statistical interpretation. The performance of our estimation approach is studied for various problems in imaging. Among others, this includes deconvolution problems that arise in Poisson nanoscale fluorescence microscopy.


Statistical multiresolution Extreme-value statistics Total-variation regularization Statistical inverse problems Statistical imaging Alternating direction method of multipliers Poisson regression