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

Authors:
Klaus Frick;Philipp Marnitz;Axel Munk
Affiliations:
Institute for Mathematical Stochastics, University of Göttingen, Göttingen, Germany 37077;Institute for Mathematical Stochastics, University of Göttingen, Göttingen, Germany 37077;Institute for Mathematical Stochastics, University of Göttingen, Göttingen, Germany 37077 and Max Planck Institute for Biophysical Chemistry, Göttingen, Germany 37077
Venue:
Journal of Mathematical Imaging and Vision
Year:
2013

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Abstract

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 驴驴-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.