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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 a"" (a)-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.
