arXiv:1101.4373v1 [stat.AP] 23 Jan 2011 STATISTICAL MULTIRESOLUTION ESTIMATION IN IMAGING: FUNDAMENTAL CONCEPTS AND ALGORITHMIC FRAMEWORK KLAUS FRICK Institute for Mathematical Stochastics University of G¨ ottingen Goldschmidtstraße 7, 37077 G¨ ottingen PHILIPP MARNITZ Institute for Mathematical Stochastics University of G¨ ottingen Goldschmidtstraße 7, 37077 G¨ ottingen AXEL MUNK Institute for Mathematical Stochastics University of G¨ ottingen Goldschmidtstraße 7, 37077 G¨ ottingen Max Planck Institute for Biophysical Chemistry Am Faßberg 11, 37077 G¨ ottingen Abstract. In this paper we introduce a general class of statistical multiresolution estimators and develop an algorithmic framework for computing those. By this we mean estimators that are defined as solutions of convex optimization problems with ℓ∞-type constraints. We employ a combination of an alternating direction augmented Lagrangian technique with Dykstra’s algorithm for computing orthogonal projections onto intersections of convex sets. The capability of the proposed method is illustrated by various examples from imaging. E-mail addresses: frick@math.uni-goettingen.de, marnitz@math.uni-goettingen.de, munk@math.uni-goettingen.de . Key words and phrases. Statistical Inverse Problems, Statistical Multiscale Analysis, Extreme-Value Sta- tistics, Dykstra’s Projection Algorithm, Total Variation, Statistical Imaging, Dantzig Selector, Alternating Direction Method. Correspondence to frick@math.uni-goettingen.de . 1