Electronic Journal of Statistics

Statistical multiresolution Dantzig estimation in imaging: Fundamental concepts and algorithmic framework

Klaus Frick, Philipp Marnitz, and Axel Munk

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Abstract

In this paper we are concerned with fully automatic and locally adaptive estimation of functions in a “signal + noise”-model where the regression function may additionally be blurred by a linear operator, e.g. by a convolution. To this end, 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 -type constraints. We employ a combination of the alternating direction method of multipliers with Dykstra’s algorithm for computing orthogonal projections onto intersections of convex sets and prove numerical convergence. The capability of the proposed method is illustrated by various examples from imaging and signal detection.

Article information

Source
Electron. J. Statist. Volume 6 (2012), 231-268.

Dates
First available in Project Euclid: 29 February 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1330524559

Digital Object Identifier
doi:10.1214/12-EJS671

Mathematical Reviews number (MathSciNet)
MR2988407

Zentralblatt MATH identifier
1314.62094

Subjects
Primary: 62G05: Estimation 90C06: Large-scale problems
Secondary: 68U10: Image processing

Keywords
alternating direction method of multipliers (ADMM) biophotonics Dantzig selector Dykstra’s projection algorithm local adaption signal detection statistical imaging statistical multiscale analysis statistical regularization

Citation

Frick, Klaus; Marnitz, Philipp; Munk, Axel. Statistical multiresolution Dantzig estimation in imaging: Fundamental concepts and algorithmic framework. Electron. J. Statist. 6 (2012), 231--268. doi:10.1214/12-EJS671. https://projecteuclid.org/euclid.ejs/1330524559


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