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

TL;DR

This paper introduces a spatially-adaptive variational image reconstruction method based on statistical multiresolution estimation, effectively balancing data fidelity and regularity, with applications demonstrated in Poisson-bioimaging.

Contribution

It presents a novel automatic balancing technique for data-fit and regularity in multiresolution estimation, with a new optimization approach and statistical interpretation.

Findings

Effective in deconvolution problems in Poisson nanoscale fluorescence microscopy

Automatic method for balancing data-fit and regularity

Convex optimization approach with proven performance

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 K, Marnitz P, and Munk A. "Statistical multiresolution Dantzig estimation in imaging: Fundamental concepts and algorithmic framework". Electron. J. Stat., 6:231-268, 2012]. It constitutes a variational regularization technique that uses an supremum-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.