Statistical Multiresolution Dantzig Estimation in Imaging: Fundamental Concepts and Algorithmic Framework

TL;DR

This paper introduces a new class of adaptive estimators for noisy, blurred signals in imaging, using convex optimization and multiresolution constraints, with proven convergence and demonstrated effectiveness.

Contribution

It develops a general framework for multiresolution estimators in imaging, combining advanced optimization algorithms with theoretical convergence guarantees.

Findings

Effective in imaging and signal detection tasks

Converges reliably with the proposed algorithm

Handles blurred signals in noisy environments

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 supremum-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.