Greedy techniques for inverse problems

TL;DR

This paper presents a new greedy sampling framework for inverse imaging problems that optimizes measurement selection to improve reconstruction quality with fewer measurements, using kernel-based methods and error bounds.

Contribution

The paper introduces a novel greedy approach for selecting measurements in inverse problems, combining interpolation, extrapolation, and explicit error bounds for improved sampling efficiency.

Findings

Achieves high-quality reconstructions with fewer measurements.

Demonstrates effectiveness in solar hard X-ray imaging.

Provides explicit error bounds for measurement propagation.

Abstract

Inverse imaging problems rely on limited and indirect measurements, making reconstruction highly dependent on both regularization and sample locations. We introduce a novel greedy framework for the optimal selection of indirect measurements in the operator codomain, specifically tailored to inverse problems. Our approach employs a two-step scheme combining kernel-based interpolation and extrapolation. Within this framework, greedy schemes can be residual-based, where points are selected according to the current approximation error for a specific target function, or error-based, where points are chosen using a priori error indicators independent of the residual. For the latter, we derive explicit error bounds that quantify the propagation of approximation errors through both interpolation and extrapolation. Numerical applications to solar hard X-ray imaging demonstrate that the proposed greedy sampling strategy achieves high-quality reconstructions using only a few available measurements.