Convergence rates for the joint solution of inverse problems with compressed sensing data

TL;DR

This paper develops convergence rate analysis for joint inverse problem solutions in compressed sensing when measurements are indirect and involves ill-posed forward operators, introducing two co-regularization methods with proven error bounds.

Contribution

It introduces two novel joint reconstruction methods for compressed sensing with indirect data and derives necessary and sufficient conditions for linear convergence.

Findings

Derived error estimates for signal and data recovery.

Established linear convergence rates under source and injectivity conditions.

Proved the necessity of conditions for linear convergence.

Abstract

Compressed sensing (CS) is a powerful tool for reducing the amount of data to be collected while maintaining high spatial resolution. Such techniques work well in practice and at the same time are supported by solid theory. Standard CS results assume measurements to be made directly on the targeted signal. In many practical applications, however, CS information can only be taken from indirect data $h_\star = W x_\star$ related to the original signal by an additional forward operator. If inverting the forward operator is ill-posed, then existing CS theory is not applicable. In this paper, we address this issue and present two joint reconstruction approaches, namely relaxed $\ell^1$ co-regularization and strict $\ell^1$ co-regularization, for CS from indirect data. As main results, we derive error estimates for recovering $x_\star$ and $h_\star$. In particular, we derive a linear convergence rate in the norm for the latter. To obtain these results, solutions are required to satisfy a source condition and the CS measurement operator is required to satisfy a restricted injectivity condition. We further show that these conditions are not only sufficient but even necessary to obtain linear convergence.