Stability and Error Estimates of BV Solutions to the Abel Inverse Problem

TL;DR

This paper analyzes the stability and error bounds of BV solutions in the Abel inverse problem, providing theoretical guarantees for image reconstruction quality using total variation regularization.

Contribution

It offers new a priori L2-stability estimates and error bounds for BV solutions in the Abel inverse problem, enhancing understanding of reconstruction reliability.

Findings

L2-stability bounds for BV solutions derived

Error estimates for total variation regularized reconstructions provided

Theoretical analysis supports improved image reconstruction accuracy

Abstract

Reconstructing images from ill-posed inverse problems often utilizes total variation regularization in order to recover discontinuities in the data while also removing noise and other artifacts. Total variation regularization has been successful in recovering images for (noisy) Abel transformed data, where object boundaries and data support will lead to sharp edges in the reconstructed image. In this work, we analyze the behavior of BV solutions to the Abel inverse problem, deriving a priori estimates on the recovery. In particular, we provide L2-stability bounds on BV solutions to the Abel inverse problem. These bounds yield error estimates on images reconstructed from a proposed total variation regularized minimization problem.