Stability of Image-Reconstruction Algorithms

TL;DR

This paper investigates the stability of image-reconstruction algorithms, providing new theoretical results for $ ext{ell}_p$ regularization that extend understanding of their robustness in medical imaging.

Contribution

It introduces novel stability results for $ ext{ell}_p$ regularized inverse problems, generalizing existing results to a broader range of $p$ and function spaces.

Findings

Lipschitz continuity for small $p$

Hölder continuity for larger $p$

Generalization to $L_p(\Omega)$ spaces

Abstract

Robustness and stability of image-reconstruction algorithms have recently come under scrutiny. Their importance to medical imaging cannot be overstated. We review the known results for the topical variational regularization strategies ($\ell_2$ and $\ell_1$ regularization) and present novel stability results for $\ell_p$-regularized linear inverse problems for $p\in(1,\infty)$. Our results guarantee Lipschitz continuity for small $p$ and H\"{o}lder continuity for larger $p$. They generalize well to the $L_p(\Omega)$ function spaces.