Stability of the non-abelian $X$-ray transform in dimension $\ge 3$

TL;DR

This paper proves a stability estimate for the non-abelian X-ray transform in dimensions three and higher, extending previous injectivity results and applying to statistical consistency in higher dimensions.

Contribution

It provides a quantitative, Hölder-type stability estimate for the non-abelian X-ray transform in dimensions ≥3, building on and extending prior injectivity results.

Findings

Established Hölder stability estimate for the transform

Extended injectivity results to higher dimensions

Generalized statistical consistency to dimensions ≥3

Abstract

Non-abelian $X$-ray tomography seeks to recover a matrix potential $\Phi:M\rightarrow \mathbb{C}^{m\times m}$ in a domain $M$ from measurements of its so called scattering data $C_\Phi$ at $\partial M$. For $\dim M\ge 3$ (and under appropriate convexity and regularity conditions), injectivity of the forward map $\Phi \mapsto C_\Phi$ was established in [arXiv:1605.07894]. In this article we extend [arXiv:1605.07894] by proving a H\"older-type stability estimate. As an application we generalise a statistical consistency result for $\dim M =2$ [arXiv:1905.00860] to higher dimensions. The injectivity proof in [arXiv:1605.07894] relies on a novel method by Uhlmann-Vasy [arXiv:1210.2084], which first establishes injectivity in a shallow layer below $\partial M$ and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular proving uniform bounds on layer-depth and stability constants.