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

Jan Bohr

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arXiv · 2008.07951

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

Jan Bohr

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Abstract

Non-abelian $X$-ray tomography seeks to recover a matrix potential $Φ:M\rightarrow \mathbb{C}^{m\times m}$ in a domain $M$ from measurements of its so called scattering data $C_Φ$ at $\partial M$. For $\dim M\ge 3$ (and under appropriate convexity and regularity conditions), injectivity of the forward map $Φ\mapsto C_Φ$ was established in [arXiv:1605.07894]. In this article we extend [arXiv:1605.07894] by proving a Hölder-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.

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Jan Bohr. 2021-04-27. Stability of the non-abelian $X$-ray transform in dimension $\ge 3$. https://arxiv.org/abs/2008.07951

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