Stability of the non-abelian X-ray transform in dimension ≥ 3

2020/08/18 by Jan Bohr, Bohr, Jan
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.2008.07951

To appear in Journal of Geometric Analysis

arxiv created 2021/04/27 · arxiv updated 2021/04/28

Abstract

Non-abelian X-ray tomography seeks to recover a matrix potential Φ:M→ ℂ^{m× m} in a domain M from measurements of its so called scattering data C_Φ at ∂ M. For dim M≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map Φ↦ 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 ∂ 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.

Citations

• The geodesic X-ray transform with matrix weights
• Consistent Inversion of Noisy Non-Abelian X-Ray Transforms
• The inverse problem for the local geodesic ray transform

Related

• Consistent Inversion of Noisy Non-Abelian X-Ray Transforms
• Discrete non-abelian X-ray transforms
• The geodesic X-ray transform with matrix weights
• Unique continuation of the normal operator of the X-ray transform and applications in geophysics
• Non-standard Sobolev scales and the mapping properties of the X-ray transform on manifolds with strictly convex boundary