The geodesic X-ray transform with matrix weights

TL;DR

This paper proves the injectivity and uniqueness of the geodesic X-ray transform with matrix weights on certain convex Riemannian manifolds, with applications to inverse problems in tomography.

Contribution

It establishes injectivity and uniqueness results for the attenuated ray transform with connections and Higgs fields on convex manifolds, extending previous results to non-negative curvature cases.

Findings

Injectivity of the attenuated ray transform modulo natural obstructions.

Uniqueness of the connection and Higgs field from scattering data.

Applicability to inverse problems in quantum state and polarization tomography.

Abstract

Consider a compact Riemannian manifold of dimension $\geq 3$ with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-forms. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo gauge transformations. The proofs involve a reduction to a local result showing that the geodesic X-ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary, and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions. As a somewhat striking corollary, we show that these integral geometry problems can be solved on strictly convex manifolds of dimension $\geq 3$ having non-negative sectional curvature (similar results were known earlier in negative sectional curvature). We also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography.