Stability estimate for the broken non-abelian x-ray transform in Minkowski space

Abstract

We study the broken non-abelian X-ray transform in Minkowski space. This transform acts on the space of Hermitian connections on a causal diamond and is known to be injective up to an infinite-dimensional gauge. We show a stability estimate that takes into account the gauge, leading to a new proof of the transform's injectivity. Our proof leads us to consider a special type of connections that we call light-sink connections. We then show that we can consistently recover a light-sink connection from noisy measurement of its X-ray transform data through Bayesian inversion.

Broken Ray Transform for Twisted Geodesics on Surfaces with a Reflecting Obstacle

“…In the case of rotational (or spherical) symmetry, one may sometimes solve these and related problems using local results and data avoiding the obstacle when the manifold satisfies the Herglotz condition [12,36,64]. Broken lens rigidity was studied recently in [13], and a broken non-Abelian ray transform in Minkowski space in [63]. Other geometric results include boundary determination from a broken ray transform [32] and a reflection approach using strong symmetry assumptions [34], for example letting to solve the broken ray transforms on flat boxes over closed billiard trajectories [29,33].…”