Stability of the Non-abelian X-ray Transform in Dimension $$\ge 3$$

Abstract

Non-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ 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.

Invariant distributions and the transport twistor space of closed surfaces

“…This lemma appears as [1,Lemma 4.4.4] and completes [2, Proposition 4.4] by proving also the stated mapping properties of 𝜕 ω; we include a proof of the latter for the reader's convenience.…”

“…Any function 𝑓 ∈ 𝐶 ∞ (𝑆𝑀) can be decomposed in Fourier modes by simply freezing the basepoint 𝑥 ∈ 𝑀 and considering the Fourier decomposition on 𝑆 𝑥 𝑀 ≃ 𝕊 1 . More precisely, any 𝑢 ∈ 𝐶 ∞ (𝑆𝑀) can be expanded as:…”

“…The biholomorphism Φ ∶ 𝔻 → ℍ = {𝑧 ∈ ℂ | ℑ(𝑧) > 0} defined by Φ(𝑦) ∶= 𝑖(1 + 𝑦)(1 − 𝑦) −1 identifies the open unit disk with the Poincaré half-space. The function Φ * (1∕𝑧) is a holomorphic function in 𝔻 blowing up polynomially near the boundary 𝜕𝔻 ≃ 𝑆 1 as in (1.8) (more precisely, it only blows up near the point −1 = Φ −1 (0)). Hence, the trace of Φ * (1∕𝑧) on 𝑆 1 defines a distribution 𝑢 ∈  ′ (𝑆 1 ) and it is straightforward to check that Φ * ((𝑥 + 𝑖0) −1 ) = 𝑢.…”

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

“…The non-abelian x-ray transform has also been studied on simple surfaces [PS20, MNP21] and compact manifolds with strictly convex boundary [Boh21] where the transport equation is now solved along unit-speed geodesics with endpoints on the boundary of the manifold. For more details and background on the two-dimensional problem, see [PSU21].…”

“…Depending on the inverse problem, can we get estimates such as (23) and can we guarantee that the posterior mean indeed converges to θ , legitimising Bayesian inversion? This question has been studied for a range of different inverse problems and is an active area of research, see [MNP21] as well as [AN19,Boh21,GN20] for some examples.…”