Consistent Inversion of Noisy Non-Abelian X-Ray Transforms
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For M a simple surface, the non-linear and non-convex statistical inverse problem of recovering a matrix field Φ: M →so(n) from discrete, noisy measurements of the SO(n)-valued scattering data C_Φ of a solution of a matrix ODE is considered (n≥ 2). Injectivity of the map Φ C_Φ was established by [Paternain, Salo, Uhlmann; Geom. Funct. Anal. 2012]. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors is proposed, and it is shown how it can be implemented by infinite-dimensional MCMC methods. It is further shown that as the number N of measurements of point-evaluations of C_Φ increases, the statistical error in the recovery of Φ converges to zero in L^2(M)-distance at a rate that is algebraic in 1/N, and approaches 1/√(N) for smooth matrix fields Φ. The proof relies, among other things, on a new stability estimate for the inverse map C_Φ→Φ. Key applications of our results are discussed in the case n=3 to polarimetric neutron tomography, see [Desai et al., Nature Sc. Rep. 2018] and [Hilger et al., Nature Comm. 2018].
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