Consistent Inversion of Noisy NonAbelian XRay Transforms
Abstract
For $M$ a simple surface, the nonlinear statistical inverse problem of recovering a matrix field $\Phi: M \to \mathfrak{so}(n)$ from discrete, noisy measurements of the $SO(n)$valued scattering data $C_\Phi$ of a solution of a matrix ODE is considered ($n\geq 2$). Injectivity of the map $\Phi \mapsto C_\Phi$ 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 infinitedimensional MCMC methods. It is further shown that as the number $N$ of measurements of pointevaluations of $C_\Phi$ increases, the statistical error in the recovery of $\Phi$ converges to zero in $L^2(M)$distance at a rate that is algebraic in $1/N$, and approaches $1/\sqrt N$ for smooth matrix fields $\Phi$. The proof relies, among other things, on a new stability estimate for the inverse map $C_\Phi \to \Phi$. 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]
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.00860
 Bibcode:
 2019arXiv190500860M
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Statistics Theory
 EPrint:
 51 pages, 5 figures