EFFICIENT NONPARAMETRIC BAYESIAN INFERENCE FOR X-RAY TRANSFORMS

• 成果类型: Article
• 署名作者: Monard, Francois; Nickl, Richard; Paternain, Gabriel P.
• 署名单位: University of California System; University of California Santa Cruz; University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1708
• 发表日期: 2019
• 页码: 1113-1147
• 关键词: INVERSE PROBLEMS; posterior distributions; mu-transmission; contraction; functionals; tomography; rates

摘要

We consider the statistical inverse problem of recovering a function f : M -> R, where M is a smooth compact Riemannian manifold with boundary, from measurements of general X-ray transforms I-a(f) of f, corrupted by additive Gaussian noise. For M equal to the unit disk with flat geometry and a = 0 this reduces to the standard Radon transform, but our general setting allows for anisotropic media M and can further model local attenuation effects-both highly relevant in practical imaging problems such as SPECT tomography. We study a nonparametric Bayesian inference method based on standard Gaussian process priors for f. The posterior reconstruction of f corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty that does not require the calculation of the singular value decomposition of the forward operator I-a. We prove Bernsteinvon Mises theorems for a large family of one-dimensional linear functionals of f, and they entail that posterior-based inferences such as credible sets are valid and optimal from a frequentist point of view. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which attains the semiparametric information lower bound. The proofs rely on an invertibility result for the Fisher information operator I-a*I-a between suitable function spaces, a result of independent interest that relies on techniques from microlocal analysis. We illustrate the performance of the proposed method via simulations in various settings.