Efficient Nonparametric Bayesian Inference For X-Ray Transforms

2017/08/21 by Monard, François, Nickl, Richard, Paternain, Gabriel P. · 7 citations
#58J40 #62F15 (Secondary) #62G20 (Primary) #65R10 #Analysis of PDEs (math.AP) #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1708.06332

Abstract

We consider the statistical inverse problem of recovering a function f: M → \mathbb 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 propose a nonparametric Bayesian inference approach 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 Bernstein-von Mises theorems that entail that posterior-based inferences such as credible sets are valid and optimal from a frequentist point of view for a large family of semi-parametric aspects of f. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which is shown to attain the semi-parametric Cramér-Rao information 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.

Cited by

• Singularly Weighted X-ray Tensor Tomography
• Discrete to continuum limits in Bayesian inverse problems
• Local Characterization of Noise in Iterative Reconstruction of the Generalized Radon Transform
• A unified theory of the high-dimensional Laplace approximation with application to Bayesian inverse problems
• Consistency of variational inference for Besov priors in non-linear inverse problems
• Data assimilation with the 2D Navier-Stokes equations: Optimal Gaussian asymptotics for the posterior measure
• Statistical microlocal analysis in two-dimensional X-ray CT

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