Efficient Nonparametric Bayesian Inference For XRay Transforms
Abstract
We consider the statistical inverse problem of recovering a function $f: M \to \mathbb R$, 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 Bernsteinvon Mises theorems that entail that posteriorbased inferences such as credible sets are valid and optimal from a frequentist point of view for a large family of semiparametric aspects of $f$. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which is shown to attain the semiparametric CramérRao 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 arXiv:
 arXiv:1708.06332
 Bibcode:
 2017arXiv170806332M
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Analysis of PDEs;
 Statistics  Methodology;
 62G20 (Primary);
 58J40;
 65R10;
 62F15 (Secondary)
 EPrint:
 38 pages, 6 figures