dc.contributor.author Monard, F en dc.contributor.author Nickl, Richard en dc.contributor.author Paternain, Gabriel en dc.date.accessioned 2018-09-18T08:36:02Z dc.date.available 2018-09-18T08:36:02Z dc.date.issued 2019-04-01 en dc.identifier.issn 0090-5364 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/280315 dc.description.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 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 Cram\'er-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. dc.publisher Institute of Mathematical Statistics dc.title Efficient nonparametric bayesian inference for X-ray transforms en dc.type Article prism.endingPage 1147 prism.issueIdentifier 2 en prism.publicationDate 2019 en prism.publicationName Annals of Statistics en prism.startingPage 1113 prism.volume 47 en dc.identifier.doi 10.17863/CAM.27688 dcterms.dateAccepted 2018-04-04 en rioxxterms.versionofrecord 10.1214/18-AOS1708 en rioxxterms.version VoR * rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved en rioxxterms.licenseref.startdate 2019-04-01 en rioxxterms.type Journal Article/Review en pubs.funder-project-id Engineering and Physical Sciences Research Council (EP/M023842/1) pubs.funder-project-id European Research Council (647812) rioxxterms.freetoread.startdate 2100-01-01
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