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dc.contributor.authorMonard, Fen
dc.contributor.authorNickl, Richarden
dc.contributor.authorPaternain, Gabrielen
dc.date.accessioned2018-09-18T08:36:02Z
dc.date.available2018-09-18T08:36:02Z
dc.date.issued2019-04-01en
dc.identifier.issn0090-5364
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/280315
dc.description.abstractWe 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.publisherInstitute of Mathematical Statistics
dc.titleEfficient nonparametric bayesian inference for X-ray transformsen
dc.typeArticle
prism.endingPage1147
prism.issueIdentifier2en
prism.publicationDate2019en
prism.publicationNameAnnals of Statisticsen
prism.startingPage1113
prism.volume47en
dc.identifier.doi10.17863/CAM.27688
dcterms.dateAccepted2018-04-04en
rioxxterms.versionofrecord10.1214/18-AOS1708en
rioxxterms.versionVoR*
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserveden
rioxxterms.licenseref.startdate2019-04-01en
rioxxterms.typeJournal Article/Reviewen
pubs.funder-project-idEngineering and Physical Sciences Research Council (EP/M023842/1)
pubs.funder-project-idEuropean Research Council (647812)
rioxxterms.freetoread.startdate2100-01-01


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