# Efficient nonparametric Bayesian inference for $X$-ray transforms

@article{Monard2017EfficientNB, title={Efficient nonparametric Bayesian inference for \$X\$-ray transforms}, author={Franccois Monard and Richard Nickl and Gabriel P. Paternain}, journal={The Annals of Statistics}, year={2017} }

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…

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## References

SHOWING 1-10 OF 70 REFERENCES

### Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

- Mathematics
- 2015

We consider nonparametric Bayesian inference in a reflected diffusion model $dX_t = b (X_t)dt + \sigma(X_t) dW_t,$ with discretely sampled observations $X_0, X_\Delta, \dots, X_{n\Delta}$. We analyse…

### MAP estimators and their consistency in Bayesian nonparametric inverse problems

- Mathematics
- 2013

We consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map G?> applied to u. We adopt a Bayesian approach to the problem and…

### The geodesic X-ray transform with matrix weights

- MathematicsAmerican Journal of Mathematics
- 2019

Abstract:Consider a compact Riemannian manifold of dimension $\geq 3$ with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray…

### Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators

- Mathematics
- 2014

A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the…

### Nonparametric Bernstein–von Mises theorems in Gaussian white noise

- Mathematics
- 2013

Bernstein-von Mises theorems for nonparametric Bayes priors in the Gaussian white noise model are proved. It is demonstrated how such results justify Bayes methods as efficient frequentist inference…

### The Bayesian Approach to Inverse Problems

- Mathematics
- 2017

These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian
approach to inverse problems in…

### Computationally Efficient Markov Chain Monte Carlo Methods for Hierarchical Bayesian Inverse Problems

- Computer Science, Mathematics
- 2016

A computationally efficient MCMC sampling scheme for ill-posed Bayesian inverse problems by employing a Metropolis-Hastings-within-Gibbs (MHwG) sampler with a proposal distribution based on a low-rank approximation of the prior-preconditioned Hessian.

### Bayesian inverse problems with non-conjugate priors

- Mathematics
- 2012

We investigate the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. A theorem is proved in a…

### Stability estimates for the X-ray transform of tensor fields and boundary rigidity

- Mathematics
- 2004

We study the boundary rigidity problem for domains in Rn: is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function g.x; y/ known for…

### The Radon transform

- Mathematics
- 2014

(Note that the improper integral converges.) Here the integral denotes the standard line integral from vector calculus. In higher dimensions the Radon transform maps a function f to its integrals…