In computed tomography, data consist of measurements of the attenuation of X-rays passing through an object. The goal is to reconstruct the linear attenuation coefficient of the object’s interior. For each position of the X-ray source, characterized by its angle with respect to a fixed coordinate system, one measures a set of data referred to as a view. A common assumption is that these view angles are known, but in some applications they are known with imprecision. We propose a framework to solve a Bayesian inverse problem that jointly estimates the view angles and an image of the object’s attenuation coefficient. We also include a few hyperparameters that characterize the likelihood and the priors. Our approach is based on a Gibbs sampler where the associated conditional densities are simulated using different sampling schemes—hence the term hybrid. In particular, the conditional distribution associated with the reconstruction is nonlinear in the image pixels, and is non-Gaussian and high-dimensional. We approach this distribution by constructing a Laplace approximation that represents the target conditional locally at each Gibbs iteration. This enables sampling of the attenuation coefficients in an efficient manner using iterative reconstruction algorithms. The numerical results show that our algorithm is able to jointly identify the image and the view angles, while also providing uncertainty estimates of both. We demonstrate our method with 2D X-ray computed tomography problems using fan beam configurations.


  1. computed tomography
  2. Bayesian inverse problems
  3. Gibbs sampler
  4. Laplace approximation
  5. stochastic Newton MCMC

MSC codes

  1. 60G60
  2. 62F15
  3. 65C05
  4. 65R32
  5. 65F22

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Information & Authors


Published In

cover image SIAM/ASA Journal on Uncertainty Quantification
SIAM/ASA Journal on Uncertainty Quantification
Pages: 1293 - 1320
ISSN (online): 2166-2525


Submitted: 14 April 2021
Accepted: 29 March 2022
Published online: 30 September 2022


  1. computed tomography
  2. Bayesian inverse problems
  3. Gibbs sampler
  4. Laplace approximation
  5. stochastic Newton MCMC

MSC codes

  1. 60G60
  2. 62F15
  3. 65C05
  4. 65R32
  5. 65F22



Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard Petersens Plads, Building 324, DK-2800 Kgs. Lyngby, Denmark ([email protected], [email protected], [email protected], [email protected]).
Johnathan M. Bardsley
Department of Mathematical Sciences, University of Montana, 32 Campus Drive, Missoula, MT 59812, USA ([email protected]).
Yiqiu Dong
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard Petersens Plads, Building 324, DK-2800 Kgs. Lyngby, Denmark ([email protected], [email protected], [email protected], [email protected]).
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard Petersens Plads, Building 324, DK-2800 Kgs. Lyngby, Denmark ([email protected], [email protected], [email protected], [email protected]).
Nicolai A. B. Riis
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard Petersens Plads, Building 324, DK-2800 Kgs. Lyngby, Denmark ([email protected], [email protected], [email protected], [email protected]).

Funding Information

This work was supported by Villum Investigator grant 25893 from The Villum Foundation.

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