TY - JOUR
T1 - A Hybrid Gibbs Sampler for Edge-Preserving Tomographic Reconstruction with Uncertain View Angles
AU - Uribe, Felipe
AU - Bardsley, Johnathan M.
AU - Dong, Yiqiu
AU - Hansen, Per Christian
AU - Riis, Nicolai A.B.
N1 - Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics.
PY - 2022
Y1 - 2022
N2 - 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 coeficient 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 coeficient. 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 coeficients 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.
AB - 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 coeficient 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 coeficient. 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 coeficients 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.
KW - Bayesian inverse problems
KW - Gibbs sampler
KW - Laplace approximation
KW - computed tomography
KW - stochastic Newton MCMC
UR - http://www.scopus.com/inward/record.url?scp=85140023341&partnerID=8YFLogxK
U2 - 10.1137/21M143251X
DO - 10.1137/21M143251X
M3 - Article
AN - SCOPUS:85140023341
SN - 2166-2525
VL - 10
SP - 1293
EP - 1320
JO - SIAM-ASA Journal on Uncertainty Quantification
JF - SIAM-ASA Journal on Uncertainty Quantification
IS - 3
ER -