Error-Splitting Forward Model for Iterative Reconstruction in X-Ray Computed Tomography and Application With Gauss–Markov–Potts Prior

Abstract

In order to enhance image quality for controlling the interior of a volume in industry, model-based iterative reconstruction (MBIR) methods in three-dimensional (3-D) X-ray computed tomography (CT) have shown their efficiency compared to analytical reconstruction methods. MBIR methods enforce a prior model on the volume to reconstruct and make fusion of the information contained in the prior model and the projection data. The projections have many uncertainties that have very different origins in 3-D CT. They are taken into account in MBIR methods but, despite of their different origins, they are in general gathered in only one term in the forward model. In this paper, we propose to derive a new forward model by adding a further error term in the Poisson statistics of photon-count, corresponding to the deviation of the monochromatic model with respect to the actual polychromacy of the rays. A Taylor expansion of the Poisson log-likelihood leads us to a new algebraic forward model accounting for two terms of uncertainties: we call it the error-splitting forward model. Different prior models are assigned to each of distinguished uncertainties: measurement uncertainties are modeled as Gaussian, while linear model uncertainties are modeled as heavy-tailed to bring robustness to the reconstruction process. We give strategies to fix the parameters of the error-splitting forward model. Next, we use it in a full MBIR method with Gauss–Markov–Potts prior model on the volume, in order to reconstruct piecewise-constant objects for non-destructive testing in industry. Compared to the conventional forward model, we show in our experiments that the use of the error-splitting forward model with Gauss–Markov–Potts prior combines better robustness and accuracy.