Bayesian 3D X-ray computed tomography image reconstruction with a scaled Gaussian mixture prior model

Abstract

In order to improve quality of 3D X-ray tomography reconstruction for Non Destructive Testing (NDT), we investigate in this paper hierarchical Bayesian methods. In NDT, useful prior information on the volume like the limited number of materials or the presence of homogeneous area can be included in the iterative reconstruction algorithms. In hierarchical Bayesian methods, not only the volume is estimated thanks to the prior model of the volume but also the hyper parameters of this prior. This additional complexity in the reconstruction methods when applied to large volumes (from 512 3 to 8192 3 voxels) results in an increasing computational cost. To reduce it, the hierarchical Bayesian methods investigated in this paper lead to an algorithm acceleration by Variational Bayesian Approximation (VBA) [1] and hardware acceleration thanks to projection and back-projection operators paralleled on many core processors like GPU [2]. In this paper, we will consider a Student-t prior on the gradient of the image implemented in a hierarchical way [3, 4, 1]. Operators H (forward or projection) and H t (adjoint or back-projection) implanted in multi-GPU [2] have been used in this study. Different methods will be evalued on synthetic volume "Shepp and Logan" in terms of quality and time of reconstruction. We used several simple regularizations of order 1 and order 2. Other prior models also exists [5]. Sometimes for a discrete image, we can do the segmentation and reconstruction at the same time, then the reconstruction can be done with less projections.

Regularization, Bayesian Inference, and Machine Learning Methods for Inverse Problems

“… If we choose for a Gaussian law, then becomes a Gauss–Markov–Potts model [ 19 ]. We can use the joint posterior to infer on : we may just do JMAP: or trying to access to the expected posterior values by using the Variational Bayesian Approximation (VBA) techniques [ 17 , 19 , 58 , 59 , 60 , 61 , 62 ]. When the iterations finished, we obtain an estimate of the reconstructed image f and its segmentation z when using JMAP and also the covariance of f as well as the parameters of the posterior laws of z …”

Regularization, Bayesian Inference and Machine Learning methods for Inverse Problems†

“…• p(g| f , z) does not depend on z, so it can be written as p(g| f ). arg max {p( f , z|g)} or trying to access to the expected posterior values by using the Variational Bayesian Approximation (VBA) techniques [19,56], [57,58], [17,55,59].…”

Bayesian Inference with Error Variable Splitting and Sparsity Enforcing Priors for Linear Inverse Problems

“…• Image restoration in RAMAN Mass Spectrometry or in Radio Astronomy [39], [37]. • Detecting and estimating unknown periodic shapes in a short duration signal [39] • 2D and 3D Industrial Computed Tomography (CT) for Non Destructive Testing (NDT) application [40], [8], [41], [42], [43], [44], [45]. • Low dose and limited angle CT for biological or medical applications [8], [46], [47], [44].…”