Bayesian 3D X-ray Computed Tomography with a Hierarchical Prior Model for Sparsity in Haar Transform Domain.

Li Wang,Nicolas Gac,Ali Mohammad-Djafari,Mircea Dumitru

doi:10.3390/e20120977

Li Wang, Nicolas Gac + Show 2 more

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https://doi.org/10.3390/e20120977

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Abstract

In this paper, a hierarchical prior model based on the Haar transformation and an appropriate Bayesian computational method for X-ray CT reconstruction are presented. Given the piece-wise continuous property of the object, a multilevel Haar transformation is used to associate a sparse representation for the object. The sparse structure is enforced via a generalized Student-t distribution (), expressed as the marginal of a normal-inverse Gamma distribution. The proposed model and corresponding algorithm are designed to adapt to specific 3D data sizes and to be used in both medical and industrial Non-Destructive Testing (NDT) applications. In the proposed Bayesian method, a hierarchical structured prior model is proposed, and the parameters are iteratively estimated. The initialization of the iterative algorithm uses the parameters of the prior distributions. A novel strategy for the initialization is presented and proven experimentally. We compare the proposed method with two state-of-the-art approaches, showing that our method has better reconstruction performance when fewer projections are considered and when projections are acquired from limited angles.

Highlights

Computed Tomography (CT) has been developed and widely used in medical diagnosis [1] and industrial Non-Destructive Testing (NDT) [2] in recent decades
We extend extensively the previous work by: (1) adapting the forward model and prior models to the 3D case, which is more appropriate for real 3D large data size applications; (2) comparing the Relative Mean Squared Error (RMSE) of the phantom reconstructed by the Hierarchical Haar transform-based Bayesian Method (HHBM) method with those by the conventional Quadratic Regularization (QR) and Total Variation (TV) methods, we show the advantages of the semi-supervised property of the HHBM method and that the HHBM method outperforms the TV method when insufficient data are estimated; (3) proposing new ideas for fixing the hyper-parameters in the proposed model; and (4)
We focus on the Joint Maximum A Posteriori (JMAP) estimation, given that in the case of the large data size of the 3D object, the computational costs for the Variational Bayesian Approximation (VBA) algorithm is too expensive

Summary

Introduction

Computed Tomography (CT) has been developed and widely used in medical diagnosis [1] and industrial Non-Destructive Testing (NDT) [2] in recent decades. In CT, objects are observed using different techniques, for example X-rays [3], ultrasound [4], microwaves [5], or infra-red [6]. X-ray CT employs the absorption of X-rays by the organs in a body or by the materials in industrial components to reconstruct the internal structure of the imaged object. When performing X-ray CT, a set of X-ray images of the measured parts is acquired. The intensity measured by the X-ray images corresponds to the intensity of the radiation passing through and attenuated by the object. CT reconstruction is typically treated as an inverse problem

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