An Automatic Regularization Method: An Application for 3-D X-Ray Micro-CT Reconstruction Using Sparse Data

X-ray tomography is a reliable tool for determining the inner structure of 3-D object with penetrating X-rays. However, traditional reconstruction methods, such as Feldkamp–Davis–Kress (FDK), require dense angular sampling in the data acquisition phase leading to long measurement times, especially in X-ray micro-tomography to obtain high-resolution scans. Acquiring less data using greater angular steps is an obvious way for speeding up the process and avoiding the need to save huge data sets. However, computing 3-D reconstruction from such a sparsely sampled data set is difficult because the measurement data are usually contaminated by errors, and linear measurement models do not contain sufficient information to solve the problem in practice. An automatic regularization method is proposed for robust reconstruction, based on enforcing sparsity in the 3-D shearlet transform domain. The inputs of the algorithm are the projection data and <italic>a priori</italic> known expected degree of sparsity, denoted as <inline-formula> <tex-math notation="LaTeX">$\textsf {0}< {\mathcal C}_{\textit {pr}}\leq \textsf {1}$ </tex-math></inline-formula>. The number <inline-formula> <tex-math notation="LaTeX">${\mathcal C}_{\textit {pr}}$ </tex-math></inline-formula> can be calibrated from a few dense-angle reconstructions and fixed. Human subchondral bone samples were tested, and morphometric parameters of the bone reconstructions were then analyzed using standard metrics. The proposed method is shown to outperform the baseline algorithm (FDK) in the case of sparsely collected data. The number of X-ray projections can be reduced up to 10% of the total amount 300 projections over 180° with uniform angular step while retaining the quality of the reconstruction images and of the morphometric parameters.