  X-ray tomography is a reliable tool for determining the inner structure of 3D
object with penetrating X-rays. However, traditional reconstruction methods
such as 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 available memory. However, computing 3D reconstruction from such
a sparsely sampled dataset is very sensitive to measurement noise and modelling
errors. An automatic regularization method is proposed for robust
reconstruction, based on enforcing sparsity in the three-dimensional shearlet
transform domain. The inputs of the algorithm are the projection data and {\it
a priori} known expected degree of sparsity, denoted $0<{\mathcal C}_{pr}\leq
1$. The number ${\mathcal C}_{pr}$ 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 while retaining the
quality of the reconstruction images and of the morphometric paramaters.
