Decomposition of tomographic reconstructions has many different practical
application. We propose two new reconstruction methods that combines the task
of tomographic reconstruction with object decomposition. We demonstrate these
reconstruction methods in the context of decomposing directional objects into
various directional components. Furthermore we propose a method for estimating
the main direction in a directional object, directly from the measured computed
tomography data. We demonstrate all the proposed methods on simulated and real
samples to show their practical applicability. The numerical tests show that
decomposition and reconstruction can combined to achieve a highly useful
fibre-crack decomposition