X-ray computed tomography technique has been used in many different practical applications. Often after reconstruction we need segment or decompose objects into different components. In this paper, we propose two new reconstruction methods that can decompose objects at the same time. By incorporating direction information, the proposed methods can decompose objects into various directional components. Furthermore, we propose an algorithm to obtain the direction information of the object directly from its CT measurements. We demonstrate the proposed methods on simulated and real samples to show their practical applicability. The numerical results show the differences between the two methods and effectiveness as dealing with fibre-crack decomposition problems.