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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 fibrecrack decomposition problems.

Classical reconstruction methods for phase-contrast tomography consist of two stages: phase retrieval and tomographic reconstruction. A novel algebraic method combining the two was suggested by Kostenko et al. (Opt. Express, 21, 12185, 2013) and preliminary results demonstrating improved reconstruction com-pared to a two-stage method given. Using simulated free-space propagation experiments with a single sample-detector distance, we thoroughly compare the novel method with the two-stage method to address limitations of the preliminary results. We demonstrate that the novel method is substantially more robust towards noise; our simulations point to a possible reduction in counting times by an order of magnitude.