Purpose: The purpose of this work is to combine two areas of active research in tomographic x-ray imaging. The first one is the use of iterative reconstruction techniques. The second one is differential phase contrast imaging (DPCI).
Method: We derive an SPS type maximum likelihood (ML) reconstruction algorithm with regularization for DPCI. Forward and back-projection are implemented using spherically symmetric basis functions (blobs) and differential footprints, thus completely avoiding the need for numerical differentiation throughout the reconstruction process. The method is applied to the problem of reconstruction of an object from sparsely sampled projection.
Results: The results show that the proposed method can handle the sparely sampled data efficiently. In particular no streak artifacts are visible which are present images obtained by filtered back-projection (FBP).
Conclusion: Iterative reconstruction algorithms have a wide spectrum of proven advantages in the area of conventional computed tomography. The present work describes for the first time, how a matched forward and back-projection can be implemented for DPCI, which is furthermore free of any heuristics. The newly developed ML reconstruction algorithm for DPCI shows that for the case of sparsely sampled projection data, an improvement in image quality is obtained that is qualitatively comparable to a corresponding situation in conventional x-ray imaging. Based on the proposed operators for forward and back-projection, a large variety of iterative reconstruction algorithms is thus made available for DPCI.