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We study the reconstruction process when the X-ray source translates along a finite straight line, the detector moving or not. This process, called linear tomosynthesis, induces a limited angle of view, which causes the vertical spatial resolution to be poor. To improve this resolution, we use iterative algebraic reconstruction methods, which are commonly used for tomographic reconstruction from a reduced number of projections. With noisy projections, such algorithms produce poor quality reconstructions. To prevent this, we use a first object prior knowledge, consisting of piecewise smoothness constraint. To reduce the computation time associated with both reconstruction and regularization processes, we introduce a second geometrical prior knowledge, based on the linear trajectory of the X-ray source. This linear source trajectory allows us to reconstruct a series of two-dimensional (2-D) planes in a fan organization of the volume. Using this adapted fan volume sampling scheme, we reduce the computation time by transforming the initial three-dimensional (3-D) problem into a series of 2-D problems. Obviously, the algorithm becomes directly parallelizable. Focusing on a particular region of interest becomes easier too. The regularization process can easily be implemented with this scheme. We test the algorithm using experimental projections. The quality of the reconstructed object is conserved, while the computation time is considerably reduced, even without any parallelization of the algorithm.