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We present a tomographic reconstruction algorithm based on a frequential decomposition of the data. We show that the frequential components of the attenuation function to be identified can be reconstructed from the frequential decomposition of the data. Moreover, downsampling techniques added to the identification of components and coupled to compression techniques, speed up the reconstruction time up to six compared to the classical FBP. We identify the optimal number of frequential components. We show reconstructions from real data. A parallel implementation of our new algorithm is then proposed and evaluated on two small PC clusters.