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In this paper, we consider sampling theory for irregular sampling sets of minimal density which are structured. An efficient method for the reconstruction of band-limited discrete signals from sampling sets which are unions of shifted lattices is developed. These sets are not necessarily periodic. A signal can be reconstructed from its samples provided the sampling set and the spectrum of the signal satisfy certain compatibility conditions. While explicit reconstruction formulas for unions of sampling lattices are possible, it is more convenient to use a recursive algorithm. A numerical example utilizing nonperiodic sampling sets implemented in MATLAB is given. The theory and the related algorithm has been applied to speech and image processing examples.