The theory of multidimensional persistent homology was initially developed in the discrete setting , and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of multidimensional persistence have been proved to hold when topological spaces are filtered by continuous functions, i.e. for continuous data. This talk aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. More precisely, we develop a method to compare persistent homologies of vector functions obtained from discrete data in such a way that stability is preserved. These advances support the appropriateness of multidimensional persistent homology for shape comparison in computer vision and computer graphics applications. This is a joint work with N. Cavazza, M. Ethier, P. Frosini, and T. Kaczynski.