A complete characterization of the (m,n)(m,n)-cubes and combinatorial applications in imaging, vision and discrete geometry

The aim of this work is to provide a complete characterization of a (m,n)(m,n)-cube. The latter are the pieces of discrete planes appearing in Theoretical Computer Science, Discrete Geometry and Combinatorics. This characterization in three dimensions is the exact equivalent of the preimage for a discrete segment as it has been introduced by McIlroy. Further this characterization, which avoids the redundancies, reduces the combinatorial problem of determining the cardinality of the (m,n)(m,n)-cubes to a new combinatorial problem consisting of determining the volumic regions formed by the crossing of planes. This work can find applications in Imaging, Vision, and pattern recognition for instance.