Rigid transformations (compositions of rotations and translations) are involved in many image processing applications. In this context, the rigid transformations are generally calculated in their continuous space, before applying a digitization process in order to obtain a result in Z². As a result, the induced rigid transformations in Z² exhibit different geometric and topological properties compared to their continuous analogues.

This work consists in studying the rigid transformations and their properties, geometric and topological, in the discrete space of Z². More precisely, we study conditions and characterizations allowing the preservation of the topology and geometry of discrete objects defined on Z² by arbitrary rigid transformations. We also present a rigid transformation model on Z² allowing to preserve these properties. The approach is based in particular on a polygonal representation of the discrete object, the rigid transformation is applied to the polygon then is followed by the process of discretization to obtain a result in Z².