Rigid image registration is an essential image processing task, with a large body of applications. This problem is usually formulated in the continuous domain, most often in an optimization framework. This strategy is suitable whenever the estimated transformation is associated with interpolation. In the case of purely discrete applications, e.g., for template-based segmentation or classification, it is however preferable to avoid digitizing the result again after transformation. In this article, we deal with this point of view in the 2D case. Based on a fully discrete framework, we explicitly explore the parameter space of rigid transformations. This exploration leads to a local search scheme that can be involved in combinatorial optimization strategies.