Honeycomb geometry: Rigid motions on the hexagonal grid

International audienceEuclidean rotations in R^2 are bijective and isometric maps, but they lose generally these properties when digitized in discrete spaces. In particular, the topological and geometrical defects of digitized rigid motions on the square grid have been studied. In this context, the main problem is related to the incompatibility between the square grid and rotations; in general, one has to accept either relatively high loss of information or non-exactness of the applied digitized rigid motion. Motivated by these considerations, we study digitized rigid motions on the hexagonal grid. We establish a framework for studying digitized rigid motions in the hexagonal grid---previously proposed for the square grid and known as neighborhood motion maps. This allows us to study non-injective digitized rigid motions on the hexagonal grid and to compare the loss of information between digitized rigid motions defined on the two grids

Rigid motions in the cubic grid: A discussion on topological issues

International audienceRigid motions on 2D digital images were recently investigated with the purpose of preserving geometric and topological properties. From the application point of view, such properties are crucial in image processing tasks, for instance image registration. The known ideas behind preserving geometry and topology rely on connections between the 2D continuous and 2D digital geometries that were established via multiple notions of regularity on digital and continuous sets. We start by recalling these results; then we discuss the difficulties that arise when extending them from $\mathbb{Z}^2$ to $\mathbb{Z}^3$. On the one hand, we aim to provide a discussion on strategies that proved to be successful in $\mathbb{Z}^2$ and remain valid in $\mathbb{Z}^3$; on the other hand, we explain why certain strategies cannot be extended to the 3D framework of digitized rigid motions. We also emphasize the relationships that may exist between certain concepts initially proposed in $\mathbb{Z}^2$. Overall, our objective is to initiate an investigation about the most promising approaches for extending the 2D results to higher dimensions