Finite rotations on Z2: A hierarchical framework for bijectivity analysis
Résumé
In Euclidean spaces ($\mathbb R^d$, $d \geqslant 2$), rotations preserve distances and angles; they are also bijective. These important properties are no longer guaranteed when rotations are considered in discrete spaces. This is especially the case in the Cartesian spaces ($\mathbb Z^d$, $d \geqslant 2$), where rotations are however crucial for various applications, from image processing to computer graphics. In this article, we deal with the issue of bijectivity of discrete rotations in the 2-dimensional case ($\mathbb Z^2$). We contribute to the state of the art from two points of view. First, we investigate the structure of the (finite and infinite) rotations in $\mathbb Z^2$, and we shed light on the nature of this combinatorial space, which is analogue to a watershed tree. Second, we focus on the finite rotations, i.e. rotations that act on (finite) Euclidean balls instead of the (infinite) set $\mathbb Z^2$. Under these hypotheses, we investigate the bijective rotations either as the restrictions of bijective rotations on $\mathbb Z^2$, or as injective rotations on the Euclidean balls (thus bijective from their domain to their image). We provide two algorithmic schemes for building the combinatorial space of these finite rotations.
Domaines
Informatique [cs]Origine | Fichiers produits par l'(les) auteur(s) |
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