Bijective Digitized Rigid Motions on Subsets of the Plane

Abstract

Rigid motions in $\mathbb {R}^2$ are fundamental operations in 2D image processing. They satisfy many properties: in particular, they are isometric and therefore bijective. Digitized rigid motions, however, lose these two properties. To investigate the lack of injectivity or surjectivity and more generally their local behavior, we extend the framework initially proposed by Nouvel and Rémila to the case of digitized rigid motions. Yet, for practical applications, the relevant information is not global bijectivity, which is seldom achieved, but bijectivity of the motion restricted to a given finite subset of $\mathbb {Z}^2$. We propose two algorithms testing that condition. Finally, because rotation angles are rarely given with infinite precision, we propose a third algorithm providing optimal angle intervals that preserve this restricted bijectivity.

Appendices

Appendix 1: Neighborhood Motion Maps for $\mathcal {G}_1^U$ (4-Neighborhood Case)

Neighborhood motion maps for $\mathcal {G}_1^U$ are depicted as label maps, for $\theta \in \left( 0,\frac{\pi }{6}\right)$ in Fig. 18; and $\theta \in \left( \frac{\pi }{6},\frac{\pi }{4}\right)$ in Fig. 19. Some neighborhood motion maps are symmetric with respect to the origin, i.e., neighborhood motion map of the indices (0, 0)

Fig. 19

Neighborhood motion maps of $\mathcal {G}^U_1$, as label maps, for $\theta \in \left( \frac{\pi }{6}, \frac{\pi }{4}\right)$ that differ from these for $\theta \in \big (0,\frac{\pi }{6}\big )$. Each label (i, j) corresponds to the frame $f_{i,j}^\theta$. Neighborhood motion maps which correspond to non-injective zones are marked by red dashed frames (Color figure online)

Fig. 20

Neighborhood motion maps of $\mathcal {G}^U_{2}$, as label maps, for $\theta \in (0, \alpha _1)$. Each label (i, j) corresponds to the frame $f_{i,j}^\theta$. Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames (Color figure online)

Fig. 21

Neighborhood motion maps $\mathcal {G}^U_{2}$, as label maps, for $\theta \in (\alpha _1, \alpha _2)$ that differ from those for $\theta \in (0, \alpha _1)$. Each label (i, j) corresponds to the frame $f_{i,j}^\theta$. Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames (Color figure online)

Fig. 22

Neighborhood motion maps $\mathcal {G}^U_{2}$, as label maps, for $\theta \in (\alpha _2, \alpha _3)$ that differ from those for $\theta \in (\alpha _1, \alpha _2)$. Each label (i, j) corresponds to the frame $f_{i,j}^\theta$. Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames (Color figure online)

Appendix 2: Neighborhood Motion Maps for $\mathcal {G}^U_{2}$ (8-Neighborhood Case)

Neighborhood motion maps for $\mathcal {G}^U_{2}$ are depicted as label maps, for $\theta \in (0, \alpha _1)$ in Fig. 20; $\theta \in (\alpha _1, \alpha _2)$ in Fig. 21; $\theta \in (\alpha _2, \alpha _3)$ in Fig. 22; $\theta \in (\alpha _3, \alpha _4)$ in Fig. 23; and $\theta \in \big (\alpha _4,\frac{\pi }{4}\big )$ in Fig. 24. For more information about angles $\alpha _n, n \in [1, 4]$ we refer the reader to Fig. 10. Some neighborhood motion maps are symmetric with respect to the origin, i.e., the neighborhood motion map of the indices (0, 0).

Fig. 23

Neighborhood motion maps $\mathcal {G}^U_{2}$, as label maps, for $\theta \in (\alpha _3, \alpha _4)$ that differ from those for $\theta \in (\alpha _2, \alpha _3)$. Each label (i, j) corresponds to the frame $f_{i,j}^\theta$. Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames (Color figure online)

Fig. 24

Neighborhood motion maps $\mathcal {G}^U_{2}$, as label maps, for $\theta \in \big (\alpha _4,\frac{\pi }{4}\big )$ that differ from those for $\theta \in (\alpha _3, \alpha _4)$. Each label (i, j) corresponds to the frame $f_{i,j}^\theta$. Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames (Color figure online)