Homotopic Digital Rigid Motion: An Optimization Approach on Cellular Complexes

Topology preservation is a property of rigid motions in ${\mathbb R^2}$, but not in $\mathbb Z^2$. In this article, given a binary object $\mathsf X \subset \mathbb Z^2$ and a rational rigid motion ${\mathcal R}$, we propose a method for building a binary object $\mathsf X_{\mathcal R} \subset \mathbb Z^2$ resulting from the application of ${\mathcal R}$ on a binary object $\mathsf X$. Our purpose is to preserve the homotopy-type between $\mathsf X$ and $\mathsf X_{\mathcal R}$. To this end, we formulate the construction of $\mathsf X_{\mathcal R}$ from $\mathsf X$ as an optimization problem in the space of cellular complexes with the notion of collapse on complexes. More precisely, we define a cellular space $\mathbb H$ by superimposition of two cubical spaces $\mathbb F$ and $\mathbb G$ corresponding to the canonical Cartesian grid of $\mathbb Z^2$ where $\mathsf X$ is defined, and the Cartesian grid induced by the rigid motion ${\mathcal R}$, respectively. The object $\mathsf X_{\mathcal R}$ is then computed by building a homotopic transformation within the space $\mathbb H$, starting from the cubical complex in $\mathbb G$ resulting from the rigid motion of $\mathsf X$ with respect to ${\mathcal R}$ and ending at a complex fitting $\mathsf X_{\mathcal R}$ in $\mathbb F$ that can be embedded back into $\mathbb Z^2$.