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