Homotopic Affine Transformations in the 2D Cartesian Grid

Abstract

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 \({{\mathbb {Z}}}^2\).