Homotopic Affine Transformations in the 2D Cartesian Grid

Nicolas Passat, Phuc Ngo, Yukiko Kenmochi, Hugues Talbot

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