Homotopic Digital Rigid Motion: An Optimization Approach on Cellular Complexes

Abstract

Topology preservation is a property of rigid motions in R 2 , but not in Z 2 . In this article, given a binary object X ⊂ Z 2 and a rational rigid motion R, we propose a method for building a binary object X R ⊂ Z 2 resulting from the application of R on a binary object X. Our purpose is to preserve the homotopy-type between X and X R . To this end, we formulate the construction of X R from 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 H by superimposition of two cubical spaces F and G corresponding to the canonical Cartesian grid of Z 2 where X is defined, and the Cartesian grid induced by the rigid motion R, respectively. The object X R is then computed by building a homotopic transformation within the space H, starting from the cubical complex in G resulting from the rigid motion of X with respect to R and ending at a complex fitting X R in F that can be embedded back into Z 2 .

Homotopic Affine Transformations in the 2D Cartesian Grid

“…The definition of X from X and A is then formulated as an optimization problem, which presents similarities with the topology-preserving paradigms developed in the framework of deformable models [16,17,19,53]. This article is an extended and improved version of the conference paper [47]. The new material is as follows.…”

“…In particular, we provide in Appendix A a description of the way of building the cellular space where to carry out the homotopic transformation required by the algorithm. Regarding the optimization part of the process, we provide a gradient descent approach, with a reproducible description, whereas only a sketched description of a general scheme was given in [47]. Finally, we present various experimental results and we compare the behaviour of the proposed transformation approach with other topology-preserving (rigid) transformations schemes that rely on the notions of regularity [43] and quasi-regularity [42], respectively.…”

“…In other words, the information on this putative embeddability is carried by the two functions Φ and ϕ defined in Eqs. (45,47). In addition, we can define three sets of 2-faces of…”