Geometric Preservation of 2D Digital Objects Under Rigid Motions

Rigid motions (i.e. transformations based on translations and rotations) are simple, yet important, transformations in image processing. In $$\mathbb {R}^n$$Rn, they are both topology and geometry preserving. Unfortunately, these properties are generally lost in $$\mathbb {Z}^n$$Zn. In particular, when applying a rigid motion on a digital object, one generally alters its structure but also the global shape of its boundary. These alterations are mainly caused by digitization during the transformation process. In this specific context, some solutions for the handling of topological issues were proposed in $$\mathbb {Z}^2$$Z2. In this article, we also focus on geometric issues in $$\mathbb {Z}^2$$Z2. Indeed, we propose a rigid motion scheme that preserves geometry and topology properties of the transformed digital object: a connected object will remain connected, and some geometric properties (e.g. convexity, area and perimeter) will be preserved. To reach that goal, our main contributions are twofold. First, from an algorithmic point of view, our scheme relies on (1) a polygonization of the digital object, (2) the transformation of the intermediate piecewise affine object of $$\mathbb {R}^2$$R2 and (3) a digitization step for recovering a result within $$\mathbb {Z}^2$$Z2. The intermediate modeling of a digital object of $$\mathbb {Z}^2$$Z2 as a piecewise affine object of $$\mathbb {R}^2$$R2 allows us to avoid the geometric alterations generally induced by standard pointwise rigid motions. However, the final digitization of the polygon back to $$\mathbb {Z}^2$$Z2 has to be carried out cautiously. In particular, our second, theoretical contribution is a notion of quasi-regularity that provides sufficient conditions to be fulfilled by a continuous object for guaranteeing both topology and geometry preservation during its digitization.