Geometric Preservation of 2D Digital Objects Under Rigid Motions

Abstract

Rigid motions (i.e. transformations based on translations and rotations) are simple, yet important, transformations in image processing. In R n , they are both topology and geometry preserving. Unfortunately, these properties are generally lost in Z n . 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 Z 2 . In this article, we also focus on geometric issues in Z 2 . 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) piecewise affine object of R 2 allows us to avoid the geometric alterations generally induced by standard pointwise rigid motions. However, the final digitization of the polygon back to Z 2 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.

Rigid Motions in the Cubic Grid: A Discussion on Topological Issues

“…The Gauss digitization of a quasi-r-regular set X ⊂ R 2 (namely X ∩ Z 2 ) is a wellcomposed set that remains simply connected, thus preserving its topological properties from R 2 to Z 2 . Proposition 1 ( [16]). If X ⊂ R 2 is quasi-1-regular with margin √ 2−1, then X = X ∩Z 2 and X = X ∩ Z 2 are both 4-connected.…”

“…Recently, the notion of quasi-r-regularity [16] was introduced together with an algorithmic scheme in order to perform rigid motions on digital sets in Z 2 . The scheme relies on the use of an intermediate modeling of a 2D digital set as a piecewise affine subset of R 2 , namely a polygon.…”

“…We recall the definitions of quasi-r-regularity that was initially defined in R 2 [16] and then extended to R 3 [17]. These definitions and associated results are developed in the case of simply connected (i.e.…”

“…Definition 4 (Quasi-r-regularity [16,17]). Let X ⊂ R n (n = 2, 3) be a bounded, simply connected set.…”

“…In particular, we consider an alternative approach to regularity, based on quasi-rregular polygons, which are used as intermediate continuous models of digital shapes for their rigid motions in Z 2 [16]. This approach, which relies on a mixed dicretecontinuous paradigm (see [9] for related works), relies on three steps: polygonizing the boundary of a given digital set; applying a rigid motion on the polygon; and digitizing the transformed polygon.…”