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

Abstract

Rigid motions on 2D digital images were recently investigated with the purpose of preserving geometric and topological properties. From the application point of view, such properties are crucial in image processing tasks, for instance image registration. The known ideas behind preserving geometry and topology rely on connections between the 2D continuous and 2D digital geometries that were established via multiple notions of regularity on digital and continuous sets. We start by recalling these results; then we discuss the difficulties that arise when extending them from Z 2 to Z 3 . On the one hand, we aim to provide a discussion on strategies that prove to be successful in Z 2 and remain valid in Z 3 ; on the other hand, we explain why certain strategies cannot be extended to the 3D framework of digitized rigid motions. We also emphasize the relationships that may exist between certain concepts initially proposed in Z 2 . Overall, our objective is to initiate an investigation about the most promising approaches for extending the 2D results to higher dimensions.

Morphological Hierarchies: A Unifying Framework with New Trees

“…Indeed, our initial purpose was to develop adequate tools that would allow us to carry out the topological analysis of objects in non-binary paradigms (e.g. for grey-level images or fuzzy modeling), especially for understanding the topological alterations induced on numerical images by geometric transformations [37].…”

Homotopic Affine Transformations in the 2D Cartesian Grid

“…The purposes were manifold: describing the combinatorial structure of these transformations with respect to R n versus Z n [1-3, 7, 8, 18, 21-23, 33, 38, 48, 56], guaranteeing their bijectivity [4,5,12,25,44,49,50,54] or their transitivity [45] in Z n , preserving geometrical properties [41,42] and, less frequently, ensuring their topological invariance [39,43] in Z n . These are non-trivial questions, and their difficulty increases with the dimension of the Cartesian grid [46]. Indeed, most of these works deal with Z 2 [4, 5, 7, 8, 25, 33, 38-40, 42-45, 50, 54]; fewer with Z 3 [41,49,56] or Z n [18].…”

Homotopic Digital Rigid Motion: An Optimization Approach on Cellular Complexes

“…Translations [5,19], rotations [1,2,6,13,27,28,31,37] and more generally rigid motions [22-26, 29, 32] in the Cartesian grids have been studied with various purposes: describing the combinatorial structure of these transformations with respect to R n vs. Z n [5,6,19,22,30,38], guaranteeing their bijectivity [1,2,13,27,31,32,37] or transitivity [28] in Z n , preserving geometrical properties [24,25] and, less frequently, ensuring their topological invariance [23,26] in Z n . These are non-trivial questions, and their difficulty increases with the dimension of the Cartesian grid [29]. Indeed, most of these works deal with Z 2 [1, 2, 5, 6, 13, 19, 22, 23, 25-28, 32, 37]; fewer with Z 3 [24,31,38].…”