Combinatorics of the Gauss digitization under translation in 2D

Etienne Baudrier (1) , Loïc Mazo (1)

1 ICube - Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie

The action of a translation on a continuous object before its digitization generates several digital objects. This paper focuses on the combinatorics of the generated digital objects up to integer translations. In the general case, a worst-case upper bound is exhibited and proved to be reached on an example. Another upper bound is also proposed by making a link between the number of the digital objects and the boundary curve, through its self-intersections on the torus. An upper bound, quadratic in digital perimeter, is then derived in the convex case and eventually an asymptotic upper bound, quadratic in the grid resolution, is exhibited in the polygonal case. A few signicant examples finish the paper.

Domaines: Traitement des images [eess.IV]; Mathématique discrète [cs.DM]

Etienne Baudrier, Loïc Mazo. Combinatorics of the Gauss digitization under translation in 2D. Journal of Mathematical Imaging and Vision, 2018, 61, pp.224-236. ⟨10.1007/s10851-018-0846-5⟩. ⟨hal-01853991⟩