Bijective Digitized Rigid Motions on Subsets of the Plane

Pluta, Kacper; Romon, Pascal; Kenmochi, Yukiko; Passat, Nicolas

doi:10.1007/s10851-017-0706-8

Cited by 19 publications

(15 citation statements)

References 10 publications

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“…In the case of the square grid, it is known that the equivalent of G is a lattice if and only if cosine and sine are rational numbers, i.e., rotations are given by primitive Pythagorean triples [16,18,21]. When on the contrary cosine or/and sine are irrational, G is an infinite and dense set [16,18,21].…”

Section: Eisenstein Rational Rotationsmentioning

confidence: 99%

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Characterization of Bijective Digitized Rotations on the Hexagonal Grid

et al. 2018

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Section: Eisenstein Rational Rotationsmentioning

confidence: 99%

“…It is also worth to mention that Pluta et al used an extension of the framework proposed originally by Nouvel and Rémila [16], to characterize bijective digitized rigid motions [21].…”

mentioning

confidence: 99%

Characterization of Bijective Digitized Rotations on the Hexagonal Grid

et al. 2018

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“…The simplest operations on digital images are the (digital or discretized versions of) isomorphic operations, and maybe the most elementary is the translation. Compositions of translations and rotations, on the square and hexagonal grids, have been considered and analyzed, e.g., in [3]. The isomorphic transformations mapping the grid into itself are described on the triangular grid in [4], but transformations that could map some gridpoints out of the grid were not considered.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, we call this process redigitization, when the corresponding pixels are computed after a translation (or other operation) which may result in some points (of the plane) that are not gridpoints. These types of redigitization play important roles also on the traditional grid when another operation, e.g., discrete rotation is considered, which may not be bijective [3,5]. The triangular grid is not a point lattice (there are grid vectors such that they do not translate the grid into itself, as it is shown in Figure 1), and therefore, as we show here, it is interesting to consider translations and analyze how the resulting image may change.…”

Section: Introductionmentioning

confidence: 99%

Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

Abuhmaidan¹,

Nagy

2019

Mathematics

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The triangular plane is the plane which is tiled by the regular triangular tessellation. The underlying discrete structure, the triangular grid, is not a point lattice. There are two types of triangle pixels. Their midpoints are assigned to them. By having a real-valued translation of the plane, the midpoints of the triangles may not be mapped to midpoints. This is the same also on the traditional square grid. However, the redigitized result on the square grid always gives a bijection (gridpoints of the square grid are mapped to gridpoints in a bijective way). This property does not necessarily hold on to the triangular plane, i.e., the redigitized translated points may not be mapped to the original points by a bijection. In this paper, we characterize the translation vectors that cause non bijective translations. Moreover, even if a translation by a vector results in a bijection after redigitization, the neighbor pixels of the original pixels may not be mapped to the neighbors of the resulting pixel, i.e., a bijective translation may not be digitally ‘continuous’. We call that type of translation semi-bijective. They are actually bijective but do not keep the neighborhood structure, and therefore, they seemingly destroy the original shape. We call translations strongly bijective if they are bijective and also the neighborhood structure is kept. Characterizations of semi- and strongly bijective translations are also given.

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“…It is then necessary to find a way for carrying T(p) back to Z n . The induced approximation may lead to altering the topological structure of the object X containing p. It may also modify the global shape of X by slightly moving its different points in a heterogeneous way [4].…”

Section: Introductionmentioning

confidence: 99%

Geometric Preservation of 2D Digital Objects Under Rigid Motions

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Passat

Kenmochi³

et al. 2018

J Math Imaging Vis

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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.

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Bijective Digitized Rigid Motions on Subsets of the Plane

Cited by 19 publications

References 10 publications

Characterization of Bijective Digitized Rotations on the Hexagonal Grid

Characterization of Bijective Digitized Rotations on the Hexagonal Grid

Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

Geometric Preservation of 2D Digital Objects Under Rigid Motions

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