A Tutorial on Well-Composedness

Boutry, Nicolas; Géraud, Thierry; Najman, Laurent

doi:10.1007/s10851-017-0769-6

Cited by 28 publications

(20 citation statements)

References 167 publications

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“…It is well-known that DWCness and CWCness are equivalent in 2D and 3D (see, for example, [4]). In this section, we prove that there exists at least one set X ⊂ Z 4 which is DWC but not CWC.…”

Section: Dwcness Does Not Imply Cwcnessmentioning

confidence: 99%

A 4D Counter-Example Showing that DWCness Does Not Imply CWCness in nD

Boutry¹,

González-Dı́az²,

Najman³

et al. 2020

Lecture Notes in Computer Science

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In this paper, we prove that the two flavors of well-composedness called Continuous Well-Composedness (shortly CWCness) and Digital Well-Composedness (shortly DWCness) are not equivalent in dimension 4 thanks to an example of a configuration of 8 tesseracts (4D cubes) sharing a common corner (vertex), which is DWC but not CWC. This result is surprising since we know that CWCness and DWCness are equivalent in 2D and 3D. To prove this new result, local (and then relative) homology are used. This paper has been submitted to IWCIA.

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Section: Dwcness Does Not Imply Cwcnessmentioning

confidence: 99%

A 4D Counter-Example Showing that DWCness Does Not Imply CWCness in nD

Boutry¹,

González-Dı́az²,

Najman³

et al. 2020

Lecture Notes in Computer Science

Self Cite

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show abstract

“…In particular, a definition of well-composedness in Z n , n ≥ 3, is based on this (n − 1)-manifoldness characterization. This discussion is out of the scope of this paper; the interested reader is referred to [4,10] for more details.…”

Section: Digital Topology and Well-composed Imagesmentioning

confidence: 99%

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

Passat

Kenmochi²,

Ngo

et al. 2019

Discrete Geometry for Computer Imagery

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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.

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“…In the sequel, we will consider the same paradigm. However, it is worth mentioning that in the 2D case and for digital objects whose continuous analogues have a manifold boundary (this will be our case with well-composed objects, see below), most topological invariants are indeed equivalent, namely homotopy type, adjacency tree and homeomorphism [15][16][17].…”

Section: Digitization and Topology Preservationmentioning

confidence: 99%

“…The half-planes can then be deduced from the consecutive vertices of the computed convex hull, from Eqs. (15)(16)(17). An example of convex hull and half-plane modeling of an H-convex digital object is illustrated in Fig.…”

Section: Polygonization Of H-convex Digital Objectsmentioning

confidence: 99%

Geometric Preservation of 2D Digital Objects Under Rigid Motions

Ngo

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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A Tutorial on Well-Composedness

Cited by 28 publications

References 167 publications

A 4D Counter-Example Showing that DWCness Does Not Imply CWCness in nD

A 4D Counter-Example Showing that DWCness Does Not Imply CWCness in nD

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

Geometric Preservation of 2D Digital Objects Under Rigid Motions

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