Well-Composed Sets

Latecki, Longin Jan; Eckhardt, Ulrich; Rosenfeld, Azriel

doi:10.1006/cviu.1995.1006

Cited by 97 publications

(27 citation statements)

References 21 publications

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“…Such curves arise naturally from the cellular decomposition (putting it simply the inter-pixel contour) of the Gauss digitization of simple Euclidean objects, provided they are well-composed [13]. In our framework we consider that the digital border of a shape is made of linel (which can also be seen as the so-called Freeman moves), that is the side of a square whose centre has the coordinates of a digital point.…”

Section: Test Shapes and Experimental Protocolmentioning

confidence: 99%

Comparison and improvement of tangent estimators on digital curves

Lachaud¹

2009

Pattern Recognition

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Section: Test Shapes and Experimental Protocolmentioning

confidence: 99%

Comparison and improvement of tangent estimators on digital curves

Lachaud¹

2009

Pattern Recognition

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“…The above problems were studied in [18] and led to the notion of regularity in Z 2 defined in the frameworks of digital topology [7] and well-composed images [11] (see Sec. 3.2).…”

Section: Regular Images and Topological Invariance Under Rigid Motionsmentioning

confidence: 99%

“…The notion of well-composedness [11] was then introduced to characterize some digital sets X whose structure intrinsically avoids the topological issues of the Jordan theorem in Z 2 and further in higher dimensions.…”

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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“…Let us start by recalling some seminal definitions from the paper [15] by Latecki, Eckhardt, and Rosenfeld.…”

Section: Well-composed Sets and Imagesmentioning

confidence: 99%

Self-duality and Digital Topology: Links Between the Morphological Tree of Shapes and Well-Composed Gray-Level Images

Géraud

Carlinet

Crozet

2015

Lecture Notes in Computer Science

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International audienceIn digital topology, the use of a pair of connectivities is required to avoid topological paradoxes. In mathematical morphology, self-dual operators and methods also rely on such a pair of connectivities. There are several major issues: self-duality is impure, the image graph structure depends on the image values, it impacts the way small objects and texture are processed, and so on. A sub-class of images defined on the cubical grid, well-composed images, has been proposed, where all connectivities are equivalent, thus avoiding many topological problems. In this paper we unveil the link existing between the notion of well-composed images and the morphological tree of shapes. We prove that a well-composed image has a well-defined tree of shapes. We also prove that the only self-dual well-composed interpolation of a 2D image is obtained by the median operator. What follows from our results is that we can have a purely self-dual representation of images, and consequently, purely self-dual operators

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Well-Composed Sets

Cited by 97 publications

References 21 publications

Comparison and improvement of tangent estimators on digital curves

Comparison and improvement of tangent estimators on digital curves

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

Self-duality and Digital Topology: Links Between the Morphological Tree of Shapes and Well-Composed Gray-Level Images

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