Topology on Digital Label Images

Mazo, Loïc; Passat, Nicolas; Couprie, Michel; Ronse, Christian

doi:10.1007/s10851-011-0325-8

Cited by 8 publications

(16 citation statements)

References 47 publications

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“…However, it remains relevant whenever a notion of simple point (or more generally a local characterisation of topology preservation) is available. This is verified, for instance, in the context of n-ary images [14]. On the other hand, only the Eulerian (backwards) model has been considered in this study.…”

Section: Resultsmentioning

confidence: 55%

Sufficient Conditions for Topological Invariance of 2D Images under Rigid Transformations

Ngo¹,

Kenmochi²,

Passat

et al. 2013

Discrete Geometry for Computer Imagery

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Abstract. In R 2 , rigid transformations are topology-preserving operations. However, this property is generally no longer true when considering digital images instead of continuous ones, due to digitization effects. In this article, we investigate this issue by studying discrete rigid transformations (DRTs) on Z 2 . More precisely, we define conditions under which digital images preserve their topological properties under any arbitrary DRTs. Based on the recently introduced notion of DRT graph and the classical notion of simple point, we first identify a family of local patterns that authorize topological invariance under DRTs. These patterns are then involved in a local analysis process that guarantees topological invariance of whole digital images in linear time.

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Section: Resultsmentioning

confidence: 55%

Sufficient Conditions for Topological Invariance of 2D Images under Rigid Transformations

Ngo¹,

Kenmochi²,

Passat

et al. 2013

Discrete Geometry for Computer Imagery

Self Cite

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show abstract

“…This is depicted in Figure 6: on the left, the 4-curve separates the plane into two 8-components, and on the right, the 8-curve separates the plane into two 4-components. The gray set is made of one or two components depending on the associated connectivity [118].…”

Section: Using Dual Pairs Of Connectivitiesmentioning

confidence: 99%

“…However, this approach can be problematic: some label can be associated with different connectivities if it is considered as the foreground or part of the background, leading to ambiguities. For example, it can be made of one component at a given time and of two components at another time [118] (see Figure 7). Furthermore, in this way we consider only each label relatively to its associated background; the topological relationship between the different labels is then lost [142,11,111].…”

Section: Using Dual Pairs Of Connectivitiesmentioning

confidence: 99%

“…This way, the new representation of the image is "isomorphic" to a binary image; this approach has then been called binary modeling [114,122]. However, this last method assumes that the topological structure of the connected components of the labels has no ambiguity, and then can only be applied on "simple images" [118].…”

Section: Using Dual Pairs Of Connectivitiesmentioning

confidence: 99%

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A Tutorial on Well-Composedness

2017

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Due to digitization, usual discrete signals generally present topological paradoxes, such as the connectivity paradoxes of Rosenfeld. To get rid of those paradoxes, and to restore some topological properties to the objects contained in the image, like manifoldness, Latecki proposed a new class of images, called wellcomposed images, with no topological issues. Furthermore, well-composed images have some other interesting properties: for example, the Euler number is locally computable, boundaries of objects separate background from foreground, the tree of shapes is well-defined, and so on. Last, but not the least, some recent works in mathematical morphology have shown that very nice practical results can be obtained thanks to well-composed images. Believing in its prime importance in digital topology, we then propose this state-of-the-art of well-composedness, summarizing its different flavours, the different methods existing to produce well-composed signals, and the various topics that are related to well-composedness.

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“…In this work, we follow a recent and general proposal [39], [40], that consists of considering the values of L as "proto-labels", and any subsets of such values as the actual labels of the image. In other words, 2 A notion of grey-level well-composedness has also been proposed in [33].…”

Section: F Topological Invariance: the Label Casementioning

confidence: 99%