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

Géraud, Thierry; Carlinet, Edwin; Crozet, Sébastien

doi:10.1007/978-3-319-18720-4_48

Cited by 10 publications

(7 citation statements)

References 21 publications

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“…2. In the 2D case-and not in nD with n > 2-the only self-dual local interpolation verifying strong invariant properties leading to DWC images is obtained with the median operator [13]. It is depicted in Fig.…”

Section: Solving a Discrete Topology Issuementioning

confidence: 94%

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Introducing the Dahu Pseudo-Distance

Géraud

Carlinet

et al. 2017

Lecture Notes in Computer Science

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The minimum barrier (MB) distance is defined as the minimal interval of gray-level values in an image along a path between two points, where the image is considered as a vertex-valued graph. Yet this definition does not fit with the interpretation of an image as an elevation map, i.e. a somehow continuous landscape. In this paper, based on the discrete set-valued continuity setting, we present a new discrete definition for this distance, which is compatible with this interpretation, while being free from digital topology issues. Amazingly, we show that the proposed distance is related to the morphological tree of shapes, which in addition allows for a fast and exact computation of this distance. That contrasts with the classical definition of the MB distance, where its fast computation is only an approximation.

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Section: Solving a Discrete Topology Issuementioning

confidence: 94%

“…Several subdivision-based interpolations of nD scalar images are known to produce DWC images. It is the case of the interpolation with the max operator, used in [13] for the quasi-linear computation of the tree of shapes of a scalar image, as defined by Eq. 2.…”

Section: Solving a Discrete Topology Issuementioning

confidence: 99%

Introducing the Dahu Pseudo-Distance

Géraud

Carlinet

et al. 2017

Lecture Notes in Computer Science

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“…This last method has been slightly modified by Gé-raud et al [62] in 2015 where the new pixels added at the center of a square of 4 pixels is always the median of these four primary pixels, since the median is a good solution to make an image well-composed in 2D (and only in 2D [27]). This method does not create any extrema.…”

Section: Then In 2006 Stelldinger Proposed a Methods Calledmentioning

confidence: 99%

“…This operator is based on connectivities, and then, in Rosenfeld's framework, we have to associate one connectivity to the upper threshold sets and one of its dual connectivities to the lower threshold sets to avoid incoherences [9,34]: see Figures 73 and 74. By contrast, using an EWC image, connectivities are equivalent, and then we can compute the tree of shapes of an image or of its negative with the same pair of connectivities: no switch of connectivities is needed [62]; Géraud and Najman [63] call this phenomenon "pure self-duality".…”

Section: Thin Topological Maps Thanks To Well-composednessmentioning

confidence: 99%

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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“…This is especially true when considering self-duality (recall that a transform ϕ is self-dual iff ϕ(−u) = −ϕ(u), where ϕ acts on the space of functions): peaks and valleys are not processed with the same connectivity, and as a consequence the self-duality property is not "perfectly pure". The companion paper [4] discusses at length these questions, which have been largely ignored in the literature on self-duality.…”

Section: Introductionmentioning

confidence: 99%

How to Make nD Functions Digitally Well-Composed in a Self-dual Way

Boutry

Géraud

Najman

2015

Lecture Notes in Computer Science

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Abstract. Latecki et al. introduced the notion of 2D and 3D wellcomposed images, i.e., a class of images free from the "connectivities paradox" of digital topology. Unfortunately natural and synthetic images are not a priori well-composed. In this paper we extend the notion of "digital well-composedness" to nD sets, integer-valued functions (graylevel images), and interval-valued maps. We also prove that the digital well-composedness implies the equivalence of connectivities of the level set components in nD. Contrasting with a previous result stating that it is not possible to obtain a discrete nD self-dual digitally well-composed function with a local interpolation, we then propose and prove a selfdual discrete (non-local) interpolation method whose result is always a digitally well-composed function. This method is based on a sub-part of a quasi-linear algorithm that computes the morphological tree of shapes.

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

Cited by 10 publications

References 21 publications

Introducing the Dahu Pseudo-Distance

Introducing the Dahu Pseudo-Distance

A Tutorial on Well-Composedness

How to Make nD Functions Digitally Well-Composed in a Self-dual Way

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