A Topological Tree of Shapes

Passat, Nicolas; Kenmochi, Yukiko

doi:10.1007/978-3-031-19897-7_18

Cited by 4 publications

(5 citation statements)

References 29 publications

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“…In this article, we continued our study of the topological tree of shapes initiated in [18,19]. Here, we focused on the design of an efficient algorithm for building the topological tree of shapes from the tree of shapes.…”

Section: Discussionmentioning

confidence: 99%

“…At the confluence of these two subfamilies, the topological tree of shapes was recently introduced [18]. It was initially designed for modelling the two orders-inclusion and nesting-associated to component-trees and trees of shapes.…”

Section: Introductionmentioning

confidence: 99%

“…The topological tree of shapes and the tree of shapes 3 derive (by a decreasing homeomorphism) from the complete tree of shapes, which is also introduced in [18]. This led to a construction scheme of the topological tree of shapes from the tree of shapes, via the complete tree of shapes.…”

Section: Introductionmentioning

confidence: 99%

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Building the Topological Tree of Shapes from the Tree of Shapes

Mendes Forte,

Passat,

Kenmochi

2024

Lecture Notes in Computer Science

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The topological tree of shapes was recently introduced as a new hierarchical structure within the family of morphological trees. Morphological trees are efficient models for image processing and analysis. For such applications, it is of paramount importance that these structures be built and manipulated with optimal complexity. In this article, we focus on the construction of the topological tree of shapes. We propose an algorithm for building the topological tree of shapes from the tree of shapes. In particular, a cornerstone of this algorithm is the construction of the complete tree of shapes, another recently introduced tree unifying both the tree of shapes and the topological tree of shapes. We also discuss the cost of the computation of these structures.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Building the Topological Tree of Shapes from the Tree of Shapes

Mendes Forte,

Passat,

Kenmochi

2024

Lecture Notes in Computer Science

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“…In this article, which is an extended and improved version of the conference paper [36], we introduce a new family of hierarchical structures, including one DAG and three trees, namely:…”

Section: Contributionsmentioning

confidence: 99%

“…For the sake of readability, they are gathered and indexed in Tables 1-2. For the sake of coherence, some notations used in this article may sometimes differ from those used in [36].…”

Section: Notations and Basic Notionsmentioning

confidence: 99%

Morphological Hierarchies: A Unifying Framework with New Trees

Passat,

Mendes Forte,

Kenmochi

2023

J Math Imaging Vis

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Morphological hierarchies constitute a rich and powerful family of graph-based structures that can be used for image modeling, processing and analysis. In this article, we focus on an important subfamily of morphological hierarchies, namely the trees that model partial partitions of the image support. This subfamily includes in particular the component-tree and the tree of shapes. In this context, we provide some new graph-based structures (one directed acyclic graph and three trees): the graph of valued shapes, the tree of valued shapes, the complete tree of shapes and the topological tree of shapes. These new objects create a continuum between the two notions of component-tree and tree of shapes. In particular, they allow to establish that these two trees (together with a third notion of adjacency tree generally considered in topological image analysis) can be defined and handled in a unified framework. In addition, this framework enables to enrich the component-tree with additional information, leading on the one hand to a topological description of grey-level images that relies on the same paradigm as persistent homology, and on the other hand to the proposal of a topological version of tree of shapes. This article provides a theoretical analysis of these new morphological hierarchies and their links with the usual ones. It also proposes an algorithmic description of two ways of building these new morphological hierarchies, and a discussion on the links that exist between these morphological hierarchies and certain topological invariants and descriptors.

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