research-article

Morphological Hierarchies: A Unifying Framework with New Trees

Authors:

Nicolas Passat,

Julien Mendes�Forte,

Yukiko KenmochiAuthors Info & Claims

Journal of Mathematical Imaging and Vision, Volume 65, Issue 5

Pages 718 - 753

https://doi.org/10.1007/s10851-023-01154-x

Published: 16 August 2023 Publication History

Abstract

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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Cited By

Perrin RLeborgne APassat NNaegel BWemmert C(2024)Multi-scale Component-Tree: A Hierarchical Representation for�Sparse ObjectsDiscrete Geometry and Mathematical Morphology10.1007/978-3-031-57793-2_24(312-324)Online publication date: 15-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-57793-2_24
Mendes Forte JPassat NKenmochi Y(2024)Building the�Topological Tree of�Shapes from�the�Tree of�ShapesDiscrete Geometry and Mathematical Morphology10.1007/978-3-031-57793-2_21(271-285)Online publication date: 15-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-57793-2_21

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cover image Journal of Mathematical Imaging and Vision

Journal of Mathematical Imaging and Vision Volume 65, Issue 5

Oct 2023

131 pages

ISSN:0924-9907

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© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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Kluwer Academic Publishers

United States

Publication History

Published: 16 August 2023

Accepted: 22 June 2023

Received: 08 January 2023

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Cited By

Perrin RLeborgne APassat NNaegel BWemmert C(2024)Multi-scale Component-Tree: A Hierarchical Representation for�Sparse ObjectsDiscrete Geometry and Mathematical Morphology10.1007/978-3-031-57793-2_24(312-324)Online publication date: 15-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-57793-2_24
Mendes Forte JPassat NKenmochi Y(2024)Building the�Topological Tree of�Shapes from�the�Tree of�ShapesDiscrete Geometry and Mathematical Morphology10.1007/978-3-031-57793-2_21(271-285)Online publication date: 15-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-57793-2_21

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