Building the Topological Tree of Shapes from the Tree of Shapes
Pages 271 - 285
Abstract
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.
References
[1]
Blin N, Carlinet E, Lemaitre F, Lacassagne L, and Géraud T Max-tree computation on GPUs IEEE Trans. Parallel Distrib. Syst. 2022 33 3520-3531
[2]
Breen EJ and Jones R Attribute openings, thinnings, and granulometries Comput. Vis. Image Underst. 1996 64 3 377-389
[3]
Carlinet, E., Crozet, S., Géraud, T.: The tree of shapes turned into a max-tree: a simple and efficient linear algorithm. In: ICIP, pp. 1488–1492 (2018)
[4]
Carlinet E and Géraud T A comparative review of component tree computation algorithms IEEE Trans. Image Process. 2014 23 3885-3895
[5]
Carlinet E and Géraud T MToS: a tree of shapes for multivariate images IEEE Trans. Image Process. 2015 24 5330-5342
[6]
Caselles V, Meinhardt E, and Monasse P Constructing the tree of shapes of an image by fusion of the trees of connected components of upper and lower level sets Positivity 2008 12 55-73
[7]
Crozet, S., Géraud, T.: A first parallel algorithm to compute the morphological tree of shapes of D images. In: ICIP, pp. 2933–2937 (2014)
[8]
Gazagnes S and Wilkinson MHF Distributed connected component filtering and analysis in 2D and 3D tera-scale data sets IEEE Trans. Image Process. 2021 30 3664-3675
[9]
Géraud, T., Carlinet, E., Crozet, S., Najman, L.: A quasi-linear algorithm to compute the tree of shapes of D images. In: Hendriks, C.L.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 98–110. Springer, Berlin (2013).
[10]
Götz M, Cavallaro G, Géraud T, Book M, and Riedel M Parallel computation of component trees on distributed memory machines IEEE Trans. Parallel Distrib. Syst. 2018 29 2582-2598
[11]
Kurtz C, Naegel B, and Passat N Connected filtering based on multivalued component trees IEEE Trans. Image Process. 2014 23 5152-5164
[12]
Monasse P Kerautret B, Colom M, Lopresti D, Monasse P, and Talbot H A root-to-leaf algorithm computing the tree of shapes of an image Reproducible Research in Pattern Recognition 2019 Cham Springer 43-54
[13]
Monasse P and Guichard F Scale-space from a level lines tree J. Vis. Commun. Image Represent. 2000 11 2 224-236
[14]
Ngoc MOV, Boutry N, Fabrizio J, and Géraud T A minimum barrier distance for multivariate images with applications Comput. Vis. Image Underst. 2020 197–198
[15]
Passat, N., Kenmochi, Y.: A Topological Tree of Shapes. In: Baudrier, É., Naegel, B., Krähenbühl, A., Tajine, M. (eds.) DGMM 2022. LNCS, vol. 13493, pp. 221–235. Springer, Cham (2022).
[16]
Passat N, Mendes Forte J, and Kenmochi Y Morphological hierarchies: a unifying framework with new trees J. Math. Imaging Vis. 2023 65 5 718-753
[17]
Perret B, Chierchia G, Cousty J, Ferzoli Guimarães SJ, Kenmochi Y, and Najman L Higra: hierarchical graph analysis SoftwareX 2019 10
[18]
Perret B and Cousty J Baudrier É, Naegel B, Krähenbühl A, and Tajine M Component tree loss function: definition and optimization Discrete Geometry and Mathematical Morphology 2022 Cham Springer 248-260
[19]
Ronse C A topological characterization of thinning Theor. Comput. Sci. 1986 43 31-41
[20]
Rosenfeld A Adjacency in digital pictures Inf. Control 1974 26 24-33
[21]
Salembier P, Oliveras A, and Garrido L Anti-extensive connected operators for image and sequence processing IEEE Trans. Image Process. 1998 7 555-570
[22]
Salembier P and Serra J Flat zones filtering, connected operators, and filters by reconstruction IEEE Trans. Image Process. 1995 4 1153-1160
[23]
Song Y and Zhang A Braquelaire A, Lachaud J-O, and Vialard A Monotonic tree Discrete Geometry for Computer Imagery 2002 Heidelberg Springer 114-123
[24]
Tao R and Qiao J Fast component tree computation for images of limited levels IEEE Trans. Pattern Anal. Mach. Intell. 2023 45 3 3059-3071
Recommendations
Tropical Geometric Variation of Tree Shapes
AbstractWe study the behavior of phylogenetic tree shapes in the tropical geometric interpretation of tree space. Tree shapes are formally referred to as tree topologies; a tree topology can also be thought of as a tree combinatorial type, which is given ...
A Topological Tree of Shapes
Discrete Geometry and Mathematical MorphologyAbstractIn this article, we enrich the framework of morphological hierarchies with new acyclic graphs and trees. These structures lie at the convergence of hierarchical models and topological descriptors. We define them in the context of digital grey-...
Approximating the selected-internal Steiner tree
In this paper, we consider a variant of the well-known Steiner tree problem. Given a complete graph G=(V,E) with a cost function c:E->R^+ and two subsets R and R^' satisfying R^'@?R@?V, a selected-internal Steiner tree is a Steiner tree which contains (...
Comments
Information & Contributors
Information
Published In
Apr 2024
461 pages
ISBN:978-3-031-57792-5
DOI:10.1007/978-3-031-57793-2
- Editors:
- Sara Brunetti,
- Andrea Frosini,
- Simone Rinaldi
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
Publisher
Springer-Verlag
Berlin, Heidelberg
Publication History
Published: 15 May 2024
Qualifiers
- Article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 16 Jan 2025