Topological analysis of shapes using Morse theory

Állili, Madjid; Corriveau, David

doi:10.1016/j.cviu.2006.10.004

Cited by 15 publications

(18 citation statements)

References 8 publications

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“…Then, knowing that F F D ′ ∂ = ∂ + ∂ , we add the projected 1-cells in the boundary of D to the boundary of F′ using the correct coefficients. For example, if we suppose that 1 d is a projected cell of dimension one that was in the boundary of D in the original complex. From the structures, we know that D appears in the projection of F with an incidence coefficient of 1 ( F F D ′ = + ).…”

Section: Building the Simplified Complexmentioning

confidence: 99%

“…In digital image analysis, topological invariants are useful in shape description, indexation, and classification. Among shape descriptors based on homology theory, there are the Morse shape descriptor [1] [2], the Morse Connection Graph [3], and the persistence barcodes for shape [4]. The necessity of improved algorithms appears evident as new applications of the homology computation arise in research for very large data sets.…”

Section: Introductionmentioning

confidence: 99%

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A Global Reduction Based Algorithm for Computing Homology of Chain Complexes

Állili¹,

Corriveau²

2016

APM

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In this paper, we propose a new algorithm to compute the homology of a finitely generated chain complex. Our method is based on grouping several reductions into structures that can be encoded as directed acyclic graphs. The organized reduction pairs lead to sequences of projection maps that reduce the number of generators while preserving the homology groups of the original chain complex. This sequencing of reduction pairs allows updating the boundary information in a single step for a whole set of reductions, which shows impressive gains in computational performance compared to existing methods. In addition, our method gives the homology generators for a small additional cost.

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Section: Building the Simplified Complexmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Global Reduction Based Algorithm for Computing Homology of Chain Complexes

Állili¹,

Corriveau²

2016

APM

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“…Consider the cocycle x 1 generated by the sum of four solid line vertical edges with [2,3] at the second coordinate, and y 1 by the sum of solid line horizontal edges with [1,2] at the first coordinate. Only the edges of the parametric square [1, 2] × [2, 3] may contribute to non-zero terms of x 1 y 1 .…”

Section: Theorem 224 Let D = Emb(x) > 1 With the Above Notation Formentioning

confidence: 99%

“…The development of a computational approach to these theories is motivated, among others, by problems in dynamical systems [18], material science [5,8], electromagnetism [9,14], geometric modeling [10], image understanding and digital image processing [1,3,12,16,22]. Conversely, that development is enabled by progress in computer science.…”

Section: Introductionmentioning

confidence: 99%

The Cubical Cohomology Ring: An Algorithmic Approach

Kaczyński

Mrożek

2012

Found Comput Math

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A cohomology ring algorithm in a dimension-independent framework of combinatorial cubical complexes is developed with the aim of applying it to the topological analysis of high-dimensional data. This approach is convenient in the cupproduct computation and motivated, among others, by interpreting pixels or voxels in digital images as cubes. The S-complex theory and so called co-reductions are adopted to build a cohomology ring algorithm speeding up the algebraic computations.

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“…It has been applied to dynamical systems [13,15], material science [4,18], electromagnetism [8,7], image understanding [1,14] and sensor networks [6].…”

Section: Introductionmentioning

confidence: 99%

Fast, Simple and Separable Computation of Betti Numbers on Three-Dimensional Cubical Complexes

Gonzalez-Lorenzo

Juda

Bac

et al. 2016

Computational Topology in Image Context

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Abstract. Betti numbers are topological invariants that count the number of holes of each dimension in a space. Cubical complexes are a class of CW complex whose cells are cubes of different dimensions such as points, segments, squares, cubes, etc. They are particularly useful for modeling structured data such as binary volumes. We introduce a fast and simple method for computing the Betti numbers of a three-dimensional cubical complex that takes advantage on its regular structure, which is not possible with other types of CW complexes such as simplicial or polyhedral complexes. This algorithm is also restricted to three-dimensional spaces since it exploits the Euler-Poincaré formula and the Alexander duality in order to avoid any matrix manipulation. The method runs in linear time on a single core CPU. Moreover, the regular cubical structure allows us to obtain an efficient implementation for a multi-core architecture.

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Topological analysis of shapes using Morse theory

Cited by 15 publications

References 8 publications

A Global Reduction Based Algorithm for Computing Homology of Chain Complexes

A Global Reduction Based Algorithm for Computing Homology of Chain Complexes

The Cubical Cohomology Ring: An Algorithmic Approach

Fast, Simple and Separable Computation of Betti Numbers on Three-Dimensional Cubical Complexes

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