Computing Homology for Surfaces with Generalized Maps: Application to 3D Images

Damiand, Guillaume; Peltier, Samuel; Fuchs, Laurent

doi:10.1007/11919629_25

Cited by 13 publications

(14 citation statements)

References 9 publications

(14 reference statements)

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“…Another way to improve the computation time is to reduce the input complex without changing its topology by applying iterative simplifications, and by computing the homology when no more simplifications are possible. This reduction approach has been mainly investigated in the context of homology computation from 3D voxel images [27,6,29,31]. Other reduction approaches apply discrete Morse theory [18] to homology computation since in many applied situations one expects the Morse complex built on the original simplicial complex to be much smaller than this latter.…”

Section: Related Workmentioning

confidence: 99%

An iterative algorithm for homology computation on simplicial shapes

Boltcheva¹,

Canino²,

Aceituno³

et al. 2011

Computer-Aided Design

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We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes, We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its sub-components. The proposed algorithm retrieves the complete homological information of an input shape including the Betti numbers, the torsion coefficients and the representative homology generators.To the best of our knowledge, this is the first algorithm based on the constructive Mayer-Vietoris sequence, which relates the homology of a topological space to the homologies of its sub-spaces, i.e. the sub-components of the input shape and their intersections. We demonstrate the validity of our approach through a specific shape decomposition, based only on topological properties, which minimizes the size of the intersections between the sub-components and increases the efficiency of the algorithm.

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Section: Related Workmentioning

confidence: 99%

An iterative algorithm for homology computation on simplicial shapes

Boltcheva¹,

Canino²,

Aceituno³

et al. 2011

Computer-Aided Design

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“…However, this method cannot directly provide a set of generators. Based on the previously mentioned work, an algorithm for computing a minimal representation of the boundary of a 3D voxel region, from which homology generators can directly be deduced has been defined in [10].…”

Section: Computing Homology Generators In a Graph Pyramidmentioning

confidence: 99%

“…This work was introduced in [9] and is build by using two operations: contraction and removal. These two operations are used also in [10] to incrementally compute homology groups and their generators of 2D closed surfaces, but a hierarchy is not build. In this paper, the complexity of the method is detailed, which shows the interest of our approach compared to Agoston's classical method.…”

Section: Introductionmentioning

confidence: 99%

Directly computing the generators of image homology using graph pyramids

Peltier

Ion

Kropatsch

et al. 2009

Image and Vision Computing

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a b s t r a c tWe introduce a method for computing homology groups and their generators of a 2D image, using a hierarchical structure, i.e. irregular graph pyramid. Starting from an image, a hierarchy of the image is built by two operations that preserve homology of each region. Instead of computing homology generators in the base where the number of entities (cells) is large, we first reduce the number of cells by a graph pyramid. Then homology generators are computed efficiently on the top level of the pyramid, since the number of cells is small. A top down process is then used to deduce homology generators in any level of the pyramid, including the base level, i.e. the initial image. The produced generators fit on the object boundaries. A unique set of generators called the minimal set, is defined and its computation is discussed. We show that the new method produces valid homology generators and present some experimental results.

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“…The idea is similar to [11,12], but all simplifications computed during the reduction process are kept in using a pyramid (Fig. 2).…”

Section: Computing Homology Generators In a Graph Pyramidmentioning

confidence: 99%

Delineating Homology Generators in Graph Pyramids

Iglesias

Ion²,

Kropatsch³

et al. 2008

Lecture Notes in Computer Science

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Abstract. Computation of homology generators using a graph pyramid can significantly increase performance, compared to the classical methods. First results in 2D exist and show the advantages of the method. Generators are computed on the upper level of a graph pyramid. Toplevel graphs may contain self loops and multiple edges, as a side product of the contraction process. Using straight lines to draw these edges would not show the full information: self loops disappear, parallel edges collapse. This paper presents a novel algorithm for correctly visualizing graph pyramids, including multiple edges and self loops which preserves the geometry and the topology of the original image. New insights about the top-down delineation of homology generators in graph pyramids are given.

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Computing Homology for Surfaces with Generalized Maps: Application to 3D Images

Cited by 13 publications

References 9 publications

An iterative algorithm for homology computation on simplicial shapes

An iterative algorithm for homology computation on simplicial shapes

Directly computing the generators of image homology using graph pyramids

Delineating Homology Generators in Graph Pyramids

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