Homology Computations via Acyclic Subspace

Brendel, Piotr; Dłotko, Paweł; Mrożek, Marian; Żelazna, Natalia

doi:10.1007/978-3-642-30238-1_13

Cited by 5 publications

(2 citation statements)

References 4 publications

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“…Some of these approaches are based on reductions and coreductions [6]- [8], others simplify the simplicial complex via acyclic subspaces [14], [15]. A similar approach for reducing the size of a complex without affecting its homology is based on the notion of tidy set [16].…”

Section: Related Workmentioning

confidence: 99%

Efficient Computation of Simplicial Homology through Acyclic Matching

Fugacci

Iuricich

Floriani

2014

2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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We consider the problem of efficiently computing homology with Z coefficients as well as homology generators for simplicial complexes of arbitrary dimension. We analyze, compare and discuss the equivalence of different methods based on combining reductions, coreductions and discrete Morse theory. We show that the combination of these methods produces theoretically sound approaches which are mutually equivalent. One of these methods has been implemented for simplicial complexes by using a compact data structure for representing the complex and a compact encoding of the discrete Morse gradient. We present experimental results and discuss further developments.

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Section: Related Workmentioning

confidence: 99%

Efficient Computation of Simplicial Homology through Acyclic Matching

Fugacci

Iuricich

Floriani

2014

2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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“…Historically, the acyclicitiy tables for cubes [71] and simplices [72] were introduced in order to speed up homology computations. In this paper we provide an even stronger result.…”

Section: -And 3-dimensional Cubes Is Represented Inmentioning

confidence: 99%

Topology Preserving Thinning of Cell Complexes

Dłotko

Specogna²

2014

IEEE Trans. on Image Process.

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Abstract. A topology preserving skeleton is a synthetic representation of an object that retains its topology and many of its significant morphological properties. The process of obtaining the skeleton, referred to as skeletonization or thinning, is a very active research area. It plays a central role in reducing the amount of information to be processed during image analysis and visualization, computer-aided diagnosis or by pattern recognition algorithms.This paper introduces a novel topology preserving thinning algorithm which removes simple cells-a generalization of simple points-of a given cell complex. The test for simple cells is based on acyclicity tables automatically produced in advance with homology computations. Using acyclicity tables render the implementation of thinning algorithms straightforward. Moreover, the fact that tables are automatically filled for all possible configurations allows to rigorously prove the generality of the algorithm and to obtain fool-proof implementations. The novel approach enables, for the first time, according to our knowledge, to thin a general unstructured simplicial complex. Acyclicity tables for cubical and simplicial complexes and an open source implementation of the thinning algorithm are provided as additional material to allow their immediate use in the vast number of practical applications arising in medical imaging and beyond.

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