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Morse Theory for Filtrations and Efficient Computation of Persistent Homology
"... We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations. ..."
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Cited by 23 (8 self)
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We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations.
Homology algorithm based on acyclic subspace
"... We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace ma ..."
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Cited by 22 (8 self)
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We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be processed in the algebraic way, and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms.
COREDUCTION HOMOLOGY ALGORITHM FOR INCLUSIONS AND PERSISTENT HOMOLOGY
"... Abstract. We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the p ..."
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Abstract. We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the persistence concept to twosided filtrations. In addition to describing the theoretical background, we present results of numerical experiments, as well as several applications to concrete problems in materials science. 1.
Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps
"... We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homology computation for a very general class of complexes. A setvalued map of topdimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) c ..."
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Cited by 10 (8 self)
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We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homology computation for a very general class of complexes. A setvalued map of topdimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) continuous function, especially when the evaluation of that function is subject to measurement errors. We introduce a new Morse theoretic preprocessing framework for deriving chain maps from such setvalued maps, and hence provide an effective scheme for computing the morphism induced on homology by the approximated continuous function.
Cech Type Approach to Computing Homology of Maps, in preparation
"... Abstract. A new approach to algorithmic computation of the homology of spaces and maps is presented. The key point of the approach is a change in the representation of sets. The proposed representation is based on a combinatorial variant of the Čech homology and the Nerve Theorem. In many situations ..."
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Cited by 8 (3 self)
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Abstract. A new approach to algorithmic computation of the homology of spaces and maps is presented. The key point of the approach is a change in the representation of sets. The proposed representation is based on a combinatorial variant of the Čech homology and the Nerve Theorem. In many situations this change of the representation of the input may help in bypassing the problems with the complexity of the standard homology algorithms by reducing the size of nedcessary input. We show that the approach is particularly advantegous in the case of homology map algorithms. 1. Introduction. Effective algorithms for computing homology of spaces and maps are needed in computer assisted proofs in dynamics based on topological tools (see [4, 17, 22, 23] and references therein). Recently, homology algorithms have also been used in robotics [30], material structure analysis [10, 11] and image recognition [3, 36],
The Efficiency of a Homology Algorithm based on Discrete Morse Theory and Coreductions
"... Abstract Two implementations of a homology algorithm based on the Forman’s discrete Morse theory combined with the coreduction method are presented. Their efficiency is compared with other implementations of homology algorithms. ..."
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Abstract Two implementations of a homology algorithm based on the Forman’s discrete Morse theory combined with the coreduction method are presented. Their efficiency is compared with other implementations of homology algorithms.
Structure of the afferent terminals in terminal ganglion of a cricket and persistent homology
 PLoS ONE
"... We use topological data analysis to investigate the three dimensional spatial structure of the locus of afferent neuron terminals in crickets Acheta domesticus. Each afferent neuron innervates a filiform hair positioned on a cercus: a protruding appendage at the rear of the animal. The hairs transdu ..."
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We use topological data analysis to investigate the three dimensional spatial structure of the locus of afferent neuron terminals in crickets Acheta domesticus. Each afferent neuron innervates a filiform hair positioned on a cercus: a protruding appendage at the rear of the animal. The hairs transduce air motion to the neuron signal that is used by a cricket to respond to the environment. We stratify the hairs (and the corresponding afferent terminals) into classes depending on hair length, along with position. Our analysis uncovers significant structure in the relative position of these terminal classes and suggests the functional relevance of this structure. Our method is very robust to the presence of significant experimental and developmental noise. It can be used to analyze a wide range of other point cloud data sets.
Distributed computation of coverage in sensor networks
"... by homological methods ..."
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Computability of Homology for Compact Absolute Neighbourhood Retracts
"... Abstract. In this note we discuss the information needed to compute the homology groups of a topological space. We argue that the natural class of spaces to consider are the compact absolute neighbourhood retracts, since for these spaces the homology groups are finite. We show that we need to specif ..."
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Abstract. In this note we discuss the information needed to compute the homology groups of a topological space. We argue that the natural class of spaces to consider are the compact absolute neighbourhood retracts, since for these spaces the homology groups are finite. We show that we need to specify both a function which defines a retraction from a neighbourhood of the space in the Hilbert cube to the space itself, and a sufficiently fine overapproximation of the set. However, neither the retraction itself, nor a description of an approximation of the set in the Hausdorff metric, is sufficient to compute the homology groups. We express the conditions in the language of computable analysis, which is a powerful framework for studying computability in topology and geometry, and use cubical homology to perform the computations.
G C I
, 2010
"... Sandia is a multiprogram laboratory operated by Sandia Corporation, ..."
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