Results 1  10
of
13
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. ..."
Abstract

Cited by 23 (8 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 10 (8 self)
 Add to MetaCart
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.
Efficient Computation of 3D MorseSmale Complexes and Persistent Homology using Discrete Morse Theory
"... We propose an efficient algorithm that computes the MorseSmale complex for 3D grayscale images. This complex allows for a efficient computation of persistent homology since it is, in general, much smaller than the input data but still contains all necessary information. Our method improves a rece ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
We propose an efficient algorithm that computes the MorseSmale complex for 3D grayscale images. This complex allows for a efficient computation of persistent homology since it is, in general, much smaller than the input data but still contains all necessary information. Our method improves a recently proposed algorithm to extract the MorseSmale complex in terms of memory consumption and running time. It also allows for a parallel computation of the complex. The computational complexity of the MorseSmale complex extraction solely depends on the topological complexity of the input data. The persistence is then computed using the MorseSmale complex by applying an existing algorithm with a good practical running time. We demonstrate that our method allows for the computation of persistent homology for large data on commodity hardware.
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. ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
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.
Coreduction Homology Algorithm for Regular CWComplexes
, 2010
"... Département de mathématiques ..."
(Show Context)
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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.
networks by homological
"... Noname manuscript No. (will be inserted by the editor) Distributed computation of coverage in sensor ..."
Abstract
 Add to MetaCart
Noname manuscript No. (will be inserted by the editor) Distributed computation of coverage in sensor
MemoryEfficient Computation of Persistent Homology for 3D Images using Discrete Morse Theory
"... Abstract—We propose a memoryefficient method that computes persistent homology for 3D grayscale images. The basic idea is to compute the persistence of the induced MorseSmale complex. Since in practice this complex is much smaller than the input data, significantly less memory is required for the ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract—We propose a memoryefficient method that computes persistent homology for 3D grayscale images. The basic idea is to compute the persistence of the induced MorseSmale complex. Since in practice this complex is much smaller than the input data, significantly less memory is required for the subsequent computations. We propose a novel algorithm that efficiently extracts the MorseSmale complex based on algorithms from discrete Morse theory. The proposed algorithm is thereby optimal with a computational complexity of O(n 2). The persistence is then computed using the MorseSmale complex by applying an existing algorithm with a good practical running time. We demonstrate that our method allows for the computation of persistent homology for large data on commodity hardware. Keywordspersistent homology, MorseSmale complex, discrete Morse theory, large data I.
Z2HOMOLOGY OF WEAK (n − 2)FACELESS nPSEUDOMANIFOLDS MAY BE COMPUTED IN O(n) TIME
"... Abstract. We consider the class of weak (n−2)faceless npseudomanifolds with bounded boundaries and coboundaries. We show that in this class the Betti numbers with Z2 coefficients may be computed in time O(n) and the Z2 homology generators in time O(nm) where n denotes the cardinality of the npseu ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We consider the class of weak (n−2)faceless npseudomanifolds with bounded boundaries and coboundaries. We show that in this class the Betti numbers with Z2 coefficients may be computed in time O(n) and the Z2 homology generators in time O(nm) where n denotes the cardinality of the npseudomanifold on input and m is the number of homology generators. 1.