Efficient computation of persistent homology for cubical data

Hubert Wagner; Chao Chen; Erald Vuçini

doi:10.1007/978-3-642-23175-9_7

Efficient computation of persistent homology for cubical data

Hubert Wagner
, Chao Chen
, Erald Vuçini

Biomedical Informatics

Jagiellonian University in Kraków
TU Wien
VRVis Research Center for Virtual Reality and Visualization

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

86 Scopus citations

Abstract

In this paper we present an efficient framework for computation of persistent homology of cubical data in arbitrary dimensions. An existing algorithm using simplicial complexes is adapted to the setting of cubical complexes. The proposed approach enables efficient application of persistent homology in domains where the data is naturally given in a cubical form. By avoiding triangulation of the data, we significantly reduce the size of the complex. We also present a data-structure designed to compactly store and quickly manipulate cubical complexes. By means of numerical experiments, we show high speed and memory efficiency of our approach. We compare our framework to other available implementations, showing its superiority. Finally, we report performance on selected 3D and 4D data-sets.

Original language	English
Title of host publication	Mathematics and Visualization
Editors	Ronald Peikert, Raphael Fuchs, Helwig Hauser, Hamish Carr
Publisher	Springer Heidelberg
Pages	91-106
Number of pages	16
ISBN (Electronic)	9783642231759
ISBN (Print)	9783319912738, 9783540250326, 9783540250760, 9783540332749, 9783540886051, 9783642150135, 9783642216077, 9783642231742, 9783642231742, 9783642273421, 9783642341403, 9783642543005
DOIs	https://doi.org/10.1007/978-3-642-23175-9_7
State	Published - 2012
Event	4th Workshop on Topology Based Methods in Data Analysis and Visualization, TopoInVis 2011 - Zurich, Switzerland Duration: Apr 4 2011 → Apr 6 2011

Publication series

Name	Mathematics and Visualization
Volume	0
ISSN (Print)	1612-3786
ISSN (Electronic)	2197-666X

Conference

Conference	4th Workshop on Topology Based Methods in Data Analysis and Visualization, TopoInVis 2011
Country/Territory	Switzerland
City	Zurich
Period	04/4/11 → 04/6/11

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10.1007/978-3-642-23175-9_7

Cite this

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title = "Efficient computation of persistent homology for cubical data",

abstract = "In this paper we present an efficient framework for computation of persistent homology of cubical data in arbitrary dimensions. An existing algorithm using simplicial complexes is adapted to the setting of cubical complexes. The proposed approach enables efficient application of persistent homology in domains where the data is naturally given in a cubical form. By avoiding triangulation of the data, we significantly reduce the size of the complex. We also present a data-structure designed to compactly store and quickly manipulate cubical complexes. By means of numerical experiments, we show high speed and memory efficiency of our approach. We compare our framework to other available implementations, showing its superiority. Finally, we report performance on selected 3D and 4D data-sets.",

author = "Hubert Wagner and Chao Chen and Erald Vu{\c c}ini",

note = "Publisher Copyright: {\textcopyright} Springer-Verlag Berlin Heidelberg 2012.; 4th Workshop on Topology Based Methods in Data Analysis and Visualization, TopoInVis 2011 ; Conference date: 04-04-2011 Through 06-04-2011",

year = "2012",

doi = "10.1007/978-3-642-23175-9\_7",

language = "English",

isbn = "9783319912738",

series = "Mathematics and Visualization",

publisher = "Springer Heidelberg",

pages = "91--106",

editor = "Ronald Peikert and Raphael Fuchs and Helwig Hauser and Hamish Carr",

booktitle = "Mathematics and Visualization",

}

Wagner, H, Chen, C & Vuçini, E 2012, Efficient computation of persistent homology for cubical data. in R Peikert, R Fuchs, H Hauser & H Carr (eds), Mathematics and Visualization. Mathematics and Visualization, vol. 0, Springer Heidelberg, pp. 91-106, 4th Workshop on Topology Based Methods in Data Analysis and Visualization, TopoInVis 2011, Zurich, Switzerland, 04/4/11. https://doi.org/10.1007/978-3-642-23175-9_7

Efficient computation of persistent homology for cubical data. / Wagner, Hubert; Chen, Chao; Vuçini, Erald.
Mathematics and Visualization. ed. / Ronald Peikert; Raphael Fuchs; Helwig Hauser; Hamish Carr. Springer Heidelberg, 2012. p. 91-106 (Mathematics and Visualization; Vol. 0).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Vuçini, Erald

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N2 - In this paper we present an efficient framework for computation of persistent homology of cubical data in arbitrary dimensions. An existing algorithm using simplicial complexes is adapted to the setting of cubical complexes. The proposed approach enables efficient application of persistent homology in domains where the data is naturally given in a cubical form. By avoiding triangulation of the data, we significantly reduce the size of the complex. We also present a data-structure designed to compactly store and quickly manipulate cubical complexes. By means of numerical experiments, we show high speed and memory efficiency of our approach. We compare our framework to other available implementations, showing its superiority. Finally, we report performance on selected 3D and 4D data-sets.

AB - In this paper we present an efficient framework for computation of persistent homology of cubical data in arbitrary dimensions. An existing algorithm using simplicial complexes is adapted to the setting of cubical complexes. The proposed approach enables efficient application of persistent homology in domains where the data is naturally given in a cubical form. By avoiding triangulation of the data, we significantly reduce the size of the complex. We also present a data-structure designed to compactly store and quickly manipulate cubical complexes. By means of numerical experiments, we show high speed and memory efficiency of our approach. We compare our framework to other available implementations, showing its superiority. Finally, we report performance on selected 3D and 4D data-sets.

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DO - 10.1007/978-3-642-23175-9_7

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SN - 9783540250760

SN - 9783540332749

SN - 9783540886051

SN - 9783642150135

SN - 9783642216077

SN - 9783642231742

SN - 9783642273421

SN - 9783642341403

SN - 9783642543005

T3 - Mathematics and Visualization

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A2 - Peikert, Ronald

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Y2 - 4 April 2011 through 6 April 2011

ER -

Efficient computation of persistent homology for cubical data

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