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Title: Measuring the Error in Approximating the Sub-Level Set Topology of Sampled Scalar Data

Abstract

This paper studies here the influence of the definition of neighborhoods and methods used for creating point connectivity on topological analysis of scalar functions. It is assumed that a scalar function is known only at a finite set of points with associated function values. In order to utilize topological approaches to analyze the scalar-valued point set, it is necessary to choose point neighborhoods and, usually, point connectivity to meaningfully determine critical-point behavior for the point set. Two distances are used to measure the difference in topology when different point neighborhoods and means to define connectivity are used: (i) the bottleneck distance for persistence diagrams and (ii) the distance between merge trees. Usually, these distances define how different scalar functions are with respect to their topology. These measures, when properly adapted to point sets coupled with a definition of neighborhood and connectivity, make it possible to understand how topological characteristics depend on connectivity. Noise is another aspect considered. Five types of neighborhoods and connectivity are discussed: (i) the Delaunay triangulation; (ii) the relative neighborhood graph; (iii) the Gabriel graph; (iv) the k-nearest-neighbor (KNN) neighborhood; and (v) the Vietoris–Rips complex. It is discussed in detail how topological characterizations depend on the chosenmore » connectivity.« less

Authors:
ORCiD logo [1];  [2];  [3]; ORCiD logo [3];  [4]
  1. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); National Lab. Astana (Kazakhstan)
  2. Univ. of California, Los Angeles, CA (United States). Dept. of Mathematics; Kazakh-British Technical Univ., Almaty (Kazakhstan)
  3. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
  4. Univ. of California, Davis, CA (United States). Dept. of Computer Science
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); Ministry of Education and Science of the Republic of Kazakhstan
OSTI Identifier:
1505518
Grant/Contract Number:  
AC02-05CH11231; 0115PK03029
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
International Journal of Computational Geometry & Applications
Additional Journal Information:
Journal Volume: 28; Journal Issue: 01; Journal ID: ISSN 0218-1959
Publisher:
World Scientific
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; sub-level set topology; error quantification; topological structures

Citation Formats

Beketayev, Kenes, Yeliussizov, Damir, Morozov, Dmitriy, Weber, Gunther H., and Hamann, Bernd. Measuring the Error in Approximating the Sub-Level Set Topology of Sampled Scalar Data. United States: N. p., 2018. Web. doi:10.1142/S0218195918500036.
Beketayev, Kenes, Yeliussizov, Damir, Morozov, Dmitriy, Weber, Gunther H., & Hamann, Bernd. Measuring the Error in Approximating the Sub-Level Set Topology of Sampled Scalar Data. United States. https://doi.org/10.1142/S0218195918500036
Beketayev, Kenes, Yeliussizov, Damir, Morozov, Dmitriy, Weber, Gunther H., and Hamann, Bernd. 2018. "Measuring the Error in Approximating the Sub-Level Set Topology of Sampled Scalar Data". United States. https://doi.org/10.1142/S0218195918500036. https://www.osti.gov/servlets/purl/1505518.
@article{osti_1505518,
title = {Measuring the Error in Approximating the Sub-Level Set Topology of Sampled Scalar Data},
author = {Beketayev, Kenes and Yeliussizov, Damir and Morozov, Dmitriy and Weber, Gunther H. and Hamann, Bernd},
abstractNote = {This paper studies here the influence of the definition of neighborhoods and methods used for creating point connectivity on topological analysis of scalar functions. It is assumed that a scalar function is known only at a finite set of points with associated function values. In order to utilize topological approaches to analyze the scalar-valued point set, it is necessary to choose point neighborhoods and, usually, point connectivity to meaningfully determine critical-point behavior for the point set. Two distances are used to measure the difference in topology when different point neighborhoods and means to define connectivity are used: (i) the bottleneck distance for persistence diagrams and (ii) the distance between merge trees. Usually, these distances define how different scalar functions are with respect to their topology. These measures, when properly adapted to point sets coupled with a definition of neighborhood and connectivity, make it possible to understand how topological characteristics depend on connectivity. Noise is another aspect considered. Five types of neighborhoods and connectivity are discussed: (i) the Delaunay triangulation; (ii) the relative neighborhood graph; (iii) the Gabriel graph; (iv) the k-nearest-neighbor (KNN) neighborhood; and (v) the Vietoris–Rips complex. It is discussed in detail how topological characterizations depend on the chosen connectivity.},
doi = {10.1142/S0218195918500036},
url = {https://www.osti.gov/biblio/1505518}, journal = {International Journal of Computational Geometry & Applications},
issn = {0218-1959},
number = 01,
volume = 28,
place = {United States},
year = {Thu Mar 29 00:00:00 EDT 2018},
month = {Thu Mar 29 00:00:00 EDT 2018}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Figures / Tables:

Fig. 1 Fig. 1: (a) Rotation of the methyl group of the molecule of dimer of formic and acetic acid produces the potential energy function, i.e., the real function. (b) Although the first approximated function deviates from the real function, it still preserves the correct number of minima — three — thusmore » bearing no error on the count of minima. However, the second approximated function contains only one minimum, leading to an error of 2/3.« less

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