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

## Abstract

This paper studies 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.

- Authors:

- Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA; National Laboratory Astana, 53 Kabanbay Batyr Ave, Astana, 010000, Kazakhstan
- Department of Mathematics, University of California, 405 Hilgard Ave, Los Angeles, CA 90095, USA; Kazakh-British Technical University, 59 Tole Bi St, Almaty, 050000, Kazakhstan
- Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
- Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA; Department of Computer Science, University of California, 1 Shields Ave, Davis, California 95616, USA
- Department of Computer Science, University of California, 1 Shields Ave, Davis, California 95616, USA

- 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) (SC-21)

- OSTI Identifier:
- 1505518

- Grant/Contract Number:
- AC02-05CH11231

- 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

- Country of Publication:
- United States

- Language:
- English

### 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. doi:10.1142/S0218195918500036.

```
Beketayev, Kenes, Yeliussizov, Damir, Morozov, Dmitriy, Weber, Gunther H., and Hamann, Bernd. Thu .
"Measuring the Error in Approximating the Sub-Level Set Topology of Sampled Scalar Data". United States. doi: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 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},

journal = {International Journal of Computational Geometry & Applications},

issn = {0218-1959},

number = 01,

volume = 28,

place = {United States},

year = {2018},

month = {3}

}