### Abstract

Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For multivariate (alternatively, multi-field) data, topological analysis requires simultaneous advances both mathematically and computationally. We propose a robust multivariate topology simplification method based on "lip"-pruning from the Reeb space. Mathematically, we show that the projection of the Jacobi set of multivariate data into the Reeb space produces a Jacobi structure that separates the Reeb space into simple components. We also show that the dual graph of these components gives rise to a Reeb skeleton that has properties similar to the scalar contour tree and Reeb graph, for topologically simple domains. We then introduce a range measure to give a scaling-invariant total ordering of the components or features that can be used for simplification. Computationally, we show how to compute Jacobi structure, Reeb skeleton, range and geometric measures in the Joint Contour Net (an approximation of the Reeb space) and that these can be used for visualisation similar to the contour tree or Reeb graph.

Original language | English |
---|---|

Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Computational Geometry: Theory and Applications |

Volume | 58 |

DOIs | |

Publication status | Published - Oct 1 2016 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Computational Geometry: Theory and Applications*,

*58*, 1-24. https://doi.org/10.1016/j.comgeo.2016.05.006

**Multivariate topology simplification.** / Chattopadhyay, Amit; Carr, Hamish; Duke, David; Geng, Zhao; Saeki, Osamu.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 58, pp. 1-24. https://doi.org/10.1016/j.comgeo.2016.05.006

}

TY - JOUR

T1 - Multivariate topology simplification

AU - Chattopadhyay, Amit

AU - Carr, Hamish

AU - Duke, David

AU - Geng, Zhao

AU - Saeki, Osamu

PY - 2016/10/1

Y1 - 2016/10/1

N2 - Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For multivariate (alternatively, multi-field) data, topological analysis requires simultaneous advances both mathematically and computationally. We propose a robust multivariate topology simplification method based on "lip"-pruning from the Reeb space. Mathematically, we show that the projection of the Jacobi set of multivariate data into the Reeb space produces a Jacobi structure that separates the Reeb space into simple components. We also show that the dual graph of these components gives rise to a Reeb skeleton that has properties similar to the scalar contour tree and Reeb graph, for topologically simple domains. We then introduce a range measure to give a scaling-invariant total ordering of the components or features that can be used for simplification. Computationally, we show how to compute Jacobi structure, Reeb skeleton, range and geometric measures in the Joint Contour Net (an approximation of the Reeb space) and that these can be used for visualisation similar to the contour tree or Reeb graph.

AB - Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For multivariate (alternatively, multi-field) data, topological analysis requires simultaneous advances both mathematically and computationally. We propose a robust multivariate topology simplification method based on "lip"-pruning from the Reeb space. Mathematically, we show that the projection of the Jacobi set of multivariate data into the Reeb space produces a Jacobi structure that separates the Reeb space into simple components. We also show that the dual graph of these components gives rise to a Reeb skeleton that has properties similar to the scalar contour tree and Reeb graph, for topologically simple domains. We then introduce a range measure to give a scaling-invariant total ordering of the components or features that can be used for simplification. Computationally, we show how to compute Jacobi structure, Reeb skeleton, range and geometric measures in the Joint Contour Net (an approximation of the Reeb space) and that these can be used for visualisation similar to the contour tree or Reeb graph.

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UR - http://www.scopus.com/inward/citedby.url?scp=84975317615&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2016.05.006

DO - 10.1016/j.comgeo.2016.05.006

M3 - Article

AN - SCOPUS:84975317615

VL - 58

SP - 1

EP - 24

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

ER -