Multivariate Topology Simplification

TL;DR

This paper introduces a robust multivariate topology simplification method using lip-pruning from the Reeb Space, leveraging mathematical properties of Jacobi Sets and Reeb Skeletons for analyzing complex multi-field data.

Contribution

It presents a novel multivariate topology simplification approach based on Jacobi Structures and Reeb Skeletons, with computational techniques in the Joint Contour Net.

Findings

Jacobi Structure separates Reeb Space into simple components.

Reeb Skeleton has properties similar to scalar contour trees.

Method enables visualization of complex multivariate data.

Abstract

Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For 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.