Multivariate Topology Simplification (1509.04465v1)

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.

• The paper presents a robust method for simplifying multivariate topologies by lip-pruning components of the Reeb Space.
• It develops a Jacobi Structure and Reeb Skeleton by projecting the Jacobi Set, effectively decomposing complex multivariate data.
• The algorithm demonstrates linear performance and enhanced noise reduction in real-world datasets, including nuclear scission data.

An Analytical Overview of "Multivariate Topology Simplification"

The paper "Multivariate Topology Simplification" presents a comprehensive framework for the simplification of multivariate topological structures through the development of a robust simplification method based on "lip"-pruning from the Reeb Space. This work addresses the complexities inherent in dealing with multi-field data, which require advancements in both mathematical formulations and computational processing. It introduces several novel concepts, including the Jacobi Structure, Reeb Skeleton, and range measures, which are essential for simplifying and visualizing multivariate topologies.

Key Contributions and Methodology

The authors propose a methodology that involves the projection of the Jacobi Set of multivariate data into the Reeb Space, resulting in a Jacobi Structure that differentiates the Reeb Space into simple components. They establish that the dual graph of these components forms a Reeb Skeleton, which exhibits properties akin to the scalar contour tree and the Reeb Graph, particularly in topologically simple domains.

The central contributions of the paper are summarized as follows:

1. Theoretical Foundation: The paper provides a mathematical basis for the simplification of multivariate structures by clarifying the relationships between the Reeb Space of a multivariate map, the Jacobi Set, and fiber topology. The introduction of the Jacobi Structure in the Reeb Space enables the decomposition of the space into regular and singular components, facilitating further reduction to a Reeb Skeleton.
2. Algorithmic Approach: The authors develop an algorithm to extract the Jacobi Structure from the Joint Contour Net (JCN), an approximation of the Reeb Space, and subsequently compute the Reeb Skeleton. This approach allows for the efficient simplification of a Reeb Space by computing the range and geometric measures using the JCN.
3. Simplification Strategy: The simplification of the Reeb Skeleton is driven by the removal of lip-like components, analogous to leaf-pruning in contour trees. This process is guided by newly introduced measures, including the range measure, which provides a scaling-invariant total ordering of the features.

Numerical Results and Implications

The paper presents a series of demonstrations and applications, including real-world datasets, which exhibit the practical utility of the proposed simplification method. For instance, the simplification of nuclear scission data notably highlights the method's capability to detect primary topological features while mitigating noise. The implementation of these algorithms shows promising performance, with linear complexity related to the number of regular components in the Reeb Space.

Future Directions

The paper lays a foundation for further research in multivariate topological simplification. Although the method demonstrates significant capability, challenges remain, such as handling false lips and discontinuities in the Jacobi Structure due to degeneracies or quantization levels. Future work could explore higher-order structures in the Reeb Skeleton and alternative methods for Reeb Space computation. Additionally, the application range of these techniques might be expanded to include more complex and diverse datasets across various scientific domains.

In conclusion, this paper contributes a robust theoretical and computational framework for the simplification of complex multivariate topologies, enhancing both the analysis and visualization capabilities for multi-field datasets.