2D Asymmetric Tensor Field Topology

Abstract

In this chapter we define the topology of 2D asymmetric tensor fields in terms of two graphs corresponding to the eigenvalue and eigenvector analysis for the tensor fields, respectively. Asymmetric tensor field topology can not only yield a concise representation of the field, but also provide a framework for spatial-temporal tracking of field features. Furthermore, inherent topological constraints in asymmetric tensor fields can be identified unambiguously through these graphs. We also describe efficient algorithms to compute the topology of a given 2D asymmetric tensor field. We demonstrate the utility of our graph representations for asymmetric tensor field topology with fluid simulation data sets.

Multi-Scale Topological Analysis of Asymmetric Tensor Fields on Surfaces

“…One of the exceptions is the work by Lin et al [21], who propose to use two topological graphs to represent the structure in an asymmetric tensor field. Their research is exploratory in nature and is limited to planar data at a single scale without clear physical interpretation.…”

“…In this section, we review the two graphs defined by Lin et al [21] to describe the topology of 2D asymmetric tensor fields.…”

“…The inverse of this map leads to a classification of the tensor field into the aforementioned four types of regions. Lin et al [21] define the eigenvector graph as follows. A node in the graph is a maximal, connected region in the domain inside which the points have the same tensor behavior, i.e.…”

“…Lin et al [21] introduce two graph-based representations for planar asymmetric tensor fields and apply them to slices of a diesel engine simulation [19]. However, their work is limited to planar datasets, at a single scale, and without physical interpretation.…”

A Survey of Topology‐based Methods in Visualization

“…Zheng and Pang [ZP04] describe the topology of asymmetric tensor fields by introducing the concept of the dual eigenvector to define lines in areas, where the tensors are not symmetric. Lin et al [LYL*12] derive two graph‐based representations of tensor fields called eigenvector and eigenvalue graph. They use the concepts introduce by Zhen and Pang to define the eigenvector graph of a tensor field as a graph with nodes for each area on the eigenvector manifold, which segments the field, and a node for each degenerate point.…”