Abstract

Abstract3D symmetric tensor fields have a wide range of applications in medicine, science, and engineering. The topology of tensor fields can provide key insight into their structures. In this paper we study the number of possible topological bifurcations in 3D linear tensor fields. Using the linearity/planarity classification and wedge/trisector classification, we explore four types of bifurcations that can change the number and connectivity in the degenerate curves as well as the number and location of transition points on these degenerate curves. This leads to four types of bifurcations among nine scenarios of 3D linear tensor fields.

Highlights

  • Tensor field visualization is an important topic in visualization, with many applications in medical imaging, solid and fluid mechanics, material science, earthquake engineering, and computer graphics.Recent advances on tensor field visualization focus in topology-driven analysis and visualization of 3D symmetric tensor fields

  • Degenerate curves are one of the most fundamental topological features in a tensor field, and much research has focused on the understanding and efficient extraction of degenerate curves from piecewise linear tensor fields defined on a tetrahedral mesh [6–8]

  • Degenerate curves and transition points play an important role in describing the behavior of a 3D tensor field

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Summary

Introduction

Tensor field visualization is an important topic in visualization, with many applications in medical imaging, solid and fluid mechanics, material science, earthquake engineering, and computer graphics. Recent advances on tensor field visualization focus in topology-driven analysis and visualization of 3D symmetric tensor fields. Degenerate curves are one of the most fundamental topological features in a tensor field, and much research has focused on the understanding and efficient extraction of degenerate curves from piecewise linear tensor fields defined on a tetrahedral mesh [6–8]. We focus on a problem that has received relatively little attention: bifurcations in tensor field topology. To make our investigation effective with potential application to real datasets, we focus on 3D linear tensor fields. We explore all the theoretically possible bifurcations. we review relevant mathematical background and results on tensor fields. we report the findings of our exploration before concluding in Sect.

Previous Work
Tensors
Tensor Field Topology
Bifurcations
Degenerate Curve Removal and Generation
Degenerate Curve Reconnection
Transition Point Pair Cancellation and Generation
Transition Point Relocation
Conclusion
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