Topological features in 2D symmetric higher-order tensor fields

Authors:
T. Schultz
Affiliations:
Computation Institute, University of Chicago
Venue:
EuroVis'11 Proceedings of the 13th Eurographics / IEEE - VGTC conference on Visualization
Year:
2011

Citing 21
Cited 0

A note on the gradient of a multi-image

Computer Vision, Graphics, and Image Processing - Lectures notes in computer science, Vol. 201 (G. Goos and J. Hartmanis, Eds.)
Representation and Display of Vector Field Topology in Fluid Flow Data Sets

Computer
Decomposition of quantics in sums of powers of linear forms

Signal Processing - Special issue on higher order statistics
Visualization of scalar topology for structural enhancement

Proceedings of the conference on Visualization '98
Edge Detection and Ridge Detection with Automatic Scale Selection

International Journal of Computer Vision
Generic structure of two-dimensional dimages under Gaussian blurring

SIAM Journal on Applied Mathematics
Visualizing Second-Order Tensor Fields with Hyperstreamlines

IEEE Computer Graphics and Applications
The topology of symmetric, second-order tensor fields

VIS '94 Proceedings of the conference on Visualization '94
Topological Lines in 3D Tensor Fields and Discriminant Hessian Factorization

IEEE Transactions on Visualization and Computer Graphics
Rotational symmetry field design on surfaces

ACM SIGGRAPH 2007 papers
Topological Visualization of Brain Diffusion MRI Data

IEEE Transactions on Visualization and Computer Graphics
Local invariant feature detectors: a survey

Foundations and Trends® in Computer Graphics and Vision
Symmetric Tensors and Symmetric Tensor Rank

SIAM Journal on Matrix Analysis and Applications
Asymmetric Tensor Analysis for Flow Visualization

IEEE Transactions on Visualization and Computer Graphics
Estimating Crossing Fibers: A Tensor Decomposition Approach

IEEE Transactions on Visualization and Computer Graphics
Invariant Crease Lines for Topological and Structural Analysis of Tensor Fields

IEEE Transactions on Visualization and Computer Graphics
Tensor Decompositions and Applications

SIAM Review
Nonlinear structure tensors

Image and Vision Computing
Letters: Numerical optimization of a sum-of-rank-1 decomposition for n-dimensional order-p symmetric tensors

Neurocomputing
Decomposition and visualization of fourth-order elastic-plastic tensors

SPBG'08 Proceedings of the Fifth Eurographics / IEEE VGTC conference on Point-Based Graphics
A maximum enhancing higher-order tensor glyph

EuroVis'10 Proceedings of the 12th Eurographics / IEEE - VGTC conference on Visualization

Quantified Score

Hi-index	0.00

Visualization

Abstract

The topological structure of scalar, vector, and second-order tensor fields provides an important mathematical basis for data analysis and visualization. In this paper, we extend this framework towards higher-order tensors. First, we establish formal uniqueness properties for a geometrically constrained tensor decomposition. This allows us to define and visualize topological structures in symmetric tensor fields of orders three and four. We clarify that in 2D, degeneracies occur at isolated points, regardless of tensor order. However, for orders higher than two, they are no longer equivalent to isotropic tensors, and their fractional Poincaré index prevents us from deriving continuous vector fields from the tensor decomposition. Instead, sorting the terms by magnitude leads to a new type of feature, lines along which the resulting vector fields are discontinuous. We propose algorithms to extract these features and present results on higher-order derivatives and higher-order structure tensors.