Scale-Space Approaches to FTLE Ridges

The finite-time Lyapunov Exponent (FTLE) is useful for the visualization of time-dependent velocity fields. The ridges of this derived scalar field have been shown to correspond well to attracting or repelling material structures, so-called Lagrangian coherent structures (LCS). There are two issues involved in the computation of FTLE for this purpose. Firstly, it is often not practically possible to refine the grid for sampling the flow map until convergence of FTLE is reached. Slow conversion is mostly caused by gradient underestimation. Secondly, there is a parameter, the integration time, which has to be chosen sensibly. Both of these problems call for an examination in scale-space. We show that a scale-space approach solves the problem of gradient underestimation. We test optimal-scale ridges for their usefulness with FTLE fields, obtaining a negative result. However, we propose an optimization of the time parameter for a given scale of observation. Finally, an incremental method for computing smoothed flow maps is presented.

[1]  Carl-Fredrik Westin,et al.  Sampling and Visualizing Creases with Scale-Space Particles , 2009, IEEE Transactions on Visualization and Computer Graphics.

[2]  G. Haller.  Finding finite-time invariant manifolds in two-dimensional velocity fields. , 2000, Chaos.

[3]  Hans Hagen,et al.  Efficient Computation and Visualization of Coherent Structures in Fluid Flow Applications , 2007, IEEE Transactions on Visualization and Computer Graphics.

[4]  Kenneth I. Joy,et al.  Lagrangian Visualization of Flow‐Embedded Surface Structures , 2008, Comput. Graph. Forum.

[5]  David H. Eberly,et al.  Ridges in Image and Data Analysis , 1996, Computational Imaging and Vision.

[6]  Hans-Christian Hege,et al.  Localized Finite-time Lyapunov Exponent for Unsteady Flow Analysis , 2009, VMV.

[7]  Filip Sadlo,et al.  Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction , 2007, IEEE Transactions on Visualization and Computer Graphics.

[8]  Steven L Brunton,et al.  Fast computation of finite-time Lyapunov exponent fields for unsteady flows. , 2010, Chaos.

[9]  Ronald Peikert,et al.  Vortex Tracking in Scale-Space , 2002, VisSym.

[10]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory , 1980 .

[11]  Filip Sadlo,et al.  Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection , 2011, Topological Methods in Data Analysis and Visualization.

[12]  G. Haller.  Lagrangian coherent structures from approximate velocity data , 2002 .

[13]  G. Haller.  A variational theory of hyperbolic Lagrangian Coherent Structures , 2010 .

[14]  Tony Lindeberg,et al.  Scale-Space Theory in Computer Vision , 1993, Lecture Notes in Computer Science.

[15]  J. Marsden,et al.  Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows , 2005 .

[16]  G. Haller.  Distinguished material surfaces and coherent structures in three-dimensional fluid flows , 2001 .

[17]  Luc Florack,et al.  The Topological Structure of Scale-Space Images , 2000, Journal of Mathematical Imaging and Vision.

[18]  Holger Theisel,et al.  Ridge Concepts for the Visualization of Lagrangian Coherent Structures , 2012 .

[19]  Gerik Scheuermann,et al.  Topology-based Methods in Visualization , 2007, Topology-based Methods in Visualization.

[20]  Tony Lindeberg,et al.  Edge Detection and Ridge Detection with Automatic Scale Selection , 1996, Proceedings CVPR IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[21]  Thomas Ertl,et al.  Scale-Space Tracking of Critical Points in 3D Vector Fields , 2007, Topology-based Methods in Visualization.

[22]  Francois Lekien,et al.  The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds. , 2010, Chaos.

[23]  Tony Lindeberg Kth.  Scale-space: A framework for handling image structures at multiple scales , 1996 .

[24]  Kamran Mohseni,et al.  A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures. , 2010, Chaos.

[25]  Tony Lindeberg,et al.  Scale-space theory : A framework for handling image structures at multiple scales , 1996 .

[26]  Christer Sjöström,et al.  State-of-the-art report , 1997 .

[27]  F. Sadlo.  Computational visualization of physics and topology in unsteady flow , 2010 .

[28]  A. Spence,et al.  Scale-space ridge detection with GPU acceleration , 2008, 2008 Canadian Conference on Electrical and Computer Engineering.

[29]  Holger Theisel,et al.  On the Way Towards Topology-Based Visualization of Unsteady Flow , 2010, Eurographics.

[30]  G. Haller,et al.  Lagrangian coherent structures and mixing in two-dimensional turbulence , 2000 .

[31]  Robert S. Laramee,et al.  The State of the Art , 2015 .

[32]  Thomas Peacock,et al.  Introduction to Focus Issue: Lagrangian Coherent Structures. , 2010, Chaos.