Fast Combinatorial Vector Field Topology

This paper introduces a novel approximation algorithm for the fundamental graph problem of combinatorial vector field topology (CVT). CVT is a combinatorial approach based on a sound theoretical basis given by Forman's work on a discrete Morse theory for dynamical systems. A computational framework for this mathematical model of vector field topology has been developed recently. The applicability of this framework is however severely limited by the quadratic complexity of its main computational kernel. In this work, we present an approximation algorithm for CVT with a significantly lower complexity. This new algorithm reduces the runtime by several orders of magnitude and maintains the main advantages of CVT over the continuous approach. Due to the simplicity of our algorithm it can be easily parallelized to improve the runtime further.

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

[2]  Bernd Hamann,et al.  A topological approach to simplification of three-dimensional scalar functions , 2006, IEEE Transactions on Visualization and Computer Graphics.

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

[4]  Thomas Lewiner,et al.  Applications of Forman's discrete Morse theory to topology visualization and mesh compression , 2004, IEEE Transactions on Visualization and Computer Graphics.

[5]  H. Edelsbrunner,et al.  Persistent Homology — a Survey , 2022 .

[6]  Lambertus Hesselink,et al.  Representation and display of vector field topology in fluid flow data sets , 1989, Computer.

[7]  Valerio Pascucci,et al.  Morse-smale complexes for piecewise linear 3-manifolds , 2003, SCG '03.

[8]  Tino Weinkauf.  Extraction of Topological Structures in 2D and 3D Vector Fields , 2008, Ausgezeichnete Informatikdissertationen.

[9]  R. Forman.  Morse Theory for Cell Complexes , 1998 .

[10]  P. J. Narayanan,et al.  Accelerating Large Graph Algorithms on the GPU Using CUDA , 2007, HiPC.

[11]  Hans-Peter Seidel,et al.  Extracting higher order critical points and topological simplification of 3D vector fields , 2005, VIS 05. IEEE Visualization, 2005..

[12]  S. Hanke,et al.  New Approximation Algorithms for the Weighted Matching Problem , 2003 .

[13]  R. Forman.  A USER'S GUIDE TO DISCRETE MORSE THEORY , 2002 .

[14]  H. Kuhn.  The Hungarian method for the assignment problem , 1955 .

[15]  Ingrid Hotz,et al.  Combinatorial 2D Vector Field Topology Extraction and Simplification , 2011, Topological Methods in Data Analysis and Visualization.

[16]  R. Ho.  Algebraic Topology , 2022 .

[17]  Konstantin Mischaikow,et al.  Vector Field Editing and Periodic Orbit Extraction Using Morse Decomposition , 2007, IEEE Transactions on Visualization and Computer Graphics.

[18]  Hans Hagen,et al.  Vector and Tensor Field Topology Simplification on Irregular Grids , 2001, VisSym.

[19]  Hans Hagen,et al.  Continuous topology simplification of planar vector fields , 2001, Proceedings Visualization, 2001. VIS '01..

[20]  A. Gyulassy.  Combinatorial construction of morse-smale complexes for data analysis and visualization , 2008 .

[21]  Herbert Edelsbrunner,et al.  Topological persistence and simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[22]  Hans-Peter Seidel,et al.  Grid-independent Detection of Closed Stream Lines in 2D Vector Fields , 2004, VMV.

[23]  R. Forman.  Combinatorial vector fields and dynamical systems , 1998 .

[24]  Hans-Christian Hege,et al.  TADD: A Computational Framework for Data Analysis Using Discrete Morse Theory , 2010, ICMS.

[25]  Gerik Scheuermann,et al.  Detection and Visualization of Closed Streamlines in Planar Flows , 2001, IEEE Trans. Vis. Comput. Graph..