Controlled-topology filtering

Many applications require the extraction of isolines and isosurfaces from scalar functions defined on regular grids. These scalar functions may have many different origins: from MRI and CT scan data to terrain data or results of a simulation. As a result of noise and other artifacts, curves and surfaces obtained by standard extraction algorithms often suffer from topological irregularities and geometric noise.While it is possible to remove topological and geometric noise as a post-processing step, in the case when a large number of isolines are of interest there is a considerable advantage in filtering the scalar function directly. While most smoothing filters result in gradual simplification of the topological structure of contours, new topological features typically emerge and disappear during the smoothing process.In this paper, we describe an algorithm for filtering functions defined on regular 2D grids with controlled topology changes, which ensures that the topological structure of the set of contour lines of the function is progressively simplified.

[1]  Herbert Edelsbrunner,et al.  Computing and Comprehending Topology: Persistence and Hierarchical Morse Complexes , 2001 .

[2]  Chandrajit L. Bajaj,et al.  Time-varying contour topology , 2006, IEEE Transactions on Visualization and Computer Graphics.

[3]  Jihad El-Sana,et al.  Controlled simplification of genus for polygonal models , 1997, Proceedings. Visualization '97 (Cat. No. 97CB36155).

[4]  SnoeyinkJack,et al.  Computing contour trees in all dimensions , 2003 .

[5]  Bernd Hamann,et al.  Topological segmentation in three-dimensional vector fields , 2004, IEEE Transactions on Visualization and Computer Graphics.

[6]  J. Hart,et al.  Fair morse functions for extracting the topological structure of a surface mesh , 2004, SIGGRAPH 2004.

[7]  Martin Rumpf,et al.  Anisotropic Diffusion in Vector Field Visualization on Euclidean Domains and Surfaces , 2000, IEEE Trans. Vis. Comput. Graph..

[8]  Hans Hagen,et al.  A topology simplification method for 2D vector fields , 2000, Proceedings Visualization 2000. VIS 2000 (Cat. No.00CH37145).

[9]  S. Rana,et al.  Topological data structures for surfaces: an introduction to geographical information science. , 2006 .

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

[11]  Santiago V. Lombeyda,et al.  Discrete multiscale vector field decomposition , 2003, ACM Trans. Graph..

[12]  John Hart.  Morse Theory for Implicit Surface Modeling , 1997, VisMath.

[13]  Mathieu Desbrun,et al.  Removing excess topology from isosurfaces , 2004, TOGS.

[14]  Bernd Hamann,et al.  Maximizing adaptivity in hierarchical topological models , 2005, International Conference on Shape Modeling and Applications 2005 (SMI' 05).

[15]  Afra J. Zomorodian,et al.  Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics) , 2005 .

[16]  Xavier Tricoche,et al.  Vector and tensor field topology simplification, tracking, and visualization , 2002 .

[17]  Xiao Han,et al.  A topology preserving deformable model using level sets , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

[18]  J. Damon.  Local Morse Theory for Solutions to the Heat Equation and Gaussian Blurring , 1995 .

[19]  Chandrajit L. Bajaj,et al.  Topology preserving data simplification with error bounds , 1998, Comput. Graph..

[20]  Lambertus Hesselink,et al.  Visualizing vector field topology in fluid flows , 1991, IEEE Computer Graphics and Applications.

[21]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[22]  Herbert Edelsbrunner,et al.  Topological Persistence and Simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[23]  Jack Snoeyink,et al.  Computing contour trees in all dimensions , 2000, SODA '00.

[24]  Jack Snoeyink,et al.  Simplifying flexible isosurfaces using local geometric measures , 2004, IEEE Visualization 2004.

[25]  Herbert Edelsbrunner,et al.  Hierarchical morse complexes for piecewise linear 2-manifolds , 2001, SCG '01.

[26]  Andrew P. Witkin,et al.  Scale-Space Filtering , 1983, IJCAI.

[27]  Carlos Andújar,et al.  Topology-reducing surface simplification using a discrete solid representation , 2002, TOGS.

[28]  Valerio Pascucci,et al.  Contour trees and small seed sets for isosurface traversal , 1997, SCG '97.

[29]  Marc Levoy,et al.  A volumetric method for building complex models from range images , 1996, SIGGRAPH.

[30]  Robert van Liere,et al.  Collapsing flow topology using area metrics , 1999, Proceedings Visualization '99 (Cat. No.99CB37067).

[31]  Amitabh Varshney,et al.  Controlled Topology Simplification , 1996, IEEE Trans. Vis. Comput. Graph..

[32]  Bernd Hamann,et al.  A topological hierarchy for functions on triangulated surfaces , 2004, IEEE Transactions on Visualization and Computer Graphics.

[33]  Hans Hagen,et al.  Topology tracking for the visualization of time-dependent two-dimensional flows , 2002, Comput. Graph..

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

[35]  Herbert Edelsbrunner,et al.  Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms , 1988, SCG '88.

[36]  Jesse Freeman,et al.  in Morse theory, , 1999 .

[37]  Zoë J. Wood,et al.  Topological Noise Removal , 2001, Graphics Interface.

[38]  James Demmel,et al.  A Supernodal Approach to Sparse Partial Pivoting , 1999, SIAM J. Matrix Anal. Appl..

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

[40]  Michael Garland,et al.  Fair morse functions for extracting the topological structure of a surface mesh , 2004, ACM Trans. Graph..