Summary
Scale-space techniques are very popular in image processing since they allow for the integrated analysis of image structure. The multi-scale approach enables one to distinguish between important features such as edges and small-scale features such as numerical artifacts or noise. In general, the same properties hold for vector fields such as flow data. Many flow features, e.g. vortices, can be observed on multiple scales of the data and also many features that can be detected are essentially artifacts of the employed interpolation scheme or originate from noise in the data. In this paper, we investigate an approach based on scale-space hierarchies of threedimensional vector fields. Our main interest concerns how vector field singularities can be tracked over multiple spatial scales in order to assess the importance of a critical point to the overall behavior of the underlying flow field.
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References
D. Bauer and R. Peikert. Vortex Tracking in Scale-Space. In Proceedings of the Symposium on Data Visualisation ’02, pages 233-240, 2002.
W. de Leeuw and R. van Liere. Visualization of Global Flow Structures Using Multiple Levels of Topology. In Proceedings of the Symposium on Data Visualisation ’99, pages 45-52, 1999.
L. Florack and A. Kuijper. The Topological Structure of Scale-Space Images. Journal of Mathematical Imaging and Vision, 12(1):65-79, 2000.
C. Garth, X. Tricoche, and G. Scheuermann. Tracking of Vector Field Singularities in Unstructured 3D Time-Dependent Datasets. In Proceedings of IEEE Visualization ’04, pages 329-336, 2004.
A. Globus, C. Levit, and T. Lasinski. A Tool for Visualizing the Topology of Three-Dimensional Vector Fields. In Proceedings of IEEE Visualization ’91, pages 33-40, 1991.
J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer Verlag, 1986.
J. L. Helman and L. Hesselink. Representation and Display of Vector Field Topology in Fluid Flow Data Sets. IEEE Computer, 22(8):27-36, 1989.
J. L. Helman and L. Hesselink. Visualizing Vector Field Topology in Fluid Flows. IEEE Comput. Graph. Appl., 11(3):36-46, 1991.
T. Iijima. Basic theory on normalization of a pattern (in case of typical onedimensional pattern). Bulletin of Electrical Laboratory, 26:368-388, 1962.
J. J. Koenderink. The Structure of Images. Biological Cybernetics, 50:363-370, 1984.
A. Kuijper. The Deep Structure of Gaussian Scale Space Images. PhD thesis, Utrecht University, 2002.
A. Kuijper and L. Florack. Calculations on Critical Points under Gaussian Blurring. In Proceedings of the Second International Conference on Scale-Space Theories in Computer Vision ’99, 1999.
T. Lindeberg. Scale-Space Theory in Computer Vision. Kluwer Academic Publishers, 1994.
T. Lindeberg. Edge Detection and Ridge Detection with Automatic Scale Selection. International Journal of Computer Vision, 30(2):117-156, 1998.
S. Mann and A. Rockwood. Computing Singularities of 3D Vector Fields with Geometric Algebra. In Proceedings of IEEE Visualization ’02, pages 283-290, 2002.
K. Polthier and E. Preuß. Identifying Vector Field Singularities Using a Discrete Hodge Decomposition. In Visualization and Mathematics III, pages 113-134. Springer-Verlag, 2003.
F. H. Post, B. Vrolijk, H. Hauser, R. S. Laramee, and H. Doleisch. The State of the Art in Flow Visualisation: Feature Extraction and Tracking. Computer Graphics Forum, 22(4):775-792, 2003.
G. Scheuermann, H. Hagen, H. Krüger, M. Menzel, and A. Rockwood. Visualization of Higher Order Singularities in Vector Fields. In Proceedings of IEEE Visualization ’97, pages 67-74, 1997.
G. Scheuermann, H. Krüger, M. Menzel, and A. P. Rockwood. Visualizing Nonlinear Vector Field Topology. IEEE Transactions on Visualization and Computer Graphics, 4(2):109-116, 1998.
G. Scheuermann, X. Tricoche, and H. Hagen. C1-Interpolation for Vector Field Topology Visualization. In Proceedings of IEEE Visualization ’99, 1999.
H. Theisel and H.-P. Seidel. Feature flow fields. In Proceedings of the Symposium on Data Visualisation 2003, pages 141-148, 2003.
H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Saddle Connectors - An Approach to Visualizing the Topological Skeleton of Complex 3D Vector Fields. In Proceedings of IEEE Visualization ’03, page 30, 2003.
X. Tricoche, G. Scheuermann, and H. Hagen. Continuous Topology Simplification of Planar Vector Fields. In Proceedings of IEEE Visualization ’01, pages 159-166, 2001.
X. Tricoche, T. Wischgoll, G. Scheuermann, and H. Hagen. Topology-Based Visualization of Time-Dependent 2D Vector Fields. In Proceedings of the Symposium on Data Visualisation ’01, pages 117-126, 2001.
X. Tricoche, T. Wischgoll, G. Scheuermann, and H. Hagen. Topology Tracking for the Visualization of Time-Dependent Two-Dimensional Flows. Computer & Graphics, 26(2):249-257, 2002.
T. Weinkauf, H. Theisel, H.-C. Hege, and H.-P. Seidel. Boundary Switch Connectors for Topological Visualization of Complex 3D Vector Fields. In Proc. Joint Eurographics - IEEE TCVG Symposium on Visualization (VisSym ’04), pages 183-192, 2004.
A. Wiebel, C. Garth, and G. Scheuermann. Localized Flow Analysis of 2D and 3D Vector Fields. In Proceedings of Eurographics / IEEE VGTC Symposium on Visualization ’05, pages 143-150, 2005.
A. P. Witkin. Scale-Space Filtering. In Proceedings of the 8th International Joint Conference on Artificial Intelligence, pages 1019-1022, 1983.
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Klein, T., Ertl, T. (2007). Scale-Space Tracking of Critical Points in 3D Vector Fields. In: Hauser, H., Hagen, H., Theisel, H. (eds) Topology-based Methods in Visualization. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70823-0_3
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DOI: https://doi.org/10.1007/978-3-540-70823-0_3
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