This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Moment Invariants for the Analysis of 2D Flow Fields
November/December 2007 (vol. 13 no. 6)
pp. 1743-1750
We present a novel approach for analyzing two-dimensional (2D) flow field data based on the idea of invariant moments. Moment invariants have traditionally been used in computer vision applications, and we have adapted them for the purpose of interactive exploration of flow field data. The new class of moment invariants we have developed allows us to extract and visualize 2D flow patterns, invariant under translation, scaling, and rotation. With our approach one can study arbitrary flow patterns by searching a given 2D flow data set for any type of pattern as specified by a user. Further, our approach supports the computation of moments at multiple scales, facilitating fast pattern extraction and recognition. This can be done for critical point classification, but also for patterns with greater complexity. This multi-scale moment representation is also valuable for the comparative visualization of flow field data. The specific novel contributions of the work presented are the mathematical derivation of the new class of moment invariants, their analysis regarding critical point features, the efficient computation of a novel feature space representation, and based upon this the development of a fast pattern recognition algorithm for complex flow structures.

[1] Y. S. Abu-Mostafa and D. Psaltis, Recognitive aspects of moment invariants. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (6): 698–706, November 1984.
[2] P. Billant, J.-M. Chomaz, and P. Huerre, Experimental study of vortex breakdown in swirling jets. Journal of Fluid Mechanics, 376: 183–219, Dec. 1998.
[3] S. P. Callahan, J. Freire, E. Santos, C. E. Scheidegger, C. T. Silva, and H. T. Vo, Vistrails: Visualization meets data management. In ACM SIGMOD, pages 745–747, 2006.
[4] J. Ebling and G. Scheuermann, Clifford convolution and pattern matching on vector fields. In Proceedings of IEEE Visualization 2003, pages 193–200, 2003.
[5] J. Ebling and G. Scheuermann, Clifford convolution and pattern matching on irregular grids. In Scientific Visualization: The Visual Extraction of Knowledge from Data. Springer-Verlag, 2005.
[6] J. Ebling and G. Scheuermann, Clifford Fourier transform on vector fields. IEEE Trans. Vis. Comput. Graph., 11 (4): 469–479, 2005.
[7] H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, Local and global comparison of continuous functions. In IEEE Visualization, pages 275–280, 2004.
[8] G. Erlebacher, C. Garth, R. S. Laramee, H. Theisel, X. Tricoche, T. Weinkauf, and D. Weiskopf, Texture and feature-based flow visualization - methodology and application. In IEEE Visualization 06 Tutorial, 2006.
[9] J. Flusser, On the independence of rotation moment invariants. Pattern Recognition, 33 (9): 1405–1410, 2000.
[10] J. Flusser and T. Suk, Pattern recognition by affine moment invariants. Pattern Recognition, 26 (1): 167–174, January 1993.
[11] E. Heiberg, T. Ebbers, L. Wigström, and M. Karlsson, Three-dimensional flow characterization using vector pattern matching. IEEE Trans. Vis. Comput. Graph., 9 (3): 313–319, 2003.
[12] M.-K. Hu, Visual pattern recognition by moment invariants. IRE Transactions on Information Theory, 8 (2): 179–187, February 1962.
[13] C. A. Kennedy, M. H. Carpenter, and R. M. Lewis, Low-storage, explicit runge-kutta schemes for the compressible navier-stokes equations. Appl. Numer. Math., 35 (3): 177–219, 2000.
[14] Y. Li, Wavenumber-extended high-order upwind-biased finite-difference schemes for convective scalar transport. J. Comput. Phys., 133 (2): 235–255, 1997.
[15] S. Liao, Q. Lu, and K. Lee, Recognition of chinese characters by moment feature extraction. In International Conference on Computer Processing of Oriental Languages, pages 566–571, 1997.
[16] H.-G. Pagendarm and B. Walter, Competent, compact, comparative visualization of a vortical flow field. IEEE Transactions on Visualization and Computer Graphics, 1 (2): 142–150, 1995.
[17] F. Post, B. Vrolijk, H. Hauser, R. Laramee, and H. Doleisch, Feature extraction and visualization of flow fields. In In Eurographics 2002 State of the Art Reports, pages 69–100. The Eurographics Association, Saarbrücken, Germany, 2002.
[18] N. Sahasrabudhe, J. E. West, R. Machiraju, and M. Janus, Structured spatial domain image and data comparison metrics. In VIS '99: Proceedings of the conference on Visualization '99, pages 97–104, Los Alamitos, CA, USA, 1999. IEEE Computer Society Press.
[19] M. Schlemmer, I. Hotz, V. Natarajan, B. Hamann, and H. Hagen, Fast clifford fourier transformation for unstructured vector field data. In Proc. Intl. Conf. Numerical Grid Generation in Computational Field Simulations, pages 101–110, 2005.
[20] N. Svakhine, Y. Jang, D. S. Ebert, and K. P. Gaither, Illustration and Photography Inspired Visualization of Flows and Volumes. In IEEE Visualization 2005, 2005.
[21] J. C. Terrillon, M. David, and S. Akamatsu, Automatic detection of human faces in natural scene images by use of a skin color model and of invariant moments. In FG '98: Proceedings of the 3rd. International Conference on Face & Gesture Recognition, page 112, Washington, DC, USA, 1998. IEEE Computer Society.
[22] V. Verma and A. Pang, Comparative flow visualization. IEEE Transactions on Visualization and Computer Graphics, 10 (6): 609–624, 2004.
[23] R. Winkler, Stochastic differential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math., 163 (2): 435–463, 2004.

Index Terms:
Flow Visualization, Feature Detection, Pattern Recognition, Image Processing
Citation:
Michael Schlemmer, Manuel Heringer, Florian Morr, Ingrid Hotz, Martin Hering-Bertram, Christoph Garth, Wolfgang Kollmann, Bernd Hamann, Hans Hagen, "Moment Invariants for the Analysis of 2D Flow Fields," IEEE Transactions on Visualization and Computer Graphics, vol. 13, no. 6, pp. 1743-1750, Nov.-Dec. 2007, doi:10.1109/TVCG.2007.70579
Usage of this product signifies your acceptance of the Terms of Use.