
This Article  
 
Share  
Bibliographic References  
Add to:  
Digg Furl Spurl Blink Simpy Del.icio.us Y!MyWeb  
Search  
 
ASCII Text  x  
Harald Garcke, Tobias Preußer, Martin Rumpf, Alexandru C. Telea, Ulrich Weikard, Jarke J. van Wijk, "A Phase Field Model for Continuous Clustering on Vector Fields," IEEE Transactions on Visualization and Computer Graphics, vol. 7, no. 3, pp. 230241, JulySeptember, 2001.  
BibTex  x  
@article{ 10.1109/2945.942691, author = {Harald Garcke and Tobias Preußer and Martin Rumpf and Alexandru C. Telea and Ulrich Weikard and Jarke J. van Wijk}, title = {A Phase Field Model for Continuous Clustering on Vector Fields}, journal ={IEEE Transactions on Visualization and Computer Graphics}, volume = {7}, number = {3}, issn = {10772626}, year = {2001}, pages = {230241}, doi = {http://doi.ieeecomputersociety.org/10.1109/2945.942691}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Visualization and Computer Graphics TI  A Phase Field Model for Continuous Clustering on Vector Fields IS  3 SN  10772626 SP230 EP241 EPD  230241 A1  Harald Garcke, A1  Tobias Preußer, A1  Martin Rumpf, A1  Alexandru C. Telea, A1  Ulrich Weikard, A1  Jarke J. van Wijk, PY  2001 KW  Flow visualization KW  clustering KW  CahnHilliard KW  multiscale KW  nonlinear diffusion KW  finite elements KW  skeletonization. VL  7 JA  IEEE Transactions on Visualization and Computer Graphics ER   
Abstract—A new method for the simplification of flow fields is presented. It is based on continuous clustering. A wellknown physical clustering model, the Cahn Hilliard model, which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly as connected components of the positivity set of a density function. An evolution equation for this function is obtained as a suitable gradient flow of an underlying anisotropic energy functional. Here, time serves as the scale parameter. The evolution is characterized by a successive coarsening of patterns—the actual clustering—during which the underlying simulation data specifies preferable pattern boundaries. We introduce specific physical quantities in the simulation to control the shape, orientation and distribution of the clusters as a function of the underlying flow field. In addition, the model is expanded, involving elastic effects. In the early stages of the evolution shear layer type representation of the flow field can thereby be generated, whereas, for later stages, the distribution of clusters can be influenced. Furthermore, we incorporate upwind ideas to give the clusters an oriented dropshaped appearance. Here, we discuss the applicability of this new type of approach mainly for flow fields, where the cluster energy penalizes cross streamline boundaries. However, the method also carries provisions for other fields as well. The clusters can be displayed directly as a flow texture. Alternatively, the clusters can be visualized by iconic representations, which are positioned by using a skeletonization algorithm.
[1] M.O. Bristeau, R. Glowinski, and J. Periaux, “Numerical Methods for the NavierStokes Equations: Applications to the Simulation of Compressible and Incompressible Viscous Flows,” Computer Physics Report, Research Report UH/MD4, Univ. of Houston, 1987.
[2] B. Cabral and L.C. Leedom, "Imaging Vector Fields Using Line Integral Convolution," Computer Graphics (SIGGRAPH '93 Proc.), pp. 263272, 1993.
[3] J. Cahn and J. Hilliard, “Free Energy of a NonUniform System I. Interfacial Free Energy,” J. Chemistry and Physics, vol. 28, pp. 258267, 1958.
[4] P. Ciarlet and J. Lions, Handbook of Numerical Analysis. Vol. V: Techniques of Scientific Computing, Elsevier, 1997.
[5] W. de Leeuw and J.J. van Wijk, “Enhanced Spot Noise for Vector Field Visualization,” Proc. IEEE Visualization '95, pp. 233239, 1995.
[6] C.M. Elliott, “The CahnHilliard Model for the Kinetics of Phase Separation,” Numerische Mathemtica, pp. 3573, 1988.
[7] L. Forssell, “Visualizing Flow over Curvilinear Grid Surfaces Using Line Integral Convolution,” Proc. IEEE Visualization '94, pp. 240246, 1994.
[8] H. Garcke, M. Rumpf, and U. Weikard, “The CahnHilliard Equation with Elasticity, Finite Element Approximation and Qualitative Analysis,” J. Interphases and Free Boundaries, vol. 3, pp. 101118, 2001.
[9] B. Heckel, G. Weber, B. Hamann, and K.I. Joy, “Construction of Vector Field Hierarchies,” Proc. IEEE Visualization '99, pp. 1925, 1999.
[10] B. Jobard and W. Lefer, “Creating EvenlySpaced Streamlines of Arbitrary Density,” Proc. Visualization in Scientific Computing '97, W. Lefer and M. Grave, eds., pp. 4354, 1997.
[11] F. Leymarie and M.D. Levine, “Simulating the Grassfire Transform Using an Active Contour Model,” IEEE Trans. Pattern Analysis and Machine Intellingence, vol. 14, no. 1, pp. 5675, Jan. 1992.
[12] S.G. Mallat,“A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 7, pp. 674693, 1989.
[13] N. Mayya and V.T. Rajan, “Voronoi Diagrams of Polygons: A Framework for Shape Representation,” J. Math. Imaging and Vision, vol. 4, pp. 355378, 1996.
[14] S. MüllerUrbaniak, “Eine Analyse des Zweischritt$\theta$Verfahrens zur Lösung der instationären NavierStokesGleichungen,” Preprint des SFB 359, 9401, 1994.
[15] A. NovickCohen, “The CahnHilliard Equation: Mathematical and Modelling Perspectives,” Advances in Math. and Science Applications, vol. 8, pp. 965985, 1998.
[16] L. Khouas, C. Odet, and D. Friboulet, “2D Vector Field Visualization Using Furlike Texture,” Proc. IEEE Visualization Symp. '99, pp. 3544, 1999.
[17] P. Perona and J. Malik, “Scale Space and Edge Detection Using Anisotropic Diffusion,” Proc. IEEE CS Workshop Computer Vision, 1987.
[18] F.J. Post, T. v. Walsum, F.H. Post, and D. Silver, “Iconic Techniques for Feature Visualization,” Trans. Visualization and Computer Graphics, vol. 1, pp. 288295, 1995.
[19] T. Preußer and M. Rumpf, “An Adaptive Finite Rlement Method for Large Scale Image Processing,” J. Visual Comm. and Image Representation, vol. 11, pp. 183195, 2000.
[20] T. Preußer and M. Rumpf, “Anisotropic Nonlinear Diffusion in Flow Visualization,” Proc. IEEE Visualization '99, pp. 325332, 1999.
[21] C. RezkSalama, P. Hastreiter, C. Teitzel, and T. Ertl, “Interactive Exploration of Volume Line Integral Convolution Based on 3DTexture Mapping,” Proc. IEEE Visualization '99, pp. 233240, 1999.
[22] J. Sethian, Level Set Methods and Fast Marching Methods. Cambridge Univ. Press, 1999.
[23] A.C. Telea and J.J. van Wijk, “Simplified Representation of Vector Fields,” Proc. IEEE Visualization '99, pp. 3542, 1999.
[24] V. Thomée, Galerkin—Finite Element Methods for Parabolic Problems. Springer, 1984.
[25] G. Turk and D. Banks, “ImageGuided Streamline Placement,” Computer Graphics (Proc. SIGGRAPH '93), 1996.
[26] R. Wegenkittl, E. Gröller, and W. Purgathofer, “Animating Flowfields: Rendering of Oriented Line Integral Convolution,” Proc. IEEE Visualization '97, pp. 119125, 1997.
[27] J. Weickert, Anisotropic Diffusion in Image Processing. Teubner, 1998.
[28] P. Wong and D. Bergeron, “Hierarchical Representation of Very Large Data Sets for Visualization Using Wavelets,” Scientific Visualization, G. Nielson, H. Hagen, and H. Mueller, eds., pp. 415429, 1997.
[29] M. Zöckler, D. Stalling, and H.C. Hege, “Interactive Visualization of 3DVector Fields Using Illuminated Streamlines,” Proc. IEEE Visualization '96, pp. 107113, 1996.