| | This Article | |
| |
| |
| | Share | |
| |
| |
| | Bibliographic References | |
| |
| |
| | Add to: | |
| |
 Digg
 Furl
 Spurl
 Blink
 Simpy
 Google
 Del.icio.us
 Y!MyWeb
| |
| | Search | |
| |
| |
| | |
Interactive Tensor Field Design and Visualization on Surfaces
January/February 2007 (vol. 13 no. 1)
pp. 94-107
Abstract—Designing tensor fields in the plane and on surfaces is a necessary task in many graphics applications, such as painterly rendering, pen-and-ink sketching of smooth surfaces, and anisotropic remeshing. In this article, we present an interactive design system that allows a user to create a wide variety of symmetric tensor fields over 3D surfaces either from scratch or by modifying a meaningful input tensor field such as the curvature tensor. Our system converts each user specification into a basis tensor field and combines them with the input field to make an initial tensor field. However, such a field often contains unwanted degenerate points which cannot always be eliminated due to topological constraints of the underlying surface. To reduce the artifacts caused by these degenerate points, our system allows the user to move a degenerate point or to cancel a pair of degenerate points that have opposite tensor indices. These operations provide control over the number and location of the degenerate points in the field. We observe that a tensor field can be locally converted into a vector field so that there is a one-to-one correspondence between the set of degenerate points in the tensor field and the set of singularities in the vector field. This conversion allows us to effectively perform degenerate point pair cancellation and movement by using similar operations for vector fields. In addition, we adapt the image-based flow visualization technique to tensor fields, therefore allowing interactive display of tensor fields on surfaces. We demonstrate the capabilities of our tensor field design system with painterly rendering, pen-and-ink sketching of surfaces, and anisotropic remeshing.
[1] P. Alliez, D. Cohen-Steiner, O. Devillers, B. Lévy, and M. Desbrun, “Anisotropic Polygonal Remeshing,” ACM Trans. Graphics (Proc. SIGGRAPH '03), vol. 22, no. 3, pp. 485-493, July 2003.
[2] B. Cabral and C. Leedom, “Imaging Vector Fields Using Line Integral Convolution,” Computer Graphics Proc., Ann. Conf. Series (SIGGRAPH '93), pp. 263-270, 1993.
[3] D. Cohen-Steiner and J.M. Morvan, “Restricted Delaunay Triangulations and Normal Cycle,” Proc. 19th Ann. ACM Symp. Computational Geometry, pp. 237-246, 2003.
[4] T. Delmarcelle and L. Hesselink, “Visualizing Second-Order Tensor Fields with Hyperstream Lines,” IEEE Computer Graphics and Applications, pp. 25-33, 1993.
[5] T. Delmarcelle and L. Hesselink, “The Topology of Symmetric, Second-Order Tensor Fields,” Proc. IEEE Visualization Conf., pp.140-147, 1994.
[6] T. Delmarcelle, “The Visualization of Second-Order Tensor Fields,” PhD thesis, Stanford Applied Physics, 1994.
[7] S. Dong, S. Kircher, and M. Garland, “Harmonic Functions for Quadrilateral Remeshing of Arbitrary Manifolds,” Computer Aided Geometry Design, 2005.
[8] M. Floater, “Mean Value Coordinates,” Computer Aided Geometric Design, vol. 20, no. 1, pp. 19-27, 2003.
[9] A. Girshick, V. Interrante, S. Haker, and T. Lemoine, “Line Direction Matters: An Argument for the Use of Principal Directions in 3D Line Drawings,” Proc. First Int'l Symp. Non-Photorealistic Animation and Rendering (NPAR '00), pp. 43-52, 2000.
[10] J.H. Hays and I. Essa, “Image and Video Based Painterly Animation,” Proc. Third Int'l Symp. Non-Photorealistic Animation and Rendering (NPAR '04), pp. 113-120, 2004.
[11] A. Hertzmann and D. Zorin, “Illustrating Smooth Surfaces,” Computer Graphics Proc., Ann. Conf. Series (SIGGRAPH '00), pp.517-526, 2000.
[12] A. Hertzmann, “Painterly Rendering with Curved Brush Strokes of Multiple Sizes,” Computer Graphics Proc., Ann. Conf. Series (SIGGRAPH '98), pp. 453-460, 1998.
[13] V. Interrante, “Illustrating Surface Shape in Volume Data via Principal Direction-Driven 3D Line Integral Convolution,” Computer Graphics Proc., Ann. Conf. Series (SIGGRAPH 1997), pp. 109-116, 1997.
[14] R.S. Laramee, B. Jobard, and H. Hauser, “Image Space Based Visualization of Unsteady Flow on Surfaces,” Proc. IEEE Visualization Conf., pp. 131-138, 2003.
[15] P. Litwinowicz, “Processing Images and Video for an Impressionist Effect,” Computer Graphics Proc., Ann. Conf. Series (SIGGRAPH 1997), pp. 407-414, 1997.
[16] M. Marinov and L. Kobbelt, “Direct Anisotropic Quad-Dominant Remeshing,” Proc. 12th Pacific Conf. Computer Graphics and Applications (PG '04), pp. 207-216, 2004.
[17] M. Meyer, M. Desbrun, P. Schröder, and A.H. Barr, “Discrete Differential-Geometry Operators for Triangulated 2-Manifolds,” Proc. Workshop Visualization and Math. (VisMath), 2002.
[18] K. Mischaikow and M. Mrozek, “Conley Index,” Handbook of Dynamic Systems, second ed., North-Holland, pp. 393-460, 2002.
[19] X. Ni, M. Garland, and J.C. Hart, “Fair Morse Functions for Extracting the Topological Structure of a Surface Mesh,” ACM Trans. Graphics (Proc. SIGGRAPH '04), vol. 23, no. 3, pp. 613-622, 2004.
[20] E. Praun, A. Finkelstein, and H. Hoppe, “Lapped Textures,” Computer Graphics Proc., Ann. Conf. Series (SIGGRAPH '00), pp.465-470, 2000.
[21] A. Rockwood and S. Bunderwala, “A Toy Vector Field Based on Geometric Algebra,” Proc. Application of Geometric Algebra in Computer Science and Eng. (AGACSE '01), pp. 179-185, 2001.
[22] M.P. Salisbury, M.T. Wong, J.F. Hughes, and D.H. Salesin, “Orientable Textures for Image-Based Pen-and-Ink Illustration,” Computer Graphics Proc., Ann. Conf. Series (SIGGRAPH '97), pp.401-406, 1997.
[23] D. Stalling, H.C. Hege, “Fast and Resolution Independent Line Integral Convolution,” Computer Graphics Proc., Ann. Conf. Series (SIGGRAPH '95), pp. 249-256, 1995.
[24] J. Stam, “Flows on Surfaces of Arbitrary Topology,” ACM Trans. Graphics (Proc. SIGGRAPH '03), vol. 22, no. 3, pp. 724-731, 2003.
[25] H. Theisel, “Designing 2D Vector Fields of Arbitrary Topology,” Computer Graphics Forum (Proc. Eurographics '02), vol. 21, no. 3, pp.595-604, 2002.
[26] Y. Tong, S. Lombeyda, A. Hirani, and M. Desbrun, “Discrete Multiscale Vector Field Decomposition,” ACM Trans. Graphics, (Proc. SIGGRAPH '03), vol. 22, no. 3, pp. 445-452, 2003.
[27] X. Tricoche, G. Scheuermann, and H. Hagen, “Scaling the Topology of Symmetric Second Order Tensor Fields,” Proc. US Nat'l Science Foundation/Dept. of Energy Lake Tahoe Workshop Hierarchical Approximation and Geometrical Methods for Scientific Visualization, 2001.
[28] X. Tricoche, “Vector and Tensor Field Topology Simplification, Tracking, and Visualization,” PhD thesis, Universität Kaiserslautern, 2002.
[29] X. Tricoche and G. Scheuermann, “Topology Simplification of Symmetric, Second-Order 2D Tensor Fields,” Geometric Modeling Methods in Scientific Visualization, B. Hamann, H. Müller, and H.Hagen, eds., Springer, 2003.
[30] G. Turk, “Texture Synthesis on Surfaces,” Computer Graphics Proc., Ann. Conf. Series (SIGGRAPH '01), pp. 347-354, 2001.
[31] T. Urness, V. Interrante, I. Marusic, E. Longmire, and B. Ganapathisubramani, “Effectively Visualizing Multi-Valued Flow Data Using Color and Texture,” Proc. IEEE Visualization Conf., pp.115-121, 2003.
[32] J.J. van Wijk, “Image Based Flow Visualization,” ACM Trans. Graphics (Proc. SIGGRAPH '02), vol. 21, no. 3, pp. 745-754, 2002.
[33] J.J. van Wijk, “Image Based Flow Visualization for Curved Surfaces,” Proc. IEEE Visualization Conf., G. Turk, J. van Wijk, and R. Moorhead, eds., pp. 123-130, 2003.
[34] L.Y. Wei and M. Levoy, “Texture Synthesis over Arbitrary Manifold Surfaces,” Computer Graphics Proc., Ann. Conf. Series, (SIGGRAPH '01), pp. 355-360, 2001.
[35] E. Zhang, K. Mischaikow, and G. Turk, “Vector Field Design on Surfaces,” ACM Trans. Graphics, vol. 25, no. 4, 2006.
[36] X. Zheng and A. Pang, “Hyperlic,” Proc. IEEE Visualization Conf., pp.249-256, 2003.
Index Terms:
Tensor field design, tensor field visualization, nonphotorealistic rendering, surfaces, remeshing, tensor field topology.
Citation:
Eugene Zhang, James Hays, Greg Turk, "Interactive Tensor Field Design and Visualization on Surfaces," IEEE Transactions on Visualization and Computer Graphics, vol. 13, no. 1, pp. 94-107, Jan./Feb. 2007, doi:10.1109/TVCG.2007.16
