Tensor Field Design: Algorithms and Applications
Abstract
Tensor field design has found increasing applications in computer graphics, geometry processing, and scientific visualization. In this chapter, we review recent advances in tensor field design and discuss possible future research directions.
Keywords
Vector Field Tensor Field Regular Element Texture Synthesis Field DesignNotes
Acknowledgements
I wish to thank Greg Turk, Konstantin Mischaikow, Guoning Chen, Jonathan Palacios, Robert S. Laramee, James Hays, Pawel Pilarczyk, Yu-Kun Lai, Xianfeng Gu, Bruno Lévy, Greg Esch, Peter Wonka, Pascal Müller, Matthias Nieser, and Konrad Polthier for their valuable discussions. The images shown in this survey are also courtesy of many of them. The reviewers’ constructive comments have made the survey stronger. Finally, I wish to thank my family for their support that they have provided during my attending the Dagstuhl seminar on Tensor Field Processing in 2009 as well as afterwards when I wrote the survey. The survey is partially funded by National Science Foundation grants IIS-0546881, CCF-0830808, and IIS-0917308.
References
- 1.Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., Desbrun, M.: Anisotropic polygonal remeshing. ACM Trans. Graph. (SIGGRAPH 2003) 22(3), 485–493 (2003)Google Scholar
- 2.Bhat, P., Ingram, S., Turk, G.: Geometric texture synthesis by example. In: SGP ’04: Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp. 41–44. ACM, New York (2004)Google Scholar
- 3.Cabral, B., Leedom, L.C.: Imaging vector fields using line integral convolution. In: SIGGRAPH ’93: Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, pp. 263–270. ACM, New York (1993)Google Scholar
- 4.Chen, G., Mischaikow, K., Laramee, R.S., Pilarczyk, P., Zhang, E.: Vector field editing and periodic orbit extraction using morse decomposition. IEEE Trans. Vis. Comput. Graph. 13(4), 769–785 (2007)Google Scholar
- 5.Chen, G., Esch, G., Wonka, P., Müller, P., Zhang, E.: Interactive procedural street modeling. ACM Trans. Graph. 27(3), 103:1–103:10 (2008)Google Scholar
- 6.Chenney, S.: Flow tiles. In: SCA ’04: Proceedings of the 2004 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 233–242. Eurographics Association, Aire-la-Ville (2004)Google Scholar
- 7.Delmarcelle, T., Hesselink, L.: The topology of symmetric, second-order tensor fields. In Proceedings of the Conference on Visualization ’94, VIS ’94, Washington, DC, pp. 140–147. IEEE Computer Society Press, Los Alamitos (1994)Google Scholar
- 8.Dong, S., Kircher, S., Garland, M.: Harmonic functions for quadrilateral remeshing of arbitrary manifolds. Comput. Aided Geom. Des. 22(5), 392–423 (2005)Google Scholar
- 9.Dong, S., Bremer, P.T., Garland, M., Pascucci, V., Hart, J.C.: Spectral surface quadrangulation. ACM Trans. Graph. 25(3), 1057–1066 (2006)Google Scholar
- 10.Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: FOCS ’00: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, p. 454. IEEE Computer Society, Washington, DC (2000)Google Scholar
- 11.Fisher, M., Schröder, P., Desbrun, M., Hoppe, H.: Design of tangent vector fields. In: SIGGRAPH ’07: ACM SIGGRAPH 2007 Papers, p. 56. ACM, New York (2007)Google Scholar
- 12.von Funck, W., Theisel, H., Seidel, H.P.: Vector field based shape deformations. In: SIGGRAPH ’06: ACM SIGGRAPH 2006 Papers, pp. 1118–1125. ACM, New York (2006)Google Scholar
- 13.Hays, J., Essa, I.: Image and video based painterly animation. In: NPAR ’04: Proceedings of the 3rd International Symposium on Non-photorealistic Animation and Rendering, pp. 113–120. ACM, New York (2004)Google Scholar
- 14.Hertzmann, A.: Painterly rendering with curved brush strokes of multiple sizes. In: SIGGRAPH ’98: Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, pp. 453–460. ACM, New York (1998)Google Scholar
- 15.Hertzmann, A., Zorin, D.: Illustrating smooth surfaces. In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’00, pp. 517–526. ACM Press/Addison-Wesley Publishing Co., New York (2000). doi: 10.1145/344779.345074Google Scholar
- 16.Hilaga, M., Shinagawa, Y., Kohmura, T., Kunii, T.L.: Topology matching for fully automatic similarity estimation of 3d shapes. In: SIGGRAPH ’01: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, pp. 203–212. ACM, New York (2001)Google Scholar
- 17.Hirani, A.N.: Discrete exterior calculus. Ph.D. thesis, California Institute of Technology, Pasadena (2003). Adviser-Marsden, Jerrold EGoogle Scholar
- 18.Kagaya, M., Brendel, W., Deng, Q., Kesterson, T., Todorovic, S., Neill, P.J., Zhang, E.: Video painting with space-time-varying style parameters. IEEE Trans. Vis. Comput. Graph. 17, 74–87 (2011)Google Scholar
- 19.Kälberer, F., Nieser, M., Polthier, K.: Quadcover – surface parameterization using branched coverings. Comput. Graph. Forum 26(3), 375–384 (2007)Google Scholar
- 20.Kimmel, R., Sethian, J.A.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. 95(15), 8431–8435 (1998)Google Scholar
- 21.Lai, Y.K., Jin, M., Xie, X., He, Y., Palacios, J., Zhang, E., Hu, S.M., Gu, X.: Metric-driven rosy field design and remeshing. IEEE Trans. Vis. Comput. Graph. 16, 95–108 (2010)Google Scholar
- 22.Laramee, R.S., Jobard, B., Hauser, H.: Image space based visualization of unsteady flow on surfaces. In Proceedings of the 14th IEEE Visualization 2003 (VIS ’03), p. 18. IEEE Computer Society, Washington, DC (2003). doi: 10.1109/VISUAL.2003.1250364Google Scholar
- 23.Li, Y., Bao, F., Zhang, E., Kobayashi, Y., Wonka, P.: Geometry synthesis on surfaces using field-guided shape grammars. IEEE Trans. Vis. Comput. Graph. 17, 231–243 (2011)Google Scholar
- 24.Marinov, M., Kobbelt, L.: Direct anisotropic quad-dominant remeshing. In Proceedings of the Computer Graphics and Applications, 12th Pacific Conference, PG ’04, pp. 207–216. IEEE Computer Society, Washington, DC (2004)Google Scholar
- 25.Mischaikow, K., Mrozek, M.: Conley index. In: Handbook of Dynamical Systems, vol. 2, pp. 393–460. North-Holland, Amsterdam (2002)Google Scholar
- 26.Ni, X., Garland, M., Hart, J.C.: Fair morse functions for extracting the topological structure of a surface mesh. In: SIGGRAPH ’04: ACM SIGGRAPH 2004 Papers, pp. 613–622. ACM, New York (2004)Google Scholar
- 27.Nieser, M., Palacios, J., Polthier, K., Zhang, E.: Hexagonal global parameterization of arbitrary surfaces. IEEE Trans. Vis. Comput. Graph. 18(6), 865–878 (2012). doi: 10.1109/TVCG.2011.118Google Scholar
- 28.Palacios, J., Zhang, E.: Rotational symmetry field design on surfaces. ACM Trans. Graph. 26(3), 55:1–55:10 (2007)Google Scholar
- 29.Palacios, J., Zhang, E.: Interactive visualization of rotational symmetry fields on surfaces. IEEE Trans. Vis. Comput. Graph. 17(7), 947–955 (2011)Google Scholar
- 30.Paris, S., Chang, W., Kozhushnyan, O.I., Jarosz, W., Matusik, W., Zwicker, M., Durand, F.: Hair photobooth: geometric and photometric acquisition of real hairstyles. In: SIGGRAPH ’08: ACM SIGGRAPH 2008 Papers, pp. 1–9. ACM, New York (2008)Google Scholar
- 31.Patil, S., van den Berg, J., Curtis, S., Lin, M.C., Manocha, D.: Directing crowd simulations using navigation fields. IEEE Trans. Vis. Comp. Graph. 17(2), 244–254 (2011). doi: 10.1109/TVCG.2010.33Google Scholar
- 32.Praun, E., Finkelstein, A., Hoppe, H.: Lapped textures. In: SIGGRAPH ’00: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 465–470. ACM/Addison-Wesley, New York (2000)Google Scholar
- 33.Ray, N., Li, W.C., Lévy, B., Sheffer, A., Alliez, P.: Periodic global parameterization. ACM Trans. Graph. 25(4), 1460–1485 (2006)Google Scholar
- 34.Ray, N., Vallet, B., Li, W.C., Lévy, B.: N-symmetry direction field design. ACM Trans. Graph. 27(2), 1–13 (2008)Google Scholar
- 35.Ray, N., Vallet, B., Alonso, L., Levy, B.: Geometry-aware direction field processing. ACM Trans. Graph. 29(1), 1–11 (2009)Google Scholar
- 36.Stam, J.: Flows on surfaces of arbitrary topology. ACM Trans. Graph. (SIGGRAPH 2003) 22(3), 724–731 (2003)Google Scholar
- 37.Tong, Y., Lombeyda, S., Hirani, A.N., Desbrun, M.: Discrete multiscale vector field decomposition. ACM Trans. Graph. 22(3), 445–452 (2003)Google Scholar
- 38.Treuille, A., Cooper, S., Popović, Z.: Continuum crowds. In: SIGGRAPH ’06: ACM SIGGRAPH 2006 Papers, pp. 1160–1168. ACM, New York (2006)Google Scholar
- 39.Tricoche, X., Scheuermann, G., Hagen, H.: Continuous topology simplification of planar vector fields. In: VIS ’01: Proceedings of the Conference on Visualization ’01, pp. 159–166. IEEE Computer Society, Washington, DC (2001)Google Scholar
- 40.Turk, G.: Texture synthesis on surfaces. In Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’01, pp. 347–354. ACM, New York, NY (2001). doi: 10. 1145/383259.383297Google Scholar
- 41.van Wijk, J.J.: Image based flow visualization. ACM Trans. Graph. (SIGGRAPH 2002) 21(3), 745–754 (2002)Google Scholar
- 42.van Wijk, J.J.: Image based flow visualization for curved surfaces. In Proceedings of the 14th IEEE Visualization 2003 (VIS’03), p. 17. IEEE Computer Society, Washington, DC (2003). doi: 10.1109/VISUAL.2003. 1250363Google Scholar
- 43.Wang, K., Weiwei, Tong, Y., Desbrun, M., Schröder, P.: Edge subdivision schemes and the construction of smooth vector fields. In: SIGGRAPH ’06: ACM SIGGRAPH 2006 Papers, pp. 1041–1048. ACM, New York (2006)Google Scholar
- 44.Wei, L.Y., Levoy, M.: Texture synthesis over arbitrary manifold surfaces. In: SIGGRAPH ’01: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, pp. 355–360. ACM, New York (2001)Google Scholar
- 45.Welch, W., Witkin, A.: Free-form shape design using triangulated surfaces. In: SIGGRAPH ’94: Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques, pp. 247–256. ACM, New York (1994)Google Scholar
- 46.van Wijk, J.J.: Spot noise texture synthesis for data visualization. In: SIGGRAPH ’91: Proceedings of the 18th Annual Conference on Computer Graphics and Interactive Techniques, pp. 309–318. ACM, New York (1991)Google Scholar
- 47.Wood, Z., Hoppe, H., Desbrun, M., Schröder, P.: Removing excess topology from isosurfaces. ACM Trans. Graph. 23(2), 190–208 (2004)Google Scholar
- 48.Xu, K., Cohne-Or, D., Ju, T., Liu, L., Zhang, H., Zhou, S., Xiong, Y.: Feature-aligned shape texturing. ACM Trans. Graph. (Proceedings SIGGRAPH Asia 2009) 28(5), Article 108 (2009)Google Scholar
- 49.Zhang, E., Mischaikow, K., Turk, G.: Feature-based surface parameterization and texture mapping. ACM Trans. Graph. 24, 1–27 (2005)Google Scholar
- 50.Zhang, E., Mischaikow, K., Turk, G.: Vector field design on surfaces. ACM Trans. Graph. 25(4), 1294–1326 (2006)Google Scholar
- 51.Zhang, E., Hays, J., Turk, G.: Interactive tensor field design and visualization on surfaces. IEEE Trans. Vis. Comput. Graph. 13(1), 94–107 (2007)Google Scholar
- 52.Zheng, X., Pang, A.: HyperLIC. In Proceedings of the 14th IEEE Visualization 2003 (VIS’03), p. 33. IEEE Computer Society, Washington, DC (2003)Google Scholar
- 53.Zhou, K., Huang, X., Wang, X., Tong, Y., Desbrun, M., Guo, B., Shum, H.Y.: Mesh quilting for geometric texture synthesis. ACM Trans. Graph. 25(3), 690–697 (2006)Google Scholar