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Region-Based Line Field Design Using Harmonic Functions
June 2012 (vol. 18 no. 6)
pp. 902-913
Chih-Yuan Yao, National Cheng-Kung University, Tainan
Ming-Te Chi, National Chengchi University, Taipei City
Tong-Yee Lee, National Cheng-Kung University, Tainan
Tao Ju, Washington University, St. Louis
Field design has wide applications in graphics and visualization. One of the main challenges in field design has been how to provide users with both intuitive control over the directions in the field on one hand and robust management of its topology on the other hand. In this paper, we present a design paradigm for line fields that addresses this challenge. Rather than asking users to input all singularities as in most methods that offer topology control, we let the user provide a partitioning of the domain and specify simple flow patterns within the partitions. Represented by a selected set of harmonic functions, the elementary fields within the partitions are then combined to form continuous fields with rich appearances and well-determined topology. Our method allows a user to conveniently design the flow patterns while having precise and robust control over the topological structure. Based on the method, we developed an interactive tool for designing line fields from images, and demonstrated the utility of the fields in image stylization.

[1] W.A. Barrett and E.N. Mortensen, “Interactive Live-Wire Boundary Extraction,” Medical Image Analysis, vol. 1, pp. 331-341, 1997.
[2] G. Chen, G. Esch, P. Wonka, P. Müller, and E. Zhang, “Interactive Procedural Street Modeling,” ACM Trans. Graphics, vol. 27, no. 3, pp. 103:1-103:10, 2008.
[3] G. Chen, K. Mischaikow, R.S. Laramee, P. Pilarczyk, and E. Zhang, “Vector Field Editing and Periodic Orbit Extraction Using Morse Decomposition,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 4, pp. 769-785, July 2007.
[4] M.-T. Chi and T.-Y. Lee, “Stylized and Abstract Painterly Rendering System Using a Multiscale Segmented Sphere Hierarchy,” IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 1, pp. 61-72, Jan./Feb. 2006.
[5] K. Crane, M. Desbrun, and P. Schröder, “Trivial Connections on Discrete Surfaces,” Computer Graphics Forum, vol. 29, no. 5, pp. 1525-1533, 2010.
[6] T. Delmarcelle and L. Hesselink, “The Topology of Symmetric, Second-Order Tensor Fields,” VIS '94: Proc. IEEE Conf. Visualization '94, pp. 140-147, 1994.
[7] L. Fan, S. Wang, H. Wang, and T. Guo, “Singular Points Detection Based on Zero-Pole Model in Fingerprint Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 30, no. 6, pp. 929-940, June 2008.
[8] M.S. Floater, “Mean Value Coordinates,” Computer Aided Geometric Design, vol. 20, no. 1, pp. 19-27, 2003.
[9] X.T. Gerik, X. Tricoche, G. Scheuermann, H. Hagen, and S. Clauss, “Scaling the Topology of Symmetric, Second-Order Tensor Fields,” Proc. NSF/DOE Lake Tahoe Workshop Hierarchical Approximation and Geometrical Methods for Scientific Visualization, pp. 171-184, 2002.
[10] J. Hays and I.A. Essa, “Image and Video Based Painterly Animation,” Proc. Third Int'l Symp. Non-Photorealistic Animation and Rendering (NPAR), pp. 113-120, 2004.
[11] A. Hertzmann, “Painterly Rendering with Curved Brush Strokes of Multiple Sizes,” Proc. SIGGRAPH '98, pp. 453-460, 1998.
[12] F. Kalberer, M. Nieser, and K. Polthier, “Quadcover - Surface Parameterization Using Branched Coverings,” Computer Graphics Forum, vol. 26, no. 3, pp. 375-384, 2007.
[13] H. Kang, S. Lee, and C.K. Chui, “Flow-Based Image Abstraction,” IEEE Trans. Visualization and Computer Graphics, vol. 15, no. 1, pp. 62-76, Jan./Feb. 2009.
[14] Y.-K. Lai, M. Jin, X. Xie, Y. He, J. Palacios, E. Zhang, S.-M. Hu, and X. Gu, “Metric-Driven Rosy Field Design and Remeshing,” IEEE Trans. Visualization and Computer Graphics, vol. 16, no. 1, pp. 95-108, Jan./Feb. 2010.
[15] X. Ni, M. Garland, and J.C. Hart, “Fair Morse Functions for Extracting the Topological Structure of a Surface Mesh,” ACM Trans. Graphics, vol. 23, no. 3, pp. 613-622, 2004.
[16] J. Palacios and E. Zhang, “Rotational Symmetry Field Design on Surfaces,” ACM Trans. Graphics, vol. 26, no. 3, pp. 55:1-55:10, 2007.
[17] N. Ray, W.C. Li, B. Lévy, A. Sheffer, and P. Alliez, “Periodic Global Parameterization,” ACM Trans. Graphics, vol. 25, no. 4, pp. 1460-1485, 2006.
[18] N. Ray, B. Vallet, L. Alonso, and B. Levy, “Geometry-Aware Direction Field Processing,” ACM Trans. Graphics, vol. 29, no. 1, pp. 1:1-1:11, 2009.
[19] N. Ray, B. Vallet, W.C. Li, and B. Lévy, “N-Symmetry Direction Field Design,” ACM Trans. Graphics, vol. 27, no. 2, pp. 10:1-10:13, 2008.
[20] B. Sherlock and D. Monro, “A Model for Interpreting Fingerprint Topology,” Pattern Recognition, vol. 26, no. 7, pp. 1047-1055, 1993.
[21] G. Taubin, “A Signal Processing Approach to Fair Surface Design,” Proc. SIGGRAPH '95, pp. 351-358, 1995.
[22] H. Theisel, “Designing 2D Vector Fields of Arbitrary Topology,” Computer Graphics Forum (Eurographics), vol. 21, pp. 595-604, 2002.
[23] Y. Tong, P. Alliez, D. Cohen-Steiner, and M. Desbrun, “Designing Quadrangulations with Discrete Harmonic Forms,” SGP '06: Proc. Fourth Eurographics Symp. Geometry Processing, pp. 201-210, 2006.
[24] Y. Tong, S. Lombeyda, A.N. Hirani, and M. Desbrun, “Discrete Multiscale Vector Field Decomposition,” ACM Trans. Graphics, vol. 22, no. 3, pp. 445-452, 2003.
[25] X. Tricoche, G. Scheuermann, and H. Hagen, “Continuous Topology Simplification of Planar Vector Fields,” VIS '01: Proc. Conf. Visualization '01, pp. 159-166, 2001.
[26] J.L. Walsh, The Location of Critical Points of Analytic and Harmonic Functions. Am. Math. Soc., 1950.
[27] N. Wiener, “Discontinuous Boundary Conditions and the Dirichlet Problem,” Trans. the Am. Math. Soc., vol. 25, no. 3, pp. 307-314, 1923.
[28] K. Xu, D. Cohen-Or, T. Ju, L. Liu, H. Zhang, S. Zhou, and Y. Xiong, “Feature-Aligned Shape Texturing,” ACM Trans. Graphics, vol. 28, no. 5, pp. 108:1-108:7, 2009.
[29] C.-R. Yen, M.-T. Chi, T.-Y. Lee, and W.-C. Lin, “Stylized Rendering Using Samples of a Painted Image,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 2, pp. 468-480, Mar./Apr. 2008.
[30] E. Zhang, J. Hays, and G. Turk, “Interactive Tensor Field Design and Visualization on Surfaces,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 1, pp. 94-107, Jan./Feb. 2007.
[31] E. Zhang, K. Mischaikow, and G. Turk, “Vector Field Design on Surfaces,” ACM Trans. Graphics, vol. 25, no. 4, pp. 1294-1326, 2006.

Index Terms:
Field design, line field, singularity, harmonic functions.
Citation:
Chih-Yuan Yao, Ming-Te Chi, Tong-Yee Lee, Tao Ju, "Region-Based Line Field Design Using Harmonic Functions," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 6, pp. 902-913, June 2012, doi:10.1109/TVCG.2011.112
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