Remeshing-assisted Optimization for Locally Injective Mappings
Constructing locally injective mappings for 2D triangular meshes is vital in applications such as deformations. In such a highly constrained optimization, the prescribed tessellation may impose strong restriction on the solution. As a consequence, the feasible region may be too small to contain an i...
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| Published in: | Computer graphics forum Vol. 33; no. 5; pp. 269 - 279 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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Oxford
Blackwell Publishing Ltd
01.08.2014
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | Constructing locally injective mappings for 2D triangular meshes is vital in applications such as deformations. In such a highly constrained optimization, the prescribed tessellation may impose strong restriction on the solution. As a consequence, the feasible region may be too small to contain an ideal solution, which leads to problems of slow convergence, poor solution, or even that no solution can be found. We propose to integrate adaptive remeshing into interior point method to solve this issue. We update the vertex positions via a parameter‐free relaxation enhanced geometry optimization, and then use edge‐flip operations to reduce the residual and keep a reasonable condition number for better convergence. For more robustness, when the iteration of interior point method terminates but leaves the positional constraints unsatisfied, we estimate the edges in the current tessellation that block vertices moving based on the convergence information of the optimization, and then split neighboring edges to break the restriction. The results show that our method has better performance than the solely geometric optimization approaches, especially for extreme deformations. |
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| AbstractList | Constructing locally injective mappings for 2D triangular meshes is vital in applications such as deformations. In such a highly constrained optimization, the prescribed tessellation may impose strong restriction on the solution. As a consequence, the feasible region may be too small to contain an ideal solution, which leads to problems of slow convergence, poor solution, or even that no solution can be found. We propose to integrate adaptive remeshing into interior point method to solve this issue. We update the vertex positions via a parameter‐free relaxation enhanced geometry optimization, and then use edge‐flip operations to reduce the residual and keep a reasonable condition number for better convergence. For more robustness, when the iteration of interior point method terminates but leaves the positional constraints unsatisfied, we estimate the edges in the current tessellation that block vertices moving based on the convergence information of the optimization, and then split neighboring edges to break the restriction. The results show that our method has better performance than the solely geometric optimization approaches, especially for extreme deformations. Constructing locally injective mappings for 2D triangular meshes is vital in applications such as deformations. In such a highly constrained optimization, the prescribed tessellation may impose strong restriction on the solution. As a consequence, the feasible region may be too small to contain an ideal solution, which leads to problems of slow convergence, poor solution, or even that no solution can be found. We propose to integrate adaptive remeshing into interior point method to solve this issue. We update the vertex positions via a parameter-free relaxation enhanced geometry optimization, and then use edge-flip operations to reduce the residual and keep a reasonable condition number for better convergence. For more robustness, when the iteration of interior point method terminates but leaves the positional constraints unsatisfied, we estimate the edges in the current tessellation that block vertices moving based on the convergence information of the optimization, and then split neighboring edges to break the restriction. The results show that our method has better performance than the solely geometric optimization approaches, especially for extreme deformations. [PUBLICATION ABSTRACT] |
| Author | Jin, Y. Huang, J. Tong, R. |
| Author_xml | – sequence: 1 givenname: Y. surname: Jin fullname: Jin, Y. organization: State Key Laboratory of CAD & CG, Zhejiang University, China – sequence: 2 givenname: J. surname: Huang fullname: Huang, J. email: hj@cad.zju.edu.cn organization: State Key Laboratory of CAD & CG, Zhejiang University, China – sequence: 3 givenname: R. surname: Tong fullname: Tong, R. organization: State Key Laboratory of CAD & CG, Zhejiang University, China |
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| Copyright | 2014 The Author(s) Computer Graphics Forum © 2014 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2014 The Eurographics Association and John Wiley & Sons Ltd. |
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| References_xml | – reference: Gortler S.J., Gotsman C., Thurston D.: Discrete one-forms on meshes and applications to 3d mesh parameterization. Computer Aided Geometric Design 23, 2 (2006), 83-112. 2 – reference: Trefethen L.N., Bau III D.: Numerical linear algebra, vol. 50. Siam, 1997. 5 – reference: Chen Y., Davis T.A., Hager W.W., Rajamanickam S.: Algorithm 887: Cholmod, supernodal sparse cholesky factorization and update/downdate. ACM Transactions on Mathematical Software (TOMS) 35, 3 (2008), 22. 6 – reference: Aigerman N., Lipman Y.: Injective and bounded distortion mappings in 3d. ACM Transactions on Graphics (TOG) 32, 4 (2013), 106. 1, 2, 6, 7, 8, 10 – reference: Schneider T., Hormann K.S. Floater M.: Bijective composite mean value mappings. Computer Graphics Forum 32, 5 (2013), 137-146. 2 – reference: Floater M.: One-to-one piecewise linear mappings over triangulations. Mathematics of Computation 72, 242 (2003), 685-696. 1, 2 – reference: Kraevoy V., Sheffer A., Gotsman C.: Matchmaker: constructing constrained texture maps. ACM Transactions on Graphics (TOG) 22, 3 (2003), 326-333. 1, 3 – reference: Schüller C., Kavan L., Panozzo D., Sorkine-Hornung O.: Locally injective mappings. Computer Graphics Forum (proceedings of EUROGRAPHICS/ACM SIGGRAPH Symposium on Geometry Processing) 32, 5 (2013), 125-135. 1, 2, 3, 4, 6, 8, 9 – reference: Lévy B., Petitjean S., Ray N., Maillot J.: Least squares conformal maps for automatic texture atlas generation. ACM Transactions on Graphics (TOG) 21, 3 (2002), 362-371. 6 – reference: Bommes D., Zimmer H., Kobbelt L.: Mixed-integer quadrangulation. ACM Transactions on Graphics 28, 3 (2009), 77-77. 2 – reference: Lipman Y.: Bounded distortion mapping spaces for triangular meshes. ACM Transactions on Graphics (TOG) 31, 4 (2012), 108. 1, 2, 6, 7, 8, 9, 10 – reference: Yu H., Lee T.-Y., Yeh I.-C., Yang X., Li W., Zhang J.J.: An rbf-based reparameterization method for constrained texture mapping. IEEE Transactions on Visualization and Computer Graphics 18, 7 (2012), 1115-1124. 2 – reference: Lee T.-Y., Yen S.-W., Yeh I.-C.: Texture mapping with hard constraints using warping scheme. IEEE Transactions on Visualization and Computer Graphics 14, 2 (2008), 382-395. 1, 3, 5, 6, 10 – reference: Fujimura K., Makarov M.: Foldover-free image warping. Graphical Models and Image Processing 60, 2 (1998), 100-111. 3 – reference: Liu L., Zhang L., Xu Y., Gotsman C., Gortler S.: A local/global approach to mesh parameterization. Computer Graphics Forum 27, 5 (2008), 1495-1504. 2, 6 – reference: Tutte W.T.: How to draw a graph. Proc. London Math. Soc 13, 3 (1963), 743-768. 1, 2 – reference: Sheffer A., Lévy B., Mogilnitsky M., Bogomyakov A.: Abf++: fast and robust angle based flattening. ACM Transactions on Graphics (TOG) 24, 2 (2005), 311-330. 2 – reference: Xu Y., Chen R., Gotsman C., Liu L.: Embedding a triangular graph within a given boundary. Computer Aided Geometric Design 28, 6 (2011), 349-356. 2 – reference: Eckstein I., Surazhsky V., Gotsman C.: Texture mapping with hard constraints. Computer Graphics Forum 20, 3 (2001), 95-104. 3 – reference: Kraevoy V., Sheffer A.: Cross-parameterization and compatible remeshing of 3d models. ACM Transactions on Graphics (TOG) 23, 3 (2004), 861-869. 3 – reference: Athanasiadis T., Zioupos G., Fudos I.: Efficient computation of constrained parameterizations on parallel platforms. Computers & Graphics 37, 6 (2013), 596-607. 2 – reference: Weber O., Myles A., Zorin D.: Computing extremal quasiconformal maps. 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| Snippet | Constructing locally injective mappings for 2D triangular meshes is vital in applications such as deformations. In such a highly constrained optimization, the... |
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| SubjectTerms | Analysis and systems Categories and Subject Descriptors (according to ACM CCS) Computer graphics Constrictions Convergence Deformation Graph theory I.3.5 [Computer Graphics] Computational Geometry and Object Modeling-Geometric algorithms I.3.5 [Computer Graphics] Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems languages Mapping Mathematical models Optimization Studies Tessellation |
| Title | Remeshing-assisted Optimization for Locally Injective Mappings |
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