Summary
Angle Based Flattening is a robust parameterization technique allowing a free boundary. The numerical optimisation associated with the approach yields a challenging problem. We discuss several approaches to effectively reduce the computational effort involved and propose appropriate numerical solvers. We propose a simple but effective transformation of the problem which reduces the computational cost and simplifies the implementation. We also show that fast convergence can be achieved by finding approximate solutions which yield a low angular distortion.
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References
D.P. Bertsekas. Constrained optimization and lagrange multiplier methods. Athena Scientific, 1996.
R.H. Byrd and J. Nocedal. Active set and interior methods for nonlinear optimization Doc. MATH, Extra Volume ICM III, 1998, pp. 667–676.
M. Desbrun, M. Meyer, and P. Alliez. Intrinsic parameterizations of triangle meshes. Proc. Eurographics 2002, pp. 209–218.
G. Di Battista and L. Vismara. Angles of planar triangular graphs. SIAM Journal on Discrete Mathematics, 9(3), 1996, pp. 349–359.
M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle. Multiresolution analysis of arbitrary meshes. Proc. ACM SIGGRAPH '95, pp. 173–182.
M. S. Floater. Parametrization and smooth approximation of surface triangulations. Comp. Aided Geom. Design, (14), 3, 1997, pp. 231–250.
M. S. Floater and K. Hormann. Parameterization of triangulations and unorganized points. Tutorials on Multiresolution in Geometric Modelling Springer-Verlag, Heidelberg (2002), pp. 287–315.
M. S. Floater. Mean value coordinates. Comp. Aided Geom. Design, (20), 1, 2003, pp. 19–27.
M. S. Floater and K. Hormann. Surface parameterization: a tutorial and survey, Advances in Multiresolution for Geometric Modelling, N. A. Dodgson, M. S. Floater, and M. A. Sabin (eds.), Springer, 2004, pp. 157–186 (this book).
M. R. Hestenes and E. Stiefel. Method of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49:409–436, 1952.
A. Garg. New results on drawing angle graphs. Computational Geometry (9), (1–2), 1998, pp. 43–82.
A. Greenbaum. Iterative Methods for Solving Linear Systems SIAM, Philadelphia, 1997.
S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, and M. Halle. Conformal surface parameterization for texture mapping. IEEE Transactions on Visualization and Computer Graphics, 6(2), 2000, pp. 181–189.
K. Hormann and G. Greiner. MIPS: an efficient global parametrization method. Curve and Surface Design: Saint-Malo 1999, 2000, pp. 153–162.
B. Levy, S. Petitjean, N. Ray, and J. Maillot. Least squares conformal maps for automatic texture atlas generation. Proc. ACM SIGGRAPH 2002, pp. 362–371.
J. Liesen, E. de Sturler, A. Sheffer, Y. Aydin, and C. Siefert. Preconditioners for indefinite linear systems arising in surface parameterization. Proceedings of the 10th International Meshing Round Table, 2001, pp. 71–81.
C. Paige and M. Saunders. Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal, 12, 1975, pp. 617–629.
U. Pinkall and K. Polthier. Computing discrete minimal surfaces and their conjugates. Experimental Mathematics, 2(15), 1993, pp. 15–36.
P. Sander, J. Snyder, S. Gortler, and H. Hoppe. Texture mapping progressive meshes. Proc. ACM SIGGRAPH 2001, pp. 409–416.
A. Sheffer and E. de Sturler. Parameterization of faceted surfaces for meshing using angle based flattening. Engineering with Computers, 17(3), 2001, pp. 326–337.
A. Sheffer and E. de Sturler. Smoothing an overlay grid to minimize linear distortion in texture mapping. ACM Transactions on Graphics, 21(4), 2002.
O. Sorkine, D. Cohen-Or, R. Goldenthal, and D. Lischinski. Bounded-distortion piecewise mesh parameterization. Proc. IEEE Visualization 2002, pp. 355–362.
W. T. Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 13(3), 1963, pp. 743–768.
G. Zigelmann, R. Kimmel, and N. Kiryati. Texture mapping using surface flattening via multi-dimensional scaling. IEEE Transactions on Visualization and Computer Graphics, 8(2), 2002.
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Zayer, R., Rössl, C., Seidel, HP. (2005). Variations on Angle Based Flattening. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (eds) Advances in Multiresolution for Geometric Modelling. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26808-1_10
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DOI: https://doi.org/10.1007/3-540-26808-1_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21462-5
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