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Fast iterative algorithms for solving the minimization of curvature-related functionals in surface fairing. (English) Zbl 1280.65022
Summary: A number of successful variational models for processing planar images have recently been generalized to three-dimensional (3D) surface processing. With this new dimensionality, the amount of numerical computations to solve the minimization of such new 3D formulations naturally grows up dramatically. Though the need of computationally fast and efficient numerical algorithms able to process high resolution surfaces is high, much less work has been done in this area. Recently, a two-step algorithm for the fast solution of the total curvature model was introduced by T. Tasdizen, R. Whitaker, P. Burchard and S. Osher [“Geometric surface processing via normal maps”, ACM Trans. Graph. 22(4), 1012–1033 (2003)]. In this paper, we generalize and modify this algorithm to the solution of analogues of the mean curvature model of M. Droske and M. Rumpf [Interfaces Free Bound. 6, No. 3, 361–378 (2004; Zbl 1062.35028)] and the Gaussian curvature model of M. Elsey and S. Esedoḡlu [Multiscale Model. Simul. 7, No. 4, 1549–1573 (2009; Zbl 1185.68803)]. Numerical experiments are shown to illustrate the good performance of the algorithms and test results.

MSC:
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49M30 Other numerical methods in calculus of variations (MSC2010)
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References:
[1] DOI: 10.1109/83.935036 · Zbl 1037.68771 · doi:10.1109/83.935036
[2] DOI: 10.1137/080737903 · Zbl 1205.68474 · doi:10.1137/080737903
[3] Brito-Loeza C., Int. J. Modern Math. 5 (2) pp 157– (2010)
[4] Burago Y., Geometric Inequalities (1988) · doi:10.1007/978-3-662-07441-1
[5] DOI: 10.1137/S0036139901390088 · Zbl 1028.68185 · doi:10.1137/S0036139901390088
[6] DOI: 10.1137/100788239 · Zbl 1218.65022 · doi:10.1137/100788239
[7] Desbrun , M. , Meyer , M. , Schröder , P. and Barr , A. Implicit fairing of irregular meshes using diffusion and curvature flow . Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’99) . pp. 317 – 324 . New York : ACM Press/Addison-Wesley Publishing Co .
[8] Do Carmo M., Differential Geometry of Curves and Surfaces, 1. ed. (1976)
[9] DOI: 10.4171/IFB/105 · Zbl 1062.35028 · doi:10.4171/IFB/105
[10] DOI: 10.1137/080736612 · Zbl 1185.68803 · doi:10.1137/080736612
[11] Eyre D. J., Comput. Math. Models Microstructural Evol. 53 pp 1686– (1998)
[12] DOI: 10.1016/j.cagd.2005.06.005 · Zbl 1084.53004 · doi:10.1016/j.cagd.2005.06.005
[13] DOI: 10.1109/TIP.2004.834662 · Zbl 1286.94022 · doi:10.1109/TIP.2004.834662
[14] DOI: 10.1016/0167-2789(92)90242-F · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[15] DOI: 10.1016/S0167-8396(01)00036-X · Zbl 0969.68154 · doi:10.1016/S0167-8396(01)00036-X
[16] Sumengen B., A Matlab Toolbox Implementing Level Set Methods
[17] Tasdizen , T. and Whitaker , R. Anisotropic diffusion of surface normals for feature preserving surface reconstruction . Proceedings, Fourth International Conference on 3-D Digital Imaging and Modeling: 3DIM 2003 . Boston , MA . pp. 353 – 360 .
[18] DOI: 10.1145/944020.944024 · Zbl 05457622 · doi:10.1145/944020.944024
[19] DOI: 10.1137/S0036142901396715 · Zbl 1035.65065 · doi:10.1137/S0036142901396715
[20] DOI: 10.1103/PhysRevE.68.066703 · doi:10.1103/PhysRevE.68.066703
[21] DOI: 10.1145/142920.134033 · doi:10.1145/142920.134033
[22] Welch , W. and Witkin , A. Free-form shape design using triangulated surfaces . Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’94) . pp. 247 – 256 . New York : ACM .
[23] DOI: 10.1137/090767558 · Zbl 1206.90245 · doi:10.1137/090767558
[24] DOI: 10.1016/j.apnum.2011.12.001 · Zbl 1416.65173 · doi:10.1016/j.apnum.2011.12.001
[25] DOI: 10.1137/110822268 · Zbl 1258.94021 · doi:10.1137/110822268
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