×

zbMATH — the first resource for mathematics

A finite element method for surface restoration with smooth boundary conditions. (English) Zbl 1069.65546
Summary: In surface restoration usually a damaged region of a surface has to be replaced by a surface patch which restores the region in a suitable way. In particular one aims for \(C^{1}\)-continuity at the patch boundary. The Willmore energy is considered to measure fairness and to allow appropriate boundary conditions to ensure continuity of the normal field. The corresponding \(L^{2}\)-gradient flow as the actual restoration process leads to a system of fourth order partial differential equations, which can also be written as a system of two coupled second order equations. As it is well known, fourth order problems require an implicit time discretization. Here a semi-implicit approach is presented which allows large time steps. For the discretization of the boundary condition, two different numerical methods are introduced. Finally, we show applications to different surface restoration problems.

MSC:
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
65D17 Computer-aided design (modeling of curves and surfaces)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Software:
CHARMS
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ambrosio, L.; Masnou, S., A direct variational approach to a problem arising in image reconstruction, Interfaces free bound., 5, 1, 63-81, (2003) · Zbl 1029.49037
[2] Ballester, C.; Bertalmio, M.; Caselles, V.; Sapiro, G.; Verdera, J., Filling-in by joint interpolation of vector fields and grey levels, IEEE trans. image processing, 10, 1200-1211, (2001) · Zbl 1037.68771
[3] Bertalmio, M.; Sapiro, G.; Caselles, V.; Ballester, C., Image inpainting, (), 417-424
[4] Bertalmio, M.; Bertozzi, A.; Sapiro, G., Navier – stokes, fluid dynamics, and image and video inpainting, Proceedings of the international conference on computer vision and pattern recognition, IEEE, I, 355-362, (2001)
[5] Bertalmio, M.; Cheng, L.-T.; Osher, S.; Sapiro, G., Variational problems and partial differential equations on implicit surfaces, J. comput. phys., 174, 759-780, (2001) · Zbl 0991.65055
[6] Caselles, V.; Morel, J.-M.; Sbert, C., An axiomatic approach to image interpolation, IEEE trans. image processing, 7, 3, 376-386, (1998) · Zbl 0993.94504
[7] Chan, T.F.; Kang, S.H.; Shen, J., Euler’s elastica and curvature-based inpainting, SIAM appl. math., 63, 2, 564-592, (2002) · Zbl 1028.68185
[8] Chan, T., Shen, J., On the role of the bv image model in image restoration. AMS Contemporary Mathematics, submitted for publication · Zbl 1035.94501
[9] Ciarlet, P., Mathematical elasticity, vol. III: theory of shells, (2000), North-Holland Amsterdam
[10] Davis, J.; Marschner, S.; Garr, M.; Levoy, M., Filling holes in complex surfaces using volumetric diffusion, ()
[11] Dziuk, G., An algorithm for evolutionary surfaces, Numer. math., 58, 603-611, (1991) · Zbl 0714.65092
[12] Dziuk, G.; Kuwert, E.; Schätzle, R., Evolution of elastic curves in \(R\^{}\{n\}\): existence and computation, SIAM J. math. anal., 33, 5, 1228-1245, (2002), (electronic) · Zbl 1031.53092
[13] Farin, G., Curves and surfaces for computer-aided geometric design, Computer science and scientific computing, (1997), Academic Press San Diego, CA · Zbl 0919.68120
[14] Gilbarg, D.; Trudinger, N., Elliptic partial differential equations of second order, Grundlehren der mathematischen wissenschaften, vol. 224, (1992), Springer-Verlag Berlin · Zbl 0691.35001
[15] Giusti, E., Minimal surfaces and functions of bounded variation, (1984), Birkhäuser · Zbl 0545.49018
[16] Greiner, G., Variational design and fairing of spline surfaces, Computer graphics forum (proc. eurographics ’94), 13, 3, 143-154, (1994)
[17] Greiner, G.; Loos, J.; Wesselink, W., Data dependent thin plate energy and its use in interactive surface modeling, Computer graphics forum (proc. eurographics ’96), 15, 3, 175-186, (1996)
[18] Grinspun, E.; Krysl, P.; Schröder, P., CHARMS: a simple framework for adaptive simulation, () · Zbl 1396.65043
[19] Jones, T.; Durand, F.; Desbrun, M., Non-iterative, feature-preserving mesh smoothing, (), 943-949
[20] Kobbelt, L.; Campagna, S.; Vorsatz, J.; Seidel, H.-P., Interactive multi-resolution modeling on arbitrary meshes, (), 105-114
[21] Kuwert, E.; Schätzle, R., The Willmore flow with small initial energy, J. differential geom., 57, 3, 409-441, (2001) · Zbl 1035.53092
[22] Kuwert, E.; Schätzle, R., Gradient flow for the Willmore functional, Comm. anal. geom., 10, 2, 307-339, (2002) · Zbl 1029.53082
[23] Kuwert, E., Schätzle, R., 2002b. Removability of point singularities of Willmore surfaces. Preprint SFB 611, Bonn No. 47
[24] Masnou, S.; Morel, J.-M., Level lines based disocclusion, (), 259-263
[25] Mumford, D., Elastica and computer vision, (), 491-506 · Zbl 0798.53003
[26] Museth, K.; Breen, D.; Whitaker, R.; Barr, A., Level set surface editing operators, (), 330-338
[27] Nitzberg, M.; Mumford, D.; Shiota, T., Filtering, segmentation and depth, Lecture notes in computer science, vol. 662, (1993), Springer-Verlag Berlin · Zbl 0801.68171
[28] Nitsche, J.C.C., Boundary value problems for variational integrals involving surfaces curvatures, Quaterly appl. math., LI, 2, 363-387, (1993) · Zbl 0785.35027
[29] Nitsche, J., Periodic surfaces that are extremal for energy functionals containing curvature functions, () · Zbl 0794.53007
[30] Polden, A., 1995. Closed curves of least total curvature. SFB 382 Tübingen, Preprint 13
[31] Polden, A., 1996. Curves and surfaces of least total curvature and fourth-order flows. Dissertation, Universität Tübingen
[32] Rusu, R., 2001. An algorithm for the elastic flow of surfaces. Preprint Mathematische Fakultät Freiburg 01-35
[33] Schneider, R.; Kobbelt, L., Discrete fairing of curves and surfaces based on linear curvature distribution, (), 371-380
[34] Schneider, R.; Kobbelt, L., Generating fair meshes with G1 boundary conditions, (), 251-261
[35] Sochen, N.; Kimmel, R.; Malladi, R., A general framework for low level vision, IEEE trans. image processing, 7, 3, 310-318, (1998) · Zbl 0973.94502
[36] Tasdizen, T., Whitaker, R., Burchard, P., Osher, S., Geometric surface processing via normal maps. ACM TOG, submitted for publication
[37] Tasdizen, T.; Whitaker, R.; Burchard, P.; Osher, S., Geometric surface smoothing via anisotropic diffusion of normals, (), 125-132
[38] Thomée, V., Galerkin—finite element methods for parabolic problems, (1984), Springer Berlin · Zbl 0528.65052
[39] Verdera, J., Caselles, V., Bertalmio, M., Sapiro, G., 2003. Inpainting surface holes. Preprint no. 1905, IMA, University of Minnesota
[40] von der Mosel, H., Geometrische variationsprobleme höherer ordnung, Bonner mathematische schriften, 293, (1996) · Zbl 0888.49031
[41] Willmore, T., Riemannian geometry, (1993), Clarendon Press Oxford · Zbl 0797.53002
[42] Yoshizawa, S.; Belyaev, A., Fair triangle mesh generation with discrete elastica, (), 119-123
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.