×

A semi-Lagrangian scheme with radial basis approximation for surface reconstruction. (English) Zbl 1364.65160

Summary: We propose a Semi-Lagrangian scheme coupled with Radial Basis Function interpolation for approximating a curvature-related level set model, which has been proposed by H.-K. Zhao et al. [Comput. Vis. Image Underst. 80, No. 3, 295–314 (2000; Zbl 1011.68538)] to reconstruct unknown surfaces from sparse data sets. The main advantages of the proposed scheme are the possibility to solve the level set method on unstructured grids, as well as to concentrate the reconstruction points in the neighbourhood of the data set, with a consequent reduction of the computational effort. Moreover, the scheme is explicit. Numerical tests show the accuracy and robustness of our approach to reconstruct curves and surfaces from relatively sparse data sets.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 1011.68538

Software:

Mfree2D
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Behrens, J., Iske, A.: Grid-free adaptive semi-Lagrangian advection using radial basis functions. Comput. Math. Appl. 43, 319-327 (2002) · Zbl 0999.65104 · doi:10.1016/S0898-1221(01)00289-9
[2] Carlini, E., Falcone, M., Ferretti, R.: Convergence of a large time-step scheme for mean curvature motion. Interf. Free Bound. 12, 409-441 (2010) · Zbl 1416.65304 · doi:10.4171/IFB/240
[3] Carlini, E., Ferretti, R.: A Semi-Lagrangian scheme for area-preserving flows. In: Proceedings of the conference “ICPR2012”, IEEExplore (2012) · Zbl 1240.65069
[4] Carlini, E., Ferretti, R.: A semi-Lagrangian approximation for the AMSS model of image processing. Appl. Numer. Math. 73, 16-32 (2013) · Zbl 1302.65051 · doi:10.1016/j.apnum.2012.07.003
[5] Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R., McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3d objects with radial basis functions. In: Proceedings of ACM SIGGRAPH, pp. 67-76 (2001) · Zbl 1203.65044
[6] Carr, J.C., Fright, W.R., Beatson, R.K.: Surface interpolation with radial basis functions for medical imaging. IEEE Trans. Med. Imag. 16, 96-107 (1997) · doi:10.1109/42.552059
[7] Chirokov, A.: Interpolation and approximation using radial base function (RBF). http://www.mathworks.com/
[8] Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1-67 (1992) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[9] Falcone, M., Ferretti, R.: Consistency of a Large Time-Step Scheme for Mean Curvature Motion, Numerical Mathematics and Advanced Applications—ENUMATH 2001. Springer, Milano (2003) · Zbl 1045.65072
[10] Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, Philadelphia (2013) · Zbl 1335.65001 · doi:10.1137/1.9781611973051
[11] Franchini, E., Morigi, S., Sgallari, F.: Implicit shape reconstruction of unorganized points using PDE-based deformable 3D manifolds. Numer. Math. Theory Methods Appl. 3, 405-430 (2010) · Zbl 1240.65069
[12] Goldstein, T., Bresson, X., Osher, S.: Geometric applications of the Split Bregman method: segmentation and surface reconstruction. J. Sci. Comput. 45, 272-293 (2009) · Zbl 1203.65044 · doi:10.1007/s10915-009-9331-z
[13] Li, Y., Lee, D., Lee, C., Lee, J., Lee, S., Kim, J., Kim, J., Ahn, S.: Surface embedding narrow volume reconstruction from unorganized points. Comput. Vis. Image Underst. 121, 100-107 (2014) · doi:10.1016/j.cviu.2014.02.002
[14] Liang, J., Park, F., Zhao, H.: Robust and efficient implicit surface reconstruction for point clouds based on convexified image segmentation. J. Sci. Comput. 54, 577-602 (2013) · doi:10.1007/s10915-012-9674-8
[15] Liu, G.R.: Mesh Free Methods: Moving Beyond the Finite Element Method. CRC Press, Boca Raton (2002) · doi:10.1201/9781420040586
[16] Savchenko, V.V., Pasko, A.A., Okunev, O.G., Kunii, T.L.: Function representation of solids reconstructed from scattered surface points and contours. Comput. Gr. Forum 14, 181-188 (1995) · doi:10.1111/1467-8659.1440181
[17] Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005) · Zbl 1075.65021
[18] Ye, J., Yanowsky, I., Dong, B., Gandlin, R., Brandt, A., Osher, S.: Multigrid narrow band surface reconstruction via level set functions. In: Proceedings of the 8th International Symposium on Visual Computing—ISVC 2012 (Rethymnon, Crete), pp. 61-70. Springer, Berlin (2012)
[19] Tsai, Y.R., Osher, S.: Total variation and level set methods in image science. Acta Numer. 14, 509-573 (2005) · Zbl 1119.65376 · doi:10.1017/S0962492904000273
[20] Zhao, H., Osher, S., Merriman, B., Kang, M.: Implicit and non-parametric shape reconstruction from unorganized points using variational level set method. Comput. Vis. Image Underst. 80, 295-319 (2000) · Zbl 1011.68538 · doi:10.1006/cviu.2000.0875
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.