Abstract
We consider the problem of reconstructing 3D objects via meshfree interpolation methods. In this framework, we usually deal with large data sets and thus develop an efficient local scheme via the well-known partition of unity method. The main contribution in this paper consists in constructing the local interpolants for the implicit interpolation by means of rational radial basis functions. Numerical experiments, devoted to test the accuracy of the scheme, confirm that the proposed method is particularly performing when 3D objects, or more in general implicit functions defined by scattered points, need to be approximated.
Similar content being viewed by others
References
Babuška I, Melenk JM (1997) The partition of unity method. Int J Numer Methods Eng 40:727–758
Behzad M, Chartrand G, Lesniak-Foster L (1979) Graphs and Digraphs. Prindle, Weber and Schmidt, Boston
Belton D (2008) Improving and extending the information on principal component analysis for local neighborhoods in 3D point clouds. In: Jun C et al (eds) The international archives of the photogrammetry, remote sensing and spatial information sciences-Part B5, vol 37, pp 477–484
Bergamaschi L, Gambolati G, Pini G (1997) Asymptotic convergence of conjugate gradient methods for the partial symmetric eigenproblem. Numer Linear Algebra Appl 4:69–84
Bergamaschi L, Putti M (2002) Numerical comparison of iterative eigensolvers for large sparse symmetric matrices. Comput Methods Appl Mech Eng 191:5233–5247
Carr JC, Beatson RK, Cherrie JB, Mitchell TJ, Fright WR, Mccallum BC, Evans TR (2001) Reconstruction and representation of 3D objects with radial basis functions. In: Proceedings of the 28-th annual conference on computer graphics and interactive techniques. ACM Press, New York, pp 67–76
Cavoretto R, De Rossi A (2015) A trivariate interpolation algorithm using a cube-partition searching procedure. SIAM J Sci Comput 37:A1891–A1908
Cavoretto R, De Rossi A, Perracchione E (2018) Optimal selection of local approximants in RBF-PU interpolation. J Sci Comput 74:1–22
Cuomo S, Galletti A, Giunta G, Starace A (2013) Surface reconstruction from scattered point via RBF interpolation on GPU. In: Ganzha M et al (eds) 2013 federated conference on computer science and information systems. IEEE Press, Los Alamitos, pp 433–440
Cuomo S, Galletti A, Giunta G, Marcellino L (2017) Reconstruction of implicit curves and surfaces via RBF interpolation. Appl Numer Math 116:157–171
De Marchi S, Martínez E. Perracchione A (2017) Fast and stable rational RBF-based Partition of Unity interpolation (submitted)
De Marchi S, Santin G (2015) Fast computation of orthonormal basis for RBF spaces through Krylov space methods. BIT 55:949–966
De Rossi A, Perracchione E (2017) Positive constrained approximation via RBF-based partition of unity method. J Comput Appl Math 319:338–351
Fasshauer GE, McCourt MJ (2015) Kernel-based approximation methods using Matlab. World Scientific, Singapore
Fasshauer GE (2007) Meshfree approximations methods with Matlab. World Scientific, Singapore
Fornberg B, Larsson E, Flyer N (2011) Stable computations with Gaussian radial basis functions. SIAM J Sci Comput 33:869–892
Gould RJ (2012) Graph theory. Dover Publications, Mineola
Hoppe H (1994) Surface reconstruction from unorganized points. Ph.D. Thesis, University of Washington, Washington
Hoppe H, Derose T, Duchamp T, Mcdonald J, Stuetzle W (1992) Surface reconstruction from unorganized points. In: Thomas JJ (ed) Proceedings of the 19th annual conference on computer graphics and interactive techniques, vol 26. ACM Press, New York, pp 71–78
Hu XG, Ho TS, Rabitz H (2002) Rational approximation with multidimensional scattered data. Phys Rev 65:035701-1–035701-4
Jakobsson S, Andersson B, Edelvik F (2009) Rational radial basis function interpolation with applications to antenna design. J Comput Appl Math 233:889–904
Larsson E, Lehto E, Heryudono A, Fornberg B (2013) Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. SIAM J Sci Comput 35:A2096–A2119
Lehoucq RB, Sorensen DC (1996) Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J Matrix Anal Appl 17:789–821
Peigl L, Tiller W (1996) The NURBS book. Springer, Berlin
Perracchione E (2017) RBF-based partition of unity method: theory, algorithms and applications. Ph.D. Thesis, University of Torino
Sarra SA, Bay Y (2017) A rational radial basis function method for accurately resolving discontinuities and steep gradients (preprint)
Shepard D (1968) A two-dimensional interpolation function for irregularly spaced data. In: Proceedings of 23rd national conference. Brandon/Systems Press, Princeton, pp 517–524
Wendland H (2005) Scattered data approximation, Cambridge Monogr. Appl. Comput. Math., vol 17, Cambridge Univ. Press, Cambridge
Wendland H (2002) Fast evaluation of radial basis functions: methods based on partition of unity. In: Chui CK et al (eds) Approximation theory X: wavelets, splines, and applications. Vanderbilt Univ. Press, Nashville, pp 473–483
Acknowledgements
This research has been accomplished within Rete ITaliana di Approssimazione (RITA) and supported by: GNCS-INdAM and the research project Radial basis functions approximations: stability issues and applications, No. BIRD167404.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Cristina Turner.
Rights and permissions
About this article
Cite this article
Perracchione, E. Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects. Comp. Appl. Math. 37, 4633–4648 (2018). https://doi.org/10.1007/s40314-018-0592-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-018-0592-8
Keywords
- Meshless approximation
- Partition of unity method
- Rational RBF approximation
- Reconstruction of 3D objects