CSRBF-Based Quasi-interpolation for Accurate and Fast Data Fitting

In this paper, quasi-interpolation based on compactly supported radial basis functions (CSRBFs) is presented for more accurate and efficient data fitting compared with global RBFs. Firstly, a CSRBF-based quasi-interpolator is constructed considering only the positions of the given data and their values. Then we make use of the first derivatives to propose a new quasi-interpolator which can achieve higher approximate order and better shape-preserving. Numerical examples demonstrate that the proposed CSRBF-based quasi-interpolation schemes are valid.

[1]  R. Schaback.  Creating Surfaces from Scattered Data Using Radial Basis Functions , 1995 .

[2]  M. J. D. Powell,et al.  Radial basis functions for multivariable interpolation: a review , 1987 .

[3]  L. Montefusco,et al.  Radial basis functions for the multivariate interpolation of large scattered data sets , 2002 .

[4]  Charlie C. L. Wang,et al.  Quasi-interpolation for surface reconstruction from scattered data with radial basis function , 2012, Comput. Aided Geom. Des..

[5]  Gregory E. Fasshauer,et al.  Solving differential equations with radial basis functions: multilevel methods and smoothing , 1999, Adv. Comput. Math..

[6]  Xuli Han,et al.  Quasi-interpolation for Data Fitting by the Radial Basis Functions , 2008, GMP.

[7]  Holger Wendland,et al.  Meshless Galerkin methods using radial basis functions , 1999, Math. Comput..

[8]  Holger Wendland,et al.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[9]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[10]  F. J. Sainz,et al.  Constructive approximate interpolation by neural networks , 2006 .

[11]  Holger Wendland,et al.  Scattered Data Approximation: Conditionally positive definite functions , 2004 .

[12]  Weixiang Zhang,et al.  Shape-preserving MQ-B-splines quasi-interpolation , 2004, Geometric Modeling and Processing, 2004. Proceedings.

[13]  Zongmin Wu,et al.  Compactly supported positive definite radial functions , 1995 .

[14]  Leevan Ling.  Multivariate quasi-interpolation schemes for dimension-splitting multiquadric , 2005, Appl. Math. Comput..

[15]  Ren-Hong Wang,et al.  High accuracy multiquadric quasi-interpolation , 2011 .

[16]  H. Seidel,et al.  Multi-level partition of unity implicits , 2003 .

[17]  E. Kansa,et al.  Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations , 2000 .

[18]  Hans-Peter Seidel,et al.  3D scattered data interpolation and approximation with multilevel compactly supported RBFs , 2005, Graph. Model..

[19]  Martin D. Buhmann.  Radial Basis Functions: Theory and Implementations: Radial basis functions with compact support , 2003 .

[20]  Benny Y. C. Hon,et al.  An efficient numerical scheme for Burgers' equation , 1998, Appl. Math. Comput..

[21]  Carsten Franke,et al.  Convergence order estimates of meshless collocation methods using radial basis functions , 1998, Adv. Comput. Math..

[22]  Paolo Cignoni,et al.  Metro: Measuring Error on Simplified Surfaces , 1998, Comput. Graph. Forum.

[23]  Alessandra Sestini,et al.  Local hybrid approximation for scattered data fitting with bivariate splines , 2006, Comput. Aided Geom. Des..

[24]  Y. Hon,et al.  A numerical computation for inverse boundary determination problem , 2000 .

[25]  Michael M. Kazhdan,et al.  Screened poisson surface reconstruction , 2013, TOGS.

[26]  M. Golberg,et al.  Improved multiquadric approximation for partial differential equations , 1996 .

[27]  Jun Wang,et al.  Multi-level hermite variational interpolation and quasi-interpolation , 2013, The Visual Computer.

[28]  C. Micchelli,et al.  Approximation by superposition of sigmoidal and radial basis functions , 1992 .

[29]  Chen Debao.  Degree of approximation by superpositions of a sigmoidal function , 1993, Approximation Theory and its Applications.

[30]  Zongmin Wu,et al.  Local error estimates for radial basis function interpolation of scattered data , 1993 .

[31]  R. Schaback.  Multivariate Interpolation by Polynomials and Radial Basis Functions , 2005 .

[32]  Robert Schaback,et al.  Error estimates and condition numbers for radial basis function interpolation , 1995, Adv. Comput. Math..