CS-RBF interpolation of surfaces with vertical faults from scattered data (original) (raw)
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Quasi-interpolation for surface reconstruction from scattered data with radial basis function
Computer Aided Geometric Design, 2012
Radial Basis Function (RBF) has been used in surface reconstruction methods to interpolate or approximate scattered data points, which involves solving a large linear system. The linear systems for determining coefficients of RBF may be ill-conditioned when processing a large point set, which leads to unstable numerical results. We introduce a quasiinterpolation framework based on compactly supported RBF to solve this problem. In this framework, implicit surfaces can be reconstructed without solving a large linear system. With the help of an adaptive space partitioning technique, our approach is robust and can successfully reconstruct surfaces on non-uniform and noisy point sets. Moreover, as the computation of quasi-interpolation is localized, it can be easily parallelized on multi-core CPUs.
Interpolation or approximation of scattered data is very often task in engineering problems. The Radial Basis Functions (RBF) interpolation is convenient for scattered (un-ordered) data sets in k-dimensional space, in general. This approach is convenient especially for a higher dimension k > 2 as the conversion to an ordered data set, e.g. using tessellation, is computationally very expensive. The RBF interpolation is not separable and it is based on distance of two points. It leads to a solution of a Linear System of Equations (LSE) =. There are two main groups of interpolating functions: 'global" and "local". Application of "local" functions, called Compactly Supporting RBF (CSFBF), can significantly decrease computational cost as they lead to a system of linear equations with a sparse matrix. In this paper the RBF interpolation theory is briefly introduced at the "application level" including some basic principles and computational issues and an incremental RBF computation is presented and approximation RBF as well. The RBF interpolation or approximation can be used also for image reconstruction, inpainting removal, for solution of Partial Differential Equations (PDE), in GIS systems, digital elevation model DEM etc.
Radial basis functions for the multivariate interpolation of large scattered data sets
Journal of Computational and Applied Mathematics, 2002
An e cient method for the multivariate interpolation of very large scattered data sets is presented. It is based on the local use of radial basis functions and represents a further improvement of the well known Shepard's method. Although the latter is simple and well suited for multivariate interpolation, it does not share the good reproduction quality of other methods widely used for bivariate interpolation. On the other hand, radial basis functions, which have proven to be highly useful for multivariate scattered data interpolation, have a severe drawback. They are unable to interpolate large sets in an e cient and numerically stable way and maintain a good level of reproduction quality at the same time. Both problems have been circumvented using radial basis functions to evaluate the nodal function of the modiÿed Shepard's method. This approach exploits the exibility of the method and improves its reproduction quality. The proposed algorithm has been implemented and numerical results conÿrm its e ciency.
ACM SIGGRAPH 2005 Courses on - SIGGRAPH '05, 2005
We describe algebraic methods for creating implicit surfaces using linear combinations of radial basis interpolants to form complex models from scattered surface points. Shapes with arbitrary topology are easily represented without the usual interpolation or aliasing errors arising from discrete sampling. These methods were first applied to implicit surfaces by Savchenko, et al. and later developed independently by Turk and O'Brien as a means of performing shape interpolation. Earlier approaches were limited as a modeling mechanism because of the order of the computational complexity involved. We explore and extend these implicit interpolating methods to make them suitable for systems of large numbers of scattered surface points by using compactly supported radial basis interpolants. The use of compactly supported elements generates a sparse solution space, reducing the computational complexity and making the technique practical for large models. The local nature of compactly supported radial basis functions permits the use of computational techniques and data structures such as k-d trees for spatial subdivision, promoting fast solvers and methods to divide and conquer many of the subproblems associated with these methods. Moreover, the representation of complex models permits the exploration of diverse surface geometry. This reduction in computational complexity enables the application of these methods to the study of shape properties of large complex shapes.
A PRACTICAL USE OF RADIAL BASIS FUNCTIONS INTERPOLATION AND APPROXIMATION
Interpolation and approximation methods are used across many fields. Standard interpolation and approximation methods rely on "ordering" that actually means tessellation in-dimensional space in general, like sorting, triangulation, generating of tetrahedral meshes etc. Tessellation algorithms are quite complex in-dimensional case. On the other hand, interpolation and approximation can be made using meshfree (meshless) techniques using Radial Basis Function (RBF). The RBF interpolation and approximation methods lead generally to a solution of linear system of equations. However, a similar approach can be taken for a reconstruction of a surface of scanned objects, etc. In this case this leads to a linear system of homogeneous equations, when a different approach has to be taken. In this paper we describe novel approaches based on RBFs for data interpolation and approximation generally in d-dimensional space. We will show properties and differences of "global" and "Compactly Supported RBF (CSRBF)", run-time and memory complexities. As the RBF interpolation and approximation naturally offer smoothness, we will analyze such properties as well as approaches how to decrease computational expenses. The proposed meshless interpolation and approximation will be demonstrated on different problems, e.g. inpainting removal, restoration of corrupted images with high percentage of corrupted pixels, digital terrain interpolation and approximation for GIS applications and methods for decreasing computational complexity.
Radial Basis Functions: Interpolation and Applications- An Incremental Approach
An Incremental Approach, Applied Mathematics, Simulation and Modeling - ASM 2010 conference, NAUN, pp.209-213, ISSN 1792-4332, ISBN 978-960-474-210-3, 2010
Radial Basis Functions (RBF) interpolation is primarily used for interpolation of scattered data in higher dimensions. The RBF interpolation is a non-separable interpolation which offers a smooth interpolation, generally in n-dimensional space. We present a new method for RBF computation using an incremental approach. The proposed method is especially convenient in cases when larger data sets are randomly updated as the proposed method is of O(N 2) computational complexity instead of O(N 3) for insert / remove operations only and therefore it is much faster than the standard approach. If t-varying data or vector data are to be interpolated, the proposed method offers a significant speed-up as well.
Scattered Data Interpolation in N-Dimensional Space
Radial Basis Functions (RBF) interpolation theory is briefly introduced at the "application level" including some basic principles and computational issues. The RBF interpolation is convenient for un-ordered data sets in n-dimensional space, in general. This approach is convenient especially for a higher dimension N 2 conversion to ordered data set, e.g. using tessellation, is computationally very expensive. The RBF interpolation is not separable and it is based on distance of two points. The RBF interpolation leads to a solution of a Linear System of Equations (LSE). There are two main groups of interpolating functions: 'global" and "local". Application of "local" functions, called Compactly Supporting Functions (CSFBF), can significantly decrease computational cost as they lead to a system of linear equations with a sparse matrix. The RBF interpolation can be used also for image reconstruction, inpainting removal, for solution of Partial Differential Equations (PDE) etc.
Scattered Data Interpolation in N-Dimensional Space VACLAV SKALA
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Radial Basis Functions (RBF) interpolation theory is briefly introduced at the “application level” including some basic principles and computational issues. The RBF interpolation is convenient for un-ordered data sets in n-dimensional space, in general. This approach is convenient especially for a higher dimension N 2 conversion to ordered data set, e.g. using tessellation, is computationally very expensive. The RBF interpolation is not separable and it is based on distance of two points. The RBF interpolation leads to a solution of a Linear System of Equations (LSE) . There are two main groups of interpolating functions: ‘global” and “local”. Application of “local” functions, called Compactly Supporting Functions (CSFBF), can significantly decrease computational cost as they lead to a system of linear equations with a sparse matrix. The RBF interpolation can be used also for image reconstruction, inpainting removal, for solution of Partial Differential Equations (PDE) etc. Key-Word...
Large Scattered Data Interpolation with Radial Basis Functions and Space Subdivision
Integrated Computer Aided Engineering, Vol.25, No.1, pp.49-62, ISSN 1069-2509, IOS Press, 2018
We propose a new approach for the radial basis function (RBF) interpolation of large scattered data sets. It uses the space subdivision technique into independent cells allowing processing of large data sets with low memory requirements and offering high computation speed, together with the possibility of parallel processing as each cell can be processed independently. The proposed RBF interpolation was tested on both synthetic and real data sets. It proved its simplicity, robustness and the ability to handle large data sets together with significant speed-up. In the case of parallel processing, speed-up was experimentally proved when 2 and 4 threads were used.
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IMA Journal of Numerical Analysis, 1993
... Local error estimates for radial basis function interpolation of scattered data ZONG-MIN WU Department of Mathematics, Fudan University, Shanghai, 200433 People's Republic of China ... Downloaded from Page 2. 14 ZONG-MIN WU AND ROBERT SCHABACK where plt . . . ...