Scattered Data Interpolation in N-Dimensional Space VACLAV SKALA

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.

Proceedings of the 11th EAI International Conference on Mobile Multimedia Communications, 2018

In many disciplines there is a need for efficient in terpolation of ir regular spaced data. For unsampled locations values have to be computed from the available data. Usually we are interested in smooth interpolations, and artificial o scillations should be avoided. For large data sets fully populated matrices are undesirable. The use of compactly supported basis functions appears to be attractive. In this paper radial basis functions satisfying the 2D bi-harmonic equation are used. Accurate interpolation could be achieved in the numerical test examples.

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.

7th Conference on Computer Graphics, Virtual Reality, Visualisation and Interaction in Africa, Afrigraph 2010, pp.17-20, ACM, ISBN:978-1-4503-0118-3, 2010

Interpolation based on Radial Basis Functions (RBF) is very often used for scattered scalar data interpolation in n-dimensional space in general. RBFs are used for surface reconstruction of 3D objects, reconstruction of corrupted images etc. As there is no explicit order in data sets, computations are quite time consuming that leads to limitation of usability even for static data sets. Generally the complexity of computation of RBF interpolation for N points is of O(N 3) or O(k N 2), k is a number of iterations if iterative methods are used, which is prohibitive for real applications. The inverse matrix can also be computed by the Strassen algorithm based on matrix block notation with O(N 2.807) complexity. Even worst situation occurs when interpolation has to be made over non-constant data sets, as the whole set of equations for determining RBFs has to be recomputed. This situation is typical for applications in which some points are becoming invalid and new points are acquired. In this paper a new technique for incremental RBFs computation with complexity of O(N 2) is presented. This technique enables efficient insertion of new points and removal of selected or invalid points. Due to the formulation it is possible to determine an error if one point is removed that leads to a possibility to determine the most important points from the precision of interpolation point of view and insert gradually new points, which will progressively decrease the error of interpolation using RBFs. The Progressive RBF Interpolation enables also fast interpolation on "sliding window" data due to insert/remove operations which will also lead to a faster rendering.

2021

The problem of interpolating functions comes up naturally in many areas of applied mathematics and natural sciences. Radial basis function methods provide an interpolant to values of a real function of $\textit{d}$ variables and are highly useful in many applications, especially if the function values are given at scattered data points. The need for iterative procedures arises when the number of interpolation conditions $\textit{n}$ is large, since hardly any sparsity occurs in the linear system of interpolation equations. Solving this system with direct methods would require O($\textit{n}$3) operations. This dissertation considers several iterative techniques. They were developed from an algorithm described by Beatson, Goodsell and Powell (1995), which is examined first. By gaining more and more theoretical insight into the original algorithm, new algorithms are developed and connections to known methods are made. We establish the important role a certain semi-inner product plays i...

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 . . . ...

2003 Shape Modeling International.

In this paper, we propose a hierarchical approach to 3D scattered data interpolation with compactly supported basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarse-to-fine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point cloud distributed along a surface, we first use spatial down sampling to construct a coarse-to-fine hierarchy of point sets. Then we interpolate the sets starting from the coarsest level. We interpolate a point set of the hierarchy, as an offsetting of the interpolating function computed at the previous level. Fig. 1 shows an original point set (the leftmost image) and its coarse-to-fine hierarchy of interpolated sets. According to our numerical experiments, the method is essentially faster than the state-of-art scattered data approximation with globally supported RBFs [9] and much simpler to implement.

Computers & Mathematics with Applications, 2005

Multivariate interpolation of smooth data using smooth radial basis functions is considered. The behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically. Explicit criteria for different types of limits are given. Using the results for the limits, the dependence of the error on the shape parameter of the radial basis function is investigated. The mechanisms that determine the optimal shape parameter value are studied and explained through approximate expansions of the interpolation error.

ICCS 2020, Part VI, LNCS 12142, pp. 239–250, 2020

Interpolation and approximation methods are used in many fields such as in engineering as well as other disciplines for various scientific discoveries. If the data domain is formed by scattered data, approximation methods may become very complicated as well as time-consuming. Usually, the given data is tessel-lated by some method, not necessarily the Delaunay triangulation, to produce triangular or tetrahedral meshes. After that approximation methods can be used to produce the surface. However, it is difficult to ensure the continuity and smoothness of the final interpolant along with all adjacent triangles. In this contribution , a meshless approach is proposed by using radial basis functions (RBFs). It is applicable to explicit functions of two variables and it is suitable for all types of scattered data in general. The key point for the RBF approximation is finding the important points that give a good approximation with high precision to the scattered data. Since the compactly supported RBFs (CSRBF) has limited influence in numerical computation, large data sets can be processed efficiently as well as very fast via some efficient algorithm. The main advantage of the RBF is, that it leads to a solution of a system of linear equations (SLE) Ax=b. Thus any efficient method solves the systems of linear equations that can be used. In this study is we propose a new method of determining the importance points on the scattered data that produces a very good reconstructed surface with higher accuracy while maintaining the smoothness of the surface.

Advances in Computational Mathematics, 2005

Radial basis function (RBF) interpolation can be very effective for scattered data in any number of dimensions. As one of their many applications, RBFs can provide highly accurate collocation-type numerical solutions to several classes of PDEs. To better understand the ...