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Finding Points of Importance for Radial Basis Function Approximation of Large Scattered Data

机译：大散乱数据的径向基函数逼近的重要发现点

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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 tessellated 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.
机译：内插法和逼近法被用于许多领域，例如工程学以及其他学科，用于各种科学发现。如果数据域是由分散的数据形成的，则近似方法可能会变得非常复杂且耗时。通常，给定数据通过某种方法（不一定是Delaunay三角剖分）进行细分，以生成三角形或四面体网格。之后，可以使用近似方法生成曲面。但是，很难确保最终插值以及所有相邻三角形的连续性和平滑性。在此贡献中，提出了一种使用径向基函数（RBF）的无网格方法。它适用于两个变量的显式函数，并且通常适用于所有类型的分散数据。 RBF近似的关键点是找到对散乱数据进行高精度近似的重要点。由于紧凑支持的RBF（CSRBF）在数值计算中的影响有限，因此可以通过某种高效算法高效快速地处理大型数据集。 RBF的主要优点是，它导致了线性方程组（SLE）Ax = b的解。因此，任何有效的方法都可以解决可以使用的线性方程组。在这项研究中，我们提出了一种确定散射数据上重要点的新方法，该方法可以在保持表面光滑度的同时，以更高的精度生成非常好的重构表面。

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