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Radial Basis Functions for HighDimensional Visualization
"... Abstract — Highdimensional visualization is usually connected with large data processing. Because of dimensionality, it is nearly impossible to make a tessellation, like the Delaunay tessellation in E d, followed by data interpolation. One possibility of data interpolation is the use of the Radial ..."
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Abstract — Highdimensional visualization is usually connected with large data processing. Because of dimensionality, it is nearly impossible to make a tessellation, like the Delaunay tessellation in E d, followed by data interpolation. One possibility of data interpolation is the use of the Radial Basis Functions (RBF) interpolation. The RBF interpolation supports the interpolation of scattered data in ddimensional space. The computational cost of the RBF interpolation is higher but does not increase significantly with the data dimensionality. It increases with the number of values to be processed nonlinearly. In this paper, the RBF interpolation properties will be discussed as well as how to process data incrementally. Incremental computation decreases computational complexity and decreases RBF computational cost for the given data set significantly, especially for the visualization purposes, when the interpolated/approximated data are used many times. As the proposed approach is based on a solution of a system of linear equations, the RBF interpolation is convenient especially for data sets processing using matrixvector or GPU architectures.
Scattered Data Interpolation in NDimensional Space
"... Abstract: 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 unordered data sets in ndimensional space, in general. This approach is convenient especia ..."
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Abstract: 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 unordered data sets in ndimensional 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. KeyWords: RBF interpolation, radial basis function, image reconstruction, incremental computation, RBF approximation. 1
Fast Interpolation and Approximation of Scattered Multidimensional and Dynamic Data Using Radial Basis Functions
"... Abstract: Interpolation or approximation of scattered data is very often task in engineering problems. The Radial Basis Functions (RBF) interpolation is convenient for scattered (unordered) data sets in kdimensional space, in general. This approach is convenient especially for a higher dimension k ..."
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Abstract: Interpolation or approximation of scattered data is very often task in engineering problems. The Radial Basis Functions (RBF) interpolation is convenient for scattered (unordered) data sets in kdimensional 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)