Fast evaluation of radial basis functions : theory and application. (2000)
Type of ContentTheses / Dissertations
Degree NameDoctor of Philosophy
PublisherUniversity of Canterbury. Mathematics and Statistics
Radial Basis Functions (RBFs) have proven to be successful interpolants to scattered data. However, the perceived high computational costs for fitting and evaluating the RBFs associated with large data sets have hindered their application to many real world problems. This thesis is concerned with the "fast" evaluation of RBFs: the O(N2) process of evaluation at all centres is reduced to O(N log N) or even O(N). The required theory is developed for polyharmonic RBFs in 4-dimensions and for multiquadric RBFs in arbitrary dimensions. These methods are applied to fit surfaces to scattered data containing many tens of thousands of points.
RightsCopyright J. B. Cherrie
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Cherrie, J. B.; Beatson, R. K.; Ragozin, D. L. (University of Canterbury. Department of Mathematics & Statistics, 2000)As is now well known for some basic functions ϕ, hierarchical and fast multipole like methods can greatly reduce the storage and operation counts for fitting and evaluating radial basis functions. In particular for spline ...
Cherrie, J. B.; Beatson, Richard Keith; Newsam, G.N. (University of Canterbury, 2000)A generalised multiquadric radial basis function is a function of the form s(x) = ∑ᴺ𝑖₌₁ 𝑑𝑖 𝜙 (𝗅x-t𝑖𝗅), where 𝜙(r) = (r² + 𝝉²)ᵏ/², x ∈ ℝⁿ, and k ∈ Z is odd. The direct evaluation of an N centre generalised ...
Beatson, Richard Keith; Newsam, G.N. (University of Canterbury. Dept. of Mathematics, 1995)In this paper we introduce a new algorithm for fast evaluation of univariate radial basis functions of the form s(x) = Σᶰn₌₁ dn𝜙(⃒x - xn⃒) to within accuracy 𝜖. The algorithm has a setup cost of 𝜙(N⃒log𝜖⃒log⃒log𝜖⃒) ...