Radial Basis Functions
Theory and Implementations
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This is an account of radial basis functions, a new method for estimating parameters which is easier to use and understand than wavelets, and more powerful.In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing, it is necessary to estimate parameters, usually multi-dimensional, by approximation and interpolation. Radial basis functions are a modern and powerful tool which work well in very general circumstances, and so are becoming of widespread use, as the limitations of other methods, such as least-squares, polynomial interpolati...
This is an account of radial basis functions, a new method for estimating parameters which is easier to use and understand than wavelets, and more powerful.
In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing, it is necessary to estimate parameters, usually multi-dimensional, by approximation and interpolation. Radial basis functions are a modern and powerful tool which work well in very general circumstances, and so are becoming of widespread use, as the limitations of other methods, such as least-squares, polynomial interpolation or wavelet-based, become apparent. The author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence, and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.
Table of content:
Preface; 1. Introduction; 2. Summary of methods and applications; 3. General methods for approximation and interpolation; 4. Radial basis function approximation on infinite grids; 5. Radial basis functions on scattered data; 6. Radial basis functions with compact support; 7. Implementations; 8. Least squares methods; 9. Wavelet methods with radial basis functions; 10. Further results and open problems; Appendix: some essentials on Fourier transforms; Commentary on the bibliography; Bibliography; Index.
In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing, it is necessary to estimate parameters, usually multi-dimensional, by approximation and interpolation. Radial basis functions are a modern and powerful tool which work well in very general circumstances, and so are becoming of widespread use, as the limitations of other methods, such as least-squares, polynomial interpolation or wavelet-based, become apparent. The author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence, and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.
Table of content:
Preface; 1. Introduction; 2. Summary of methods and applications; 3. General methods for approximation and interpolation; 4. Radial basis function approximation on infinite grids; 5. Radial basis functions on scattered data; 6. Radial basis functions with compact support; 7. Implementations; 8. Least squares methods; 9. Wavelet methods with radial basis functions; 10. Further results and open problems; Appendix: some essentials on Fourier transforms; Commentary on the bibliography; Bibliography; Index.
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