Changes between Version 127 and Version 128 of UvmatHelp


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Timestamp:
Dec 8, 2014, 12:47:37 AM (6 years ago)
Author:
sommeria
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  • UvmatHelp

    v127 v128  
    335335-'''Thin plate shell (tps) interpolation:'''
    336336
    337 This is a multi-dimensional generalisation of the spline interpolation/smoothing, an optimum way to interpolate data with minimal  curvature of the interpolating function. The result at an interpolation position vector ${\bf r}$ is expressed in the form, (see http://coriolis.legi.grenoble-inp.fr/spip.php?article73)
     337This is a multi-dimensional generalisation of the spline interpolation/smoothing, an optimum way to interpolate data with minimal  curvature of the interpolating function. The result at an interpolation position vector ${\bf r}$ is expressed in the form, (see ThinPlateShell)
    338338
    339339$$\label{sol_gene} f({\bf r})=\sum S_i \phi({\bf|r-r_i}|)+a_0+a_1x+a_2y\; $$ where ${\bf r_i}$ are the positions of the measurement points (the ''centres''). Each ''centre'' can be viewed as the source of an axisymmetric field  $\phi$ of the form $\phi(r)=r^2\log (r)$. The weights $S_i$ and the linar coefficients $a_0,a_1,a_2$ are the thin plate shell (tps) coefficients which determine the interpolated value at any point. The spatial derivatives are similarly obtained at any point by analytical differentiation of the source functions $\phi(r)$. These tps weights, with the corresponding centre coordinates, therefore contain all the information needed for interpolation at any point. We call that a ''tps field representation''.^