Changes between Version 2 and Version 3 of ThinPlateShell


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Timestamp:
Dec 8, 2014, 1:13:32 AM (6 years ago)
Author:
sommeria
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  • ThinPlateShell

    v2 v3  
    77Generalisation to multi-dimensional spaces can be performed as products of spline functions along each coordinate. However this choice is not optimum and depends on the coordinate axis. The proper generalisation is the ’thin plate spline’ proposed by Duchon[Duchon]. The name refers to a physical analogy with the bending of a thin sheet of metal. Practical algorithms have been first developed by Pahia Montes .It has been first used in the field of cartography and first applied to flow measurements by NguyenDuc and Sommeria(..). We use here a more recent algorithm proposed by Wahba, 1990, rely on the standard inversion matrix functions provided by Matlab.
    88
    9 {{{Case of pure interpolation}}}
     9=Case of pure interpolation=
    1010We consider a set of measurement points ${\bf r_i}, i = 1,2, \ldots,N$, and the corresponding set of measurements values for a quantity $f$ (a velocity component for PIV data).
    1111$$
     
    7474where $I_{000}$ is the NxN identity matrix extended by three columns of 0.
    7575
    76 {{{spatial derivatives}}}
     76=spatial derivatives=
    7777Spatial derivatives of the interpolated quantity $f$ can be obtained by direct differentiation of the result.
    7878For any function $\phi(r)$, with radial distance $r=|{\bf r-r_i}|$, $r^2=X^2+Y^2$, $\partial_X \phi=(d\phi/dr) \partial_X r$, and $\partial_X r=X/r$. This yields $\partial_X \phi=X (2 \log(r) +1)$, so that,
     
    8181$$\partial_y f({\bf r})=\sum S_i (2 \log|{\bf r-r_i}|+1)(y-y_i)+a_2;$$
    8282
    83 {{{Sub-domains}}}
     83=Sub-domains=
    8484The computing cost and memory required to invert the matrix [->equ. 9] increases as the cube of the number of measurement points, so that large data sets need to be partitioned in sub-domains. The interpolation is made independently in each subdomain. If sub-domain overlap, the average of the different answers is taken.
    8585