Changes between Version 4 and Version 5 of ThinPlateShell

Timestamp:

Dec 9, 2014, 5:11:27 PM (10 years ago)

Author:

sommeria

Comment:

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ThinPlateShell

@@ -3 +3 @@
 Interpolation/smoothing is an important aspect of PIV. Linear interpolation is the simplest option but it cannot smooth noise, and for 2D (or 3D) fields it does not extrapolate data beyond the convex hull of the initial data set. Global polynomial interpolation generally produce excessive smoothing, for low order polynomials, or spurious oscillations, for high order polynomials. Cubic spline is an optimum general method to interpolate a 1D data series $f_{i}$  measured at a set of abscissa $x_{i},i=1,\ldots,N.$ The spline interpolation function $f(x)$  is made of piecewise cubic polynomials matched at the measurement abscissa $x_{i}$ by the condition $f(x_{i})=f_{i} $, with the continuity of the first derivative. It can be shown that this spline function minimizes the global curvature $\int_{x_{1}}^{x_{N}}f''(x)^{2}dx$ . The name 'spline' refers to a flexible wooden strip used in ship design to draw a smooth curved line pinned on a set of reference points. Its equilibrium shape indeed minimizes the elastic energy. This is proportional to the integral of $f''^{2}$  if it represents a small transverse displacement from the absissa line. 
 
-To deal with noisy data, smoothing spline do not go strictly through the measurement values, but minimises a linear combination of distance to these values and curvature $E=\sum_{i=1}^{N}(f_{i}-f(x_{i}))^{2}+\rho\int_{x_{1}}^{x_{N}}f''(x)^{2}\,\mathrm{d}x$, where \rho  is a smoothing parameter. In the limit of small\rho , the weight of the distance constraint becomes very strong so the optimum approaches the pure interpolation spline, with $f(x_{i})=f_{i}$ . In the opposite limit of large $\rho$ , the curvature constraint becomes very strong, so the optimum tends to be linear $(f''=0 )$ and approaches the least square linear fit by the minimisation of the distance term contribution.
+To deal with noisy data, smoothing spline do not go strictly through the measurement values, but minimises a linear combination of distance to these values and curvature $E=\sum_{i=1}^{N}(f_{i}-f(x_{i}))^{2}+\rho\int_{x_{1}}^{x_{N}}f''(x)^{2}\,\mathrm{d}x$, where $\rho$  is a smoothing parameter. In the limit of small $\rho$ , the weight of the distance constraint becomes very strong so the optimum approaches the pure interpolation spline, with $f(x_{i})=f_{i}$ . In the opposite limit of large $\rho$ , the curvature constraint becomes very strong, so the optimum tends to be linear $(f''=0 )$ and approaches the least square linear fit by the minimisation of the distance term contribution.
 
 Generalisation 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.