Changes between Version 16 and Version 17 of ThinPlateShell


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Timestamp:
Feb 21, 2015, 12:19:43 AM (6 years ago)
Author:
sommeria
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  • ThinPlateShell

    v16 v17  
    11= '''Thin Plate Shell interpolation''' =
    2 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. ^
     2Interpolation/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.
    33
    44To 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.