Changes between Version 15 and Version 16 of ThinPlateShell


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Timestamp:
Dec 9, 2014, 6:00:49 PM (6 years ago)
Author:
sommeria
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  • ThinPlateShell

    v15 v16  
    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.
    55
    6 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, 1976 ('Splines minimizing rotation invariant semi-norms in Sobolev spaces. pp 85–100, In: Constructive Theory of Functions of Several Variables, Oberwolfach 1976, W. Schempp and K. Zeller, eds., Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977). The name refers to a physical analogy with the bending of a thin sheet of metal. Practical algorithms have been first developed by Paihua Montes (these UJF 1978, 'Quelques méthodes numériques pour le calcul de fonctions splines').It has been first used in the field of cartography and first applied to flow measurements by [attachment:NguyenDuc_sommeria_1988.pdf !NguyenDuc and Sommeria(1988)]. We use here a more recent algorithm proposed by Wahba, 1990 ('Spline models for observational data', SIAM bookstore), rely on the standard inversion matrix functions provided by Matlab.
     6Generalisation 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, 1976 ('Splines minimizing rotation invariant semi-norms in Sobolev spaces. pp 85–100, In: Constructive Theory of Functions of Several Variables, Oberwolfach 1976, W. Schempp and K. Zeller, eds., Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977). The name refers to a physical analogy with the bending of a thin sheet of metal. Practical algorithms have been first developed by Paihua Montes (these UJF 1978, 'Quelques méthodes numériques pour le calcul de fonctions splines').It has been first used in the field of cartography and first applied to flow measurements by [attachment:NguyenDuc_sommeria_1988.pdf NguyenDuc and Sommeria(1988)]. We use here a more recent algorithm proposed by Wahba, 1990 ('Spline models for observational data', SIAM bookstore), rely on the standard inversion matrix functions provided by Matlab.
    77
    88= Case of pure interpolation =