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 !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. |
| 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. |