fluidimage.calcul.interpolate.thin_plate_spline#

Thin plate spline#

Translated and adapted from UVmat code (Joel Sommeria, LEGI, CNRS).

This interpolation/smoothing (Duchon, 1976; NguyenDuc and Sommeria, 1988) minimises a linear combination of the squared curvature and squared difference from the initial data.

We first need to compute tps coefficients U_tps (function compute_tps_weights). Interpolated data can then be obtained as the matrix product dot(U_tps, EM) where the matrix EM is obtained by the function compute_tps_matrix. The spatial derivatives are obtained as dot(U_tps, EMDX) and dot(U_tps, EMDY), where EMDX and EMDY are obtained from the function compute_tps_matrix_dxy. A helper class is also provided.

fluidimage.calcul.interpolate.thin_plate_spline.compute_tps_weights(centers, values, smoothing_coef)[source]#

Calculate the thin plate spline (tps) coefficients

Parameters:

centersnp.array: [nb_dim, N] array representing the positions of the N centers, sources of the TPS (nb_dim = space dimension).
valuesnp.array: [N] array representing the values of the considered scalar measured at the centres centers.
smoothing_coeffloat: Smoothing parameter. The result is smoother for larger smoothing_coef.

Returns:

smoothed_valuesnp.array: Values of the quantity U at the N centres after smoothing.
tps_weightsnp.array: TPS weights of the centres and columns of the linear.

class fluidimage.calcul.interpolate.thin_plate_spline.ThinPlateSpline(new_positions, centers)[source]#

Bases: object

Helper class for thin plate interpolation.

compute_field(U_tps)[source]#: Compute the interpolated field.

compute_gradient(U_tps)[source]#: Compute the gradient (dx_U, dy_U)

fluidimage.calcul.interpolate.thin_plate_spline.compute_tps_matrix_numpy(new_pos: float64[][], centers: float64[][])[source]#

calculate the thin plate spline (tps) interpolation at a set of points

Parameters:

new_pos: np.array: [nb_dim, M] array representing the postions of the M ‘observation’ sites, with nb_dim the space dimension.
centers: np.array: [nb_dim, N] array representing the postions of the N centers, sources of the tps.

Returns:

EMnp.array

[(N+nb_dim), M] matrix representing the contributions at the M sites.

From unit sources located at each of the N centers, + (nb_dim+1) columns representing the contribution of the linear gradient part.

Notes

>>> U_interp = np.dot(U_tps, EM)

fluidimage.calcul.interpolate.thin_plate_spline.compute_tps_matrices_dxy(dsites, centers)[source]#

Calculate the derivatives of thin plate spline (tps) interpolation at a set of points (limited to the 2D case)

Parameters:

dsitesnp.array: [nb_dim, M] array of interpolation site coordinates (nb_dim = space dimension = 2, here).
centersnp.array: [nb_dim, N] array of centre coordinates (initial data).

Returns:

DMXnp.array: [(N+3), M] array representing the contributions to the X derivatives at the M sites from unit sources located at each of the N centers, + 3 columns representing the contribution of the linear gradient part.
DMYnp.array: idem for Y derivatives.

Functions

`compute_tps_matrices_dxy`(dsites, centers)	Calculate the derivatives of thin plate spline (tps) interpolation at a set of points (limited to the 2D case)
`compute_tps_matrix`(new_pos, centers)	calculate the thin plate spline (tps) interpolation at a set of points
`compute_tps_matrix_numpy`(new_pos, centers)	calculate the thin plate spline (tps) interpolation at a set of points
`compute_tps_matrix_pythran`(new_pos, centers)	calculate the thin plate spline (tps) interpolation at a set of points
`compute_tps_weights`(centers, values, ...)	Calculate the thin plate spline (tps) coefficients

Classes

`ThinPlateSpline`(new_positions, centers)	Helper class for thin plate interpolation.
`ThinPlateSplineNumpy`(new_positions, centers)