scipy.interpolate.splprep#

scipy.interpolate.splprep(x, w=None, u=None, ub=None, ue=None, k=3, task=0, s=None, t=None, full_output=0, nest=None, per=0, quiet=1)[source]#

Find the B-spline representation of an N-D curve.

Given a list of N rank-1 arrays, x, which represent a curve in N-dimensional space parametrized by u, find a smooth approximating spline curve g(u). Uses the FORTRAN routine parcur from FITPACK.

Parameters:

xarray_like: A list of sample vector arrays representing the curve.
warray_like, optional: Strictly positive rank-1 array of weights the same length as x[0]. The weights are used in computing the weighted least-squares spline fit. If the errors in the x values have standard-deviation given by the vector d, then w should be 1/d. Default is ones(len(x[0])).
uarray_like, optional: An array of parameter values. If not given, these values are calculated automatically as M = len(x[0]), where

v[0] = 0

v[i] = v[i-1] + distance(x[i], x[i-1])

u[i] = v[i] / v[M-1]
ub, ueint, optional: The end-points of the parameters interval. Defaults to u[0] and u[-1].
kint, optional: Degree of the spline. Cubic splines are recommended. Even values of k should be avoided especially with a small s-value. 1 <= k <= 5, default is 3.
taskint, optional: If task==0 (default), find t and c for a given smoothing factor, s. If task==1, find t and c for another value of the smoothing factor, s. There must have been a previous call with task=0 or task=1 for the same set of data. If task=-1 find the weighted least square spline for a given set of knots, t.
sfloat, optional: A smoothing condition. The amount of smoothness is determined by satisfying the conditions: sum((w * (y - g))**2,axis=0) <= s, where g(x) is the smoothed interpolation of (x,y). The user can use s to control the trade-off between closeness and smoothness of fit. Larger s means more smoothing while smaller values of s indicate less smoothing. Recommended values of s depend on the weights, w. If the weights represent the inverse of the standard-deviation of y, then a good s value should be found in the range (m-sqrt(2*m),m+sqrt(2*m)), where m is the number of data points in x, y, and w.
tarray, optional: The knots needed for task=-1. There must be at least 2*k+2 knots.
full_outputint, optional: If non-zero, then return optional outputs.
nestint, optional: An over-estimate of the total number of knots of the spline to help in determining the storage space. By default nest=m/2. Always large enough is nest=m+k+1.
perint, optional: If non-zero, data points are considered periodic with period x[m-1] - x[0] and a smooth periodic spline approximation is returned. Values of y[m-1] and w[m-1] are not used.
quietint, optional: Non-zero to suppress messages.

Returns:

tcktuple: A tuple, (t,c,k) containing the vector of knots, the B-spline coefficients, and the degree of the spline.
uarray: An array of the values of the parameter.
fpfloat: The weighted sum of squared residuals of the spline approximation.
ierint: An integer flag about splrep success. Success is indicated if ier<=0. If ier in [1,2,3] an error occurred but was not raised. Otherwise an error is raised.
msgstr: A message corresponding to the integer flag, ier.

See also

splrep, splev, sproot, spalde, splint
bisplrep, bisplev
UnivariateSpline, BivariateSpline
BSpline
make_interp_spline

Notes

See splev for evaluation of the spline and its derivatives. The number of dimensions N must be smaller than 11.

The number of coefficients in the c array is k+1 less than the number of knots, len(t). This is in contrast with splrep, which zero-pads the array of coefficients to have the same length as the array of knots. These additional coefficients are ignored by evaluation routines, splev and BSpline.

References

[1]

P. Dierckx, “Algorithms for smoothing data with periodic and parametric splines, Computer Graphics and Image Processing”, 20 (1982) 171-184.

[2]

P. Dierckx, “Algorithms for smoothing data with periodic and parametric splines”, report tw55, Dept. Computer Science, K.U.Leuven, 1981.

[3]

P. Dierckx, “Curve and surface fitting with splines”, Monographs on Numerical Analysis, Oxford University Press, 1993.

Examples

Generate a discretization of a limacon curve in the polar coordinates:

>>> import numpy as np
>>> phi = np.linspace(0, 2.*np.pi, 40)
>>> r = 0.5 + np.cos(phi)         # polar coords
>>> x, y = r * np.cos(phi), r * np.sin(phi)    # convert to cartesian

And interpolate:

>>> from scipy.interpolate import splprep, splev
>>> tck, u = splprep([x, y], s=0)
>>> new_points = splev(u, tck)

Notice that (i) we force interpolation by using s=0, (ii) the parameterization, u, is generated automatically. Now plot the result:

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> ax.plot(x, y, 'ro')
>>> ax.plot(new_points[0], new_points[1], 'r-')
>>> plt.show()

../../_images/scipy-interpolate-splprep-1.png