BSpline — SciPy v1.15.3 Manual (original) (raw)

scipy.interpolate.

class scipy.interpolate.BSpline(t, c, k, extrapolate=True, axis=0)[source]#

Univariate spline in the B-spline basis.

\[S(x) = \sum_{j=0}^{n-1} c_j B_{j, k; t}(x)\]

where \(B_{j, k; t}\) are B-spline basis functions of degree _k_and knots t.

Parameters:

tndarray, shape (n+k+1,)

knots

cndarray, shape (>=n, …)

spline coefficients

kint

B-spline degree

extrapolatebool or ‘periodic’, optional

whether to extrapolate beyond the base interval, t[k] .. t[n], or to return nans. If True, extrapolates the first and last polynomial pieces of b-spline functions active on the base interval. If ‘periodic’, periodic extrapolation is used. Default is True.

axisint, optional

Interpolation axis. Default is zero.

Notes

B-spline basis elements are defined via

\[ \begin{align}\begin{aligned}B_{i, 0}(x) = 1, \textrm{if tilex<ti+1t_i \le x < t_{i+1}tilex<ti+1, otherwise 000,}\\B_{i, k}(x) = \frac{x - t_i}{t_{i+k} - t_i} B_{i, k-1}(x) + \frac{t_{i+k+1} - x}{t_{i+k+1} - t_{i+1}} B_{i+1, k-1}(x)\end{aligned}\end{align} \]

Implementation details

• At least k+1 coefficients are required for a spline of degree k, so that n >= k+1. Additional coefficients, c[j] withj > n, are ignored.
• B-spline basis elements of degree k form a partition of unity on the_base interval_, t[k] <= x <= t[n].

References

[2]

Carl de Boor, A practical guide to splines, Springer, 2001.

Examples

Translating the recursive definition of B-splines into Python code, we have:

def B(x, k, i, t): ... if k == 0: ... return 1.0 if t[i] <= x < t[i+1] else 0.0 ... if t[i+k] == t[i]: ... c1 = 0.0 ... else: ... c1 = (x - t[i])/(t[i+k] - t[i]) * B(x, k-1, i, t) ... if t[i+k+1] == t[i+1]: ... c2 = 0.0 ... else: ... c2 = (t[i+k+1] - x)/(t[i+k+1] - t[i+1]) * B(x, k-1, i+1, t) ... return c1 + c2

def bspline(x, t, c, k): ... n = len(t) - k - 1 ... assert (n >= k+1) and (len(c) >= n) ... return sum(c[i] * B(x, k, i, t) for i in range(n))

Note that this is an inefficient (if straightforward) way to evaluate B-splines — this spline class does it in an equivalent, but much more efficient way.

Here we construct a quadratic spline function on the base interval2 <= x <= 4 and compare with the naive way of evaluating the spline:

from scipy.interpolate import BSpline k = 2 t = [0, 1, 2, 3, 4, 5, 6] c = [-1, 2, 0, -1] spl = BSpline(t, c, k) spl(2.5) array(1.375) bspline(2.5, t, c, k) 1.375

Note that outside of the base interval results differ. This is becauseBSpline extrapolates the first and last polynomial pieces of B-spline functions active on the base interval.

import matplotlib.pyplot as plt import numpy as np fig, ax = plt.subplots() xx = np.linspace(1.5, 4.5, 50) ax.plot(xx, [bspline(x, t, c ,k) for x in xx], 'r-', lw=3, label='naive') ax.plot(xx, spl(xx), 'b-', lw=4, alpha=0.7, label='BSpline') ax.grid(True) ax.legend(loc='best') plt.show()

Attributes:

tndarray

knot vector

cndarray

spline coefficients

kint

spline degree

extrapolatebool

If True, extrapolates the first and last polynomial pieces of b-spline functions active on the base interval.

axisint

Interpolation axis.

tcktuple

Equivalent to (self.t, self.c, self.k) (read-only).

Methods