Python/Numpy implementation of Bspline basis functions via Cox - de Boor algorithm.
Also provided are higher-order differentiation, collocation matrix generation, and a minimal procedural API (mainly for dealing with knot vectors) which may help in converting MATLAB codes.
import numpy import bspline import bspline.splinelab as splinelab ## Spline setup and evaluation p = 3 # order of spline (as-is; 3 = cubic) nknots = 11 # number of knots to generate (here endpoints count only once) tau = [0.1, 0.33] # collocation sites (i.e. where to evaluate) knots = numpy.linspace(0,1,nknots) # create a knot vector without endpoint repeats k = splinelab.augknt(knots, p) # add endpoint repeats as appropriate for spline order p B = bspline.Bspline(k, p) # create spline basis of order p on knots k A0 = B.collmat(tau) # collocation matrix for function value at sites tau A2 = B.collmat(tau, deriv_order=2) # collocation matrix for second derivative at sites tau print( A0 ) print( A2 ) D3 = B.diff(order=3) # third derivative of B as lambda x: ... print( D3(0.4) ) D = numpy.array( [D3(t) for t in tau], dtype=numpy.float64 ) # third derivative of B at sites tau ## Spline setup by defining collocation sites ncolloc = 7 tau = numpy.linspace(0,1,ncolloc) # These are the sites to which we would like to interpolate k = splinelab.aptknt(tau, p) # Given the collocation sites, generate a knot vector # (incl. endpoint repeats). To get meaningful results, # here one must choose ncolloc such that ncolloc >= p+1. B = bspline.Bspline(k, p) A0 = B.collmat(tau) print( A0 ) ## Evaluate a function expressed in the spline basis: # set up coefficients (in a real use case, fill this with something sensible, # e.g. with an L2 projection onto the spline basis) # nbasis = A0.shape # A0.shape = (num_collocation_sites, num_basis_functions) c = numpy.ones( (nbasis,), dtype=numpy.float64 ) # evaluate f(0.4) y1 = numpy.sum( B(0.4) * c ) # evaluate at each tau[k] y2 = numpy.array( [numpy.sum( B(t) * c ) for t in tau], dtype=numpy.float64 ) # equivalent, using the collocation matrix # # NOTE: the sites tau are built into the matrix when collmat() is called. # y3 = numpy.sum( A0 * c, axis=-1 )
Install as user:
pip install bspline --user
Install as admin:
sudo pip install bspline
git clone https://github.com/johntfoster/bspline.git cd bspline python setup.py install --user
As admin, change the last command to
sudo python setup.py install
splinelab.py files from the
bspline subdirectory next to your source code,
or leave them there and call it as a module.
- Python 2.7 and 3.4.
- Linux Mint.