Julia library for 1-d and 2-d splines
This is a Julia wrapper for the dierckx Fortran library, the same library underlying the spline classes in scipy.interpolate. Some of the functionality here overlaps with Interpolations.jl, a pure-Julia interpolation package. Take a look at it if you have a use case not covered here.
All new development on
Dierckx.jl will be for Julia v1.0 and above.
master branch is therefore incompatible with earlier versions
- Implements B-splines (basis splines).
- Splines from first order to fifth order; default is third order (cubic).
- Fit and evaluate 1-d and 2-d splines on irregular grids.
- Fit and evaluate 2-d splines at unstructured points.
- Fit "smooth" (non-interpolating) splines with adjustable smoothing factor s.
- Derivatives, integrals and roots of 1-d splines.
- Parametric B-splines.
Install (Julia 1.0 and later)
(v1.0) pkg> add Dierckx
] to enter package mode.) No Fortran compiler is required on
The Fortran library source code is distributed with the package, so
you need a Fortran compiler on OSX or Linux. On Ubuntu,
sudo apt-get install gfortran will do it.
gfortran comes bundled with
gcc, so after installing Homebrew,
brew install gcc should install
On Windows, a compiled dll will be downloaded.
Fit a 1-d spline to some input data (points can be unevenly spaced):
x = [0., 1., 2., 3., 4.] y = [-1., 0., 7., 26., 63.] # x.^3 - 1. spl = Spline1D(x, y)
Evaluate the spline at some new points:
spl([1.5, 2.5]) # result = [2.375, 14.625] spl(1.5) # result = 2.375
Equivalent to the above:
evaluate(spl, [1.5, 2.5]) evaluate(spl, 1.5)
Evaluate derivative, integral, or roots:
derivative(spl, 1.5) # derivate at x=1.5; result is 5.75 integrate(spl, 0., 4.) # integrate from x=0 to x=4; result is 60.0 roots(spl) # result is [1.0]
roots() only works for cubic splines (k=3).
Fit a 2-d spline to data on a (possibly irregular) grid:
x = [0.5, 2., 3., 4., 5.5, 8.] y = [0.5, 2., 3., 4.] z = [1. 2. 1. 2.; # size is (length(x), length(y)) 1. 2. 1. 2.; 1. 2. 3. 2.; 1. 2. 2. 2.; 1. 2. 1. 2.; 1. 2. 3. 1.] spline = Spline2D(x, y, z)
Note that if you consider
z as a matrix,
x refers to row
y refers to column coordinates.
Evaluate at element-wise points:
xi = [1., 1.5, 2.3, 4.5, 3.3, 3.2, 3.] yi = [1., 2.3, 5.3, 0.5, 3.3, 1.2, 3.] zi = spline(xi, yi) # 1-d array of length 7 zi = evaluate(spline, xi, yi) # equivalent to previous line
Evaluate at grid spanned by input arrays:
xi = [1., 1.5, 2.3, 4.5] yi = [1., 2.3, 5.3] zi = evalgrid(spline, xi, yi) # 2-d array of size (4, 3)
Spline1D(x, y; w=ones(length(x)), k=3, bc="nearest", s=0.0) Spline1D(x, y, xknots; w=ones(length(x)), k=3, bc="nearest")
Create a spline of degree
k(1 = linear, 2 = quadratic, 3 = cubic, up to 5) from 1-d arrays
y. The parameter
bcspecifies the behavior when evaluating the spline outside the support domain, which is
(minimum(x), maximum(x)). The allowed values are
In the first form, the number and positions of knots are chosen automatically. The smoothness of the spline is then achieved by minimizing the discontinuity jumps of the
kth derivative of the spline at the knots. The amount of smoothness is determined by the condition that
sum((w[i]*(y[i]-spline(x[i])))**2) <= s, with
sa given non-negative constant, called the smoothing factor. The number of knots is increased until the condition is satisfied. By means of this parameter, the user can control the tradeoff between closeness of fit and smoothness of fit of the approximation. if
sis too large, the spline will be too smooth and signal will be lost ; if
sis too small the spline will pick up too much noise. in the extreme cases the program will return an interpolating spline if
s=0.0and the weighted least-squares polynomial of degree
sis very large.
In the second form, the knots are supplied by the user. There is no smoothing parameter in this form. The program simply minimizes the discontinuity jumps of the
kth derivative of the spline at the given knots.
- Evaluate the 1-d spline
splat points given in
x, which can be a 1-d array or scalar. If a 1-d array, the values must be monotonically increasing.
derivative(spl, x; nu=1)
- Evaluate the
nu-th derivative of the spline at points in
integrate(spl, a, b)
- Definite integral of spline between
- For cubic splines (
k=3) only, find roots. Only up to
maxnroots are returned. A warning is issued if the spline has more roots than the number returned.
These are the B-spline representation of a curve through N-dimensional space.
ParametricSpline(X; s=0.0, ...) ParametricSpline(u, X; s=0.0, ...) ParametricSpline(X, knots, ...) ParametricSpline(u, X, knots, ...)
Xis a 2-d array with size
Nis the number of dimensions of the space (must be between 1 and 10) and
mis the number of points.
X[:, i]gives the coordinates of the
uis a 1-d array giving parameter values at each of the
mpoints. If not given, values are calculated automatically.
knotsis a 1-d array giving user-specified knots, if desired.
Keyword arguments common to all constructor methods:
w: weight applied to each point (length
k: Spline order (between 1 and 5; default 3).
bc: Boundary condition (default
periodic: Treat curve as periodic? (Default is
Spline2D(x, y, z; w=ones(length(x)), kx=3, ky=3, s=0.0) Spline2D(x, y, z; kx=3, ky=3, s=0.0)
Fit a 2-d spline to the input data.
ymust be 1-d arrays.
zis also a 1-d array, the inputs are assumed to represent unstructured data, with
z[i]being the function value at point
(x[i], y[i]). In this case, the lengths of all inputs must match.
zis a 2-d array, the data are assumed to be gridded:
z[i, j]is the function value at
(x[i], y[j]). In this case, it is required that
size(z) == (length(x), length(y)). (Note that when interpreting
zas a matrix,
xgives the row coordinates and
ygives the column coordinates.)
evaluate(spl, x, y)
- Evaluate the 2-d spline
(x[i], y[i]). Inputs can be Vectors or scalars. Points outside the domain of the spline are set to the values at the boundary.
evalgrid(spl, x, y)
Evaluate the 2-d spline
splat the grid points spanned by the coordinate arrays
y. The input arrays must be monotonically increasing. The output is a 2-d array with size
output[i, j]is the function value at
(x[i], y[j]). In other words, when interpreting the result as a matrix,
xgives the row coordinates and
ygives the column coordinates.
integral of a 2-d spline over the domain
[x0, x1]*[y0, y1]
integrate(spl, x0, x1, y0, y1)
Translation from scipy.interpolate
Spline classes in scipy.interpolate are also thin wrappers
for the Dierckx Fortran library. The performance of Dierckx.jl should
be similar or better than the scipy.interpolate classes. (Better for
small arrays where Python overhead is more significant.) The
equivalent of a specific classes in scipy.interpolate:
|scipy.interpolate class||Dierckx.jl constructor method|
|scipy.interpolate function||Dierckx.jl constructor method|
Dierckx.jl is distributed under a 3-clause BSD license. See LICENSE.md for details. The real*8 version of the Dierckx Fortran library as well as some test cases and error messages are copied from the scipy package, which is distributed under this license.