RBF Factory

Factory class for radial basis functions

class RBFFactory(fname)[source]

Bases: object

Factory class that spawns the radial basis functions.

Example:

>>> from pygem import RBFFactory
>>> import numpy as np
>>> x = np.linspace(0, 1)
>>> for fname in RBFFactory.bases:
>>>     y = RBFFactory(fname)(x)

static gaussian_spline(X, r=1)[source]

It implements the following formula:

\[\varphi(\boldsymbol{x}) = e^{-\frac{\boldsymbol{x}^2}{r^2}}\]

Parameters:

X (numpy.ndarray) – the norm x in the formula above.
r (float) – the parameter r in the formula above.

Returns:

result: the result of the formula above.

Return type:

float

static multi_quadratic_biharmonic_spline(X, r=1)[source]

It implements the following formula:

\[\varphi(\boldsymbol{x}) = \sqrt{\boldsymbol{x}^2 + r^2}\]

Parameters:

X (numpy.ndarray) – the norm x in the formula above.
r (float) – the parameter r in the formula above.

Returns:

result: the result of the formula above.

Return type:

float

static inv_multi_quadratic_biharmonic_spline(X, r=1)[source]

It implements the following formula:

\[\varphi(\boldsymbol{x}) = (\boldsymbol{x}^2 + r^2 )^{-\frac{1}{2}}\]

Parameters:

X (numpy.ndarray) – the norm x in the formula above.
r (float) – the parameter r in the formula above.

Returns:

result: the result of the formula above.

Return type:

float

static thin_plate_spline(X, r=1, k=2)[source]

It implements the following formula:

\[ \begin{align}\begin{aligned}\varphi(\boldsymbol{x}) = \left(\frac{\boldsymbol{x}}{r}\right)^k \ln\frac{\boldsymbol{x}}{r}\\With k=2 the function is "radius free", that means independent of radius value.\end{aligned}\end{align} \]

Parameters:

X (numpy.ndarray) – the norm x in the formula above.
r (float) – the parameter r in the formula above.
k (float) – the parameter k in the formula above.

Returns:

result: the result of the formula above.

Return type:

float

static beckert_wendland_c2_basis(X, r=1)[source]

It implements the following formula:

\[\varphi(\boldsymbol{x}) = \left( 1 - \frac{\boldsymbol{x}}{r}\right)^4 + \left( 4 \frac{ \boldsymbol{x} }{r} + 1 \right)\]

Parameters:

X (numpy.ndarray) – the norm x in the formula above.
r (float) – the parameter r in the formula above.

Returns:

result: the result of the formula above.

Return type:

float

static polyharmonic_spline(X, r=1, k=2)[source]

It implements the following formula:

\[\begin{split}\varphi(\boldsymbol{x}) = \begin{cases} \frac{\boldsymbol{x}}{r}^k \quad & \text{if}~k = 1,3,5,...\\ \frac{\boldsymbol{x}}{r}^{k-1} \ln(\frac{\boldsymbol{x}}{r}^ {\frac{\boldsymbol{x}}{r}}) \quad & \text{if}~\frac{\boldsymbol{x}}{r} < 1, ~k = 2,4,6,...\\ \frac{\boldsymbol{x}}{r}^k \ln(\frac{\boldsymbol{x}}{r}) \quad & \text{if}~\frac{\boldsymbol{x}}{r} \ge 1, ~k = 2,4,6,...\\ \end{cases}\end{split}\]

Parameters:

X (numpy.ndarray) – the norm x in the formula above.
r (float) – the parameter r in the formula above.

Returns:

result: the result of the formula above.

Return type:

float