def get_sampling_w(t: npt.ArrayLike,
oversampling: Optional[int] = 8,
max_freq: Optional[int] = 1) -> npt.ArrayLike:
'''
Get sampling frequency of time-series
Args:
t: Sampling times
oversampling: Oversampling factor
max_freq: Maximum frequency scaling factor. Any value over 1 will
ignore Nyquists frequency limit
Returns:
Sampling frequencies spaced by 1/(T * oversampling) where T
is the interval spanned by t
'''
T = (t.max() - t.min())
N = t.shape[0]
return np.arange(1 / (T * oversampling), max_freq * N / (2 * T),
1 / (T * oversampling))
def geometric_slerp(
start: npt.ArrayLike,
end: npt.ArrayLike,
t: npt.ArrayLike,
tol: float = 1e-7,
) -> np.ndarray:
"""
Geometric spherical linear interpolation.
The interpolation occurs along a unit-radius
great circle arc in arbitrary dimensional space.
Parameters
----------
start : (n_dimensions, ) array-like
Single n-dimensional input coordinate in a 1-D array-like
object. `n` must be greater than 1.
end : (n_dimensions, ) array-like
Single n-dimensional input coordinate in a 1-D array-like
object. `n` must be greater than 1.
t: float or (n_points,) array-like
A float or array-like of doubles representing interpolation
parameters, with values required in the inclusive interval
between 0 and 1. A common approach is to generate the array
with ``np.linspace(0, 1, n_pts)`` for linearly spaced points.
Ascending, descending, and scrambled orders are permitted.
tol: float
The absolute tolerance for determining if the start and end
coordinates are antipodes.
Returns
-------
result : (t.size, D)
An array of doubles containing the interpolated
spherical path and including start and
end when 0 and 1 t are used. The
interpolated values should correspond to the
same sort order provided in the t array. The result
may be 1-dimensional if ``t`` is a float.
Raises
------
ValueError
If ``start`` and ``end`` are antipodes, not on the
unit n-sphere, or for a variety of degenerate conditions.
Notes
-----
The implementation is based on the mathematical formula provided in [1]_,
and the first known presentation of this algorithm, derived from study of
4-D geometry, is credited to Glenn Davis in a footnote of the original
quaternion Slerp publication by Ken Shoemake [2]_.
.. versionadded:: 1.5.0
References
----------
.. [1] https://en.wikipedia.org/wiki/Slerp#Geometric_Slerp
.. [2] Ken Shoemake (1985) Animating rotation with quaternion curves.
ACM SIGGRAPH Computer Graphics, 19(3): 245-254.
See Also
--------
scipy.spatial.transform.Slerp : 3-D Slerp that works with quaternions
Examples
--------
Interpolate four linearly-spaced values on the circumference of
a circle spanning 90 degrees:
>>> from scipy.spatial import geometric_slerp
>>> import matplotlib.pyplot as plt
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> start = np.array([1, 0])
>>> end = np.array([0, 1])
>>> t_vals = np.linspace(0, 1, 4)
>>> result = geometric_slerp(start,
... end,
... t_vals)
The interpolated results should be at 30 degree intervals
recognizable on the unit circle:
>>> ax.scatter(result[...,0], result[...,1], c='k')
>>> circle = plt.Circle((0, 0), 1, color='grey')
>>> ax.add_artist(circle)
>>> ax.set_aspect('equal')
>>> plt.show()
Attempting to interpolate between antipodes on a circle is
ambiguous because there are two possible paths, and on a
sphere there are infinite possible paths on the geodesic surface.
Nonetheless, one of the ambiguous paths is returned along
with a warning:
>>> opposite_pole = np.array([-1, 0])
>>> with np.testing.suppress_warnings() as sup:
... sup.filter(UserWarning)
... geometric_slerp(start,
... opposite_pole,
... t_vals)
array([[ 1.00000000e+00, 0.00000000e+00],
[ 5.00000000e-01, 8.66025404e-01],
[-5.00000000e-01, 8.66025404e-01],
[-1.00000000e+00, 1.22464680e-16]])
Extend the original example to a sphere and plot interpolation
points in 3D:
>>> from mpl_toolkits.mplot3d import proj3d
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111, projection='3d')
Plot the unit sphere for reference (optional):
>>> u = np.linspace(0, 2 * np.pi, 100)
>>> v = np.linspace(0, np.pi, 100)
>>> x = np.outer(np.cos(u), np.sin(v))
>>> y = np.outer(np.sin(u), np.sin(v))
>>> z = np.outer(np.ones(np.size(u)), np.cos(v))
>>> ax.plot_surface(x, y, z, color='y', alpha=0.1)
Interpolating over a larger number of points
may provide the appearance of a smooth curve on
the surface of the sphere, which is also useful
for discretized integration calculations on a
sphere surface:
>>> start = np.array([1, 0, 0])
>>> end = np.array([0, 0, 1])
>>> t_vals = np.linspace(0, 1, 200)
>>> result = geometric_slerp(start,
... end,
... t_vals)
>>> ax.plot(result[...,0],
... result[...,1],
... result[...,2],
... c='k')
>>> plt.show()
"""
start = np.asarray(start, dtype=np.float64)
end = np.asarray(end, dtype=np.float64)
if start.ndim != 1 or end.ndim != 1:
raise ValueError("Start and end coordinates "
"must be one-dimensional")
if start.size != end.size:
raise ValueError("The dimensions of start and "
"end must match (have same size)")
if start.size < 2 or end.size < 2:
raise ValueError("The start and end coordinates must "
"both be in at least two-dimensional "
"space")
if np.array_equal(start, end):
return np.linspace(start, start, np.asarray(t).size)
# for points that violate equation for n-sphere
for coord in [start, end]:
if not np.allclose(np.linalg.norm(coord), 1.0, rtol=1e-9, atol=0):
raise ValueError("start and end are not" " on a unit n-sphere")
if not isinstance(tol, float):
raise ValueError("tol must be a float")
else:
tol = np.fabs(tol)
coord_dist = euclidean(start, end)
# diameter of 2 within tolerance means antipodes, which is a problem
# for all unit n-spheres (even the 0-sphere would have an ambiguous path)
if np.allclose(coord_dist, 2.0, rtol=0, atol=tol):
warnings.warn("start and end are antipodes"
" using the specified tolerance;"
" this may cause ambiguous slerp paths")
t = np.asarray(t, dtype=np.float64)
if t.size == 0:
return np.empty((0, start.size))
if t.min() < 0 or t.max() > 1:
raise ValueError("interpolation parameter must be in [0, 1]")
if t.ndim == 0:
return _geometric_slerp(start, end, np.atleast_1d(t)).ravel()
else:
return _geometric_slerp(start, end, t)