/
_bsplines.py
519 lines (421 loc) · 15.4 KB
/
_bsplines.py
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from numpy import (asarray, pi, zeros_like,
array, arctan2, tan, ones, arange, floor,
r_, atleast_1d, sqrt, exp, greater, cos, add, sin)
# From splinemodule.c
from ._spline import cspline2d, sepfir2d
from ._signaltools import lfilter, sosfilt, lfiltic
from scipy.interpolate import BSpline
__all__ = ['spline_filter', 'gauss_spline',
'cspline1d', 'qspline1d', 'cspline1d_eval', 'qspline1d_eval']
def spline_filter(Iin, lmbda=5.0):
"""Smoothing spline (cubic) filtering of a rank-2 array.
Filter an input data set, `Iin`, using a (cubic) smoothing spline of
fall-off `lmbda`.
Parameters
----------
Iin : array_like
input data set
lmbda : float, optional
spline smooghing fall-off value, default is `5.0`.
Returns
-------
res : ndarray
filtered input data
Examples
--------
We can filter an multi dimensional signal (ex: 2D image) using cubic
B-spline filter:
>>> import numpy as np
>>> from scipy.signal import spline_filter
>>> import matplotlib.pyplot as plt
>>> orig_img = np.eye(20) # create an image
>>> orig_img[10, :] = 1.0
>>> sp_filter = spline_filter(orig_img, lmbda=0.1)
>>> f, ax = plt.subplots(1, 2, sharex=True)
>>> for ind, data in enumerate([[orig_img, "original image"],
... [sp_filter, "spline filter"]]):
... ax[ind].imshow(data[0], cmap='gray_r')
... ax[ind].set_title(data[1])
>>> plt.tight_layout()
>>> plt.show()
"""
intype = Iin.dtype.char
hcol = array([1.0, 4.0, 1.0], 'f') / 6.0
if intype in ['F', 'D']:
Iin = Iin.astype('F')
ckr = cspline2d(Iin.real, lmbda)
cki = cspline2d(Iin.imag, lmbda)
outr = sepfir2d(ckr, hcol, hcol)
outi = sepfir2d(cki, hcol, hcol)
out = (outr + 1j * outi).astype(intype)
elif intype in ['f', 'd']:
ckr = cspline2d(Iin, lmbda)
out = sepfir2d(ckr, hcol, hcol)
out = out.astype(intype)
else:
raise TypeError("Invalid data type for Iin")
return out
_splinefunc_cache = {}
def gauss_spline(x, n):
r"""Gaussian approximation to B-spline basis function of order n.
Parameters
----------
x : array_like
a knot vector
n : int
The order of the spline. Must be non-negative, i.e., n >= 0
Returns
-------
res : ndarray
B-spline basis function values approximated by a zero-mean Gaussian
function.
Notes
-----
The B-spline basis function can be approximated well by a zero-mean
Gaussian function with standard-deviation equal to :math:`\sigma=(n+1)/12`
for large `n` :
.. math:: \frac{1}{\sqrt {2\pi\sigma^2}}exp(-\frac{x^2}{2\sigma})
References
----------
.. [1] Bouma H., Vilanova A., Bescos J.O., ter Haar Romeny B.M., Gerritsen
F.A. (2007) Fast and Accurate Gaussian Derivatives Based on B-Splines. In:
Sgallari F., Murli A., Paragios N. (eds) Scale Space and Variational
Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer
Science, vol 4485. Springer, Berlin, Heidelberg
.. [2] http://folk.uio.no/inf3330/scripting/doc/python/SciPy/tutorial/old/node24.html
Examples
--------
We can calculate B-Spline basis functions approximated by a gaussian
distribution:
>>> import numpy as np
>>> from scipy.signal import gauss_spline
>>> knots = np.array([-1.0, 0.0, -1.0])
>>> gauss_spline(knots, 3)
array([0.15418033, 0.6909883, 0.15418033]) # may vary
"""
x = asarray(x)
signsq = (n + 1) / 12.0
return 1 / sqrt(2 * pi * signsq) * exp(-x ** 2 / 2 / signsq)
def _cubic(x):
x = asarray(x, dtype=float)
b = BSpline.basis_element([-2, -1, 0, 1, 2], extrapolate=False)
out = b(x)
out[(x < -2) | (x > 2)] = 0
return out
def _quadratic(x):
x = abs(asarray(x, dtype=float))
b = BSpline.basis_element([-1.5, -0.5, 0.5, 1.5], extrapolate=False)
out = b(x)
out[(x < -1.5) | (x > 1.5)] = 0
return out
def _coeff_smooth(lam):
xi = 1 - 96 * lam + 24 * lam * sqrt(3 + 144 * lam)
omeg = arctan2(sqrt(144 * lam - 1), sqrt(xi))
rho = (24 * lam - 1 - sqrt(xi)) / (24 * lam)
rho = rho * sqrt((48 * lam + 24 * lam * sqrt(3 + 144 * lam)) / xi)
return rho, omeg
def _hc(k, cs, rho, omega):
return (cs / sin(omega) * (rho ** k) * sin(omega * (k + 1)) *
greater(k, -1))
def _hs(k, cs, rho, omega):
c0 = (cs * cs * (1 + rho * rho) / (1 - rho * rho) /
(1 - 2 * rho * rho * cos(2 * omega) + rho ** 4))
gamma = (1 - rho * rho) / (1 + rho * rho) / tan(omega)
ak = abs(k)
return c0 * rho ** ak * (cos(omega * ak) + gamma * sin(omega * ak))
def _cubic_smooth_coeff(signal, lamb):
rho, omega = _coeff_smooth(lamb)
cs = 1 - 2 * rho * cos(omega) + rho * rho
K = len(signal)
k = arange(K)
zi_2 = (_hc(0, cs, rho, omega) * signal[0] +
add.reduce(_hc(k + 1, cs, rho, omega) * signal))
zi_1 = (_hc(0, cs, rho, omega) * signal[0] +
_hc(1, cs, rho, omega) * signal[1] +
add.reduce(_hc(k + 2, cs, rho, omega) * signal))
# Forward filter:
# for n in range(2, K):
# yp[n] = (cs * signal[n] + 2 * rho * cos(omega) * yp[n - 1] -
# rho * rho * yp[n - 2])
zi = lfiltic(cs, r_[1, -2 * rho * cos(omega), rho * rho], r_[zi_1, zi_2])
zi = zi.reshape(1, -1)
sos = r_[cs, 0, 0, 1, -2 * rho * cos(omega), rho * rho]
sos = sos.reshape(1, -1)
yp, _ = sosfilt(sos, signal[2:], zi=zi)
yp = r_[zi_2, zi_1, yp]
# Reverse filter:
# for n in range(K - 3, -1, -1):
# y[n] = (cs * yp[n] + 2 * rho * cos(omega) * y[n + 1] -
# rho * rho * y[n + 2])
zi_2 = add.reduce((_hs(k, cs, rho, omega) +
_hs(k + 1, cs, rho, omega)) * signal[::-1])
zi_1 = add.reduce((_hs(k - 1, cs, rho, omega) +
_hs(k + 2, cs, rho, omega)) * signal[::-1])
zi = lfiltic(cs, r_[1, -2 * rho * cos(omega), rho * rho], r_[zi_1, zi_2])
zi = zi.reshape(1, -1)
y, _ = sosfilt(sos, yp[-3::-1], zi=zi)
y = r_[y[::-1], zi_1, zi_2]
return y
def _cubic_coeff(signal):
zi = -2 + sqrt(3)
K = len(signal)
powers = zi ** arange(K)
if K == 1:
yplus = signal[0] + zi * add.reduce(powers * signal)
output = zi / (zi - 1) * yplus
return atleast_1d(output)
# Forward filter:
# yplus[0] = signal[0] + zi * add.reduce(powers * signal)
# for k in range(1, K):
# yplus[k] = signal[k] + zi * yplus[k - 1]
state = lfiltic(1, r_[1, -zi], atleast_1d(add.reduce(powers * signal)))
b = ones(1)
a = r_[1, -zi]
yplus, _ = lfilter(b, a, signal, zi=state)
# Reverse filter:
# output[K - 1] = zi / (zi - 1) * yplus[K - 1]
# for k in range(K - 2, -1, -1):
# output[k] = zi * (output[k + 1] - yplus[k])
out_last = zi / (zi - 1) * yplus[K - 1]
state = lfiltic(-zi, r_[1, -zi], atleast_1d(out_last))
b = asarray([-zi])
output, _ = lfilter(b, a, yplus[-2::-1], zi=state)
output = r_[output[::-1], out_last]
return output * 6.0
def _quadratic_coeff(signal):
zi = -3 + 2 * sqrt(2.0)
K = len(signal)
powers = zi ** arange(K)
if K == 1:
yplus = signal[0] + zi * add.reduce(powers * signal)
output = zi / (zi - 1) * yplus
return atleast_1d(output)
# Forward filter:
# yplus[0] = signal[0] + zi * add.reduce(powers * signal)
# for k in range(1, K):
# yplus[k] = signal[k] + zi * yplus[k - 1]
state = lfiltic(1, r_[1, -zi], atleast_1d(add.reduce(powers * signal)))
b = ones(1)
a = r_[1, -zi]
yplus, _ = lfilter(b, a, signal, zi=state)
# Reverse filter:
# output[K - 1] = zi / (zi - 1) * yplus[K - 1]
# for k in range(K - 2, -1, -1):
# output[k] = zi * (output[k + 1] - yplus[k])
out_last = zi / (zi - 1) * yplus[K - 1]
state = lfiltic(-zi, r_[1, -zi], atleast_1d(out_last))
b = asarray([-zi])
output, _ = lfilter(b, a, yplus[-2::-1], zi=state)
output = r_[output[::-1], out_last]
return output * 8.0
def cspline1d(signal, lamb=0.0):
"""
Compute cubic spline coefficients for rank-1 array.
Find the cubic spline coefficients for a 1-D signal assuming
mirror-symmetric boundary conditions. To obtain the signal back from the
spline representation mirror-symmetric-convolve these coefficients with a
length 3 FIR window [1.0, 4.0, 1.0]/ 6.0 .
Parameters
----------
signal : ndarray
A rank-1 array representing samples of a signal.
lamb : float, optional
Smoothing coefficient, default is 0.0.
Returns
-------
c : ndarray
Cubic spline coefficients.
See Also
--------
cspline1d_eval : Evaluate a cubic spline at the new set of points.
Examples
--------
We can filter a signal to reduce and smooth out high-frequency noise with
a cubic spline:
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.signal import cspline1d, cspline1d_eval
>>> rng = np.random.default_rng()
>>> sig = np.repeat([0., 1., 0.], 100)
>>> sig += rng.standard_normal(len(sig))*0.05 # add noise
>>> time = np.linspace(0, len(sig))
>>> filtered = cspline1d_eval(cspline1d(sig), time)
>>> plt.plot(sig, label="signal")
>>> plt.plot(time, filtered, label="filtered")
>>> plt.legend()
>>> plt.show()
"""
if lamb != 0.0:
return _cubic_smooth_coeff(signal, lamb)
else:
return _cubic_coeff(signal)
def qspline1d(signal, lamb=0.0):
"""Compute quadratic spline coefficients for rank-1 array.
Parameters
----------
signal : ndarray
A rank-1 array representing samples of a signal.
lamb : float, optional
Smoothing coefficient (must be zero for now).
Returns
-------
c : ndarray
Quadratic spline coefficients.
See Also
--------
qspline1d_eval : Evaluate a quadratic spline at the new set of points.
Notes
-----
Find the quadratic spline coefficients for a 1-D signal assuming
mirror-symmetric boundary conditions. To obtain the signal back from the
spline representation mirror-symmetric-convolve these coefficients with a
length 3 FIR window [1.0, 6.0, 1.0]/ 8.0 .
Examples
--------
We can filter a signal to reduce and smooth out high-frequency noise with
a quadratic spline:
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.signal import qspline1d, qspline1d_eval
>>> rng = np.random.default_rng()
>>> sig = np.repeat([0., 1., 0.], 100)
>>> sig += rng.standard_normal(len(sig))*0.05 # add noise
>>> time = np.linspace(0, len(sig))
>>> filtered = qspline1d_eval(qspline1d(sig), time)
>>> plt.plot(sig, label="signal")
>>> plt.plot(time, filtered, label="filtered")
>>> plt.legend()
>>> plt.show()
"""
if lamb != 0.0:
raise ValueError("Smoothing quadratic splines not supported yet.")
else:
return _quadratic_coeff(signal)
def cspline1d_eval(cj, newx, dx=1.0, x0=0):
"""Evaluate a cubic spline at the new set of points.
`dx` is the old sample-spacing while `x0` was the old origin. In
other-words the old-sample points (knot-points) for which the `cj`
represent spline coefficients were at equally-spaced points of:
oldx = x0 + j*dx j=0...N-1, with N=len(cj)
Edges are handled using mirror-symmetric boundary conditions.
Parameters
----------
cj : ndarray
cublic spline coefficients
newx : ndarray
New set of points.
dx : float, optional
Old sample-spacing, the default value is 1.0.
x0 : int, optional
Old origin, the default value is 0.
Returns
-------
res : ndarray
Evaluated a cubic spline points.
See Also
--------
cspline1d : Compute cubic spline coefficients for rank-1 array.
Examples
--------
We can filter a signal to reduce and smooth out high-frequency noise with
a cubic spline:
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.signal import cspline1d, cspline1d_eval
>>> rng = np.random.default_rng()
>>> sig = np.repeat([0., 1., 0.], 100)
>>> sig += rng.standard_normal(len(sig))*0.05 # add noise
>>> time = np.linspace(0, len(sig))
>>> filtered = cspline1d_eval(cspline1d(sig), time)
>>> plt.plot(sig, label="signal")
>>> plt.plot(time, filtered, label="filtered")
>>> plt.legend()
>>> plt.show()
"""
newx = (asarray(newx) - x0) / float(dx)
res = zeros_like(newx, dtype=cj.dtype)
if res.size == 0:
return res
N = len(cj)
cond1 = newx < 0
cond2 = newx > (N - 1)
cond3 = ~(cond1 | cond2)
# handle general mirror-symmetry
res[cond1] = cspline1d_eval(cj, -newx[cond1])
res[cond2] = cspline1d_eval(cj, 2 * (N - 1) - newx[cond2])
newx = newx[cond3]
if newx.size == 0:
return res
result = zeros_like(newx, dtype=cj.dtype)
jlower = floor(newx - 2).astype(int) + 1
for i in range(4):
thisj = jlower + i
indj = thisj.clip(0, N - 1) # handle edge cases
result += cj[indj] * _cubic(newx - thisj)
res[cond3] = result
return res
def qspline1d_eval(cj, newx, dx=1.0, x0=0):
"""Evaluate a quadratic spline at the new set of points.
Parameters
----------
cj : ndarray
Quadratic spline coefficients
newx : ndarray
New set of points.
dx : float, optional
Old sample-spacing, the default value is 1.0.
x0 : int, optional
Old origin, the default value is 0.
Returns
-------
res : ndarray
Evaluated a quadratic spline points.
See Also
--------
qspline1d : Compute quadratic spline coefficients for rank-1 array.
Notes
-----
`dx` is the old sample-spacing while `x0` was the old origin. In
other-words the old-sample points (knot-points) for which the `cj`
represent spline coefficients were at equally-spaced points of::
oldx = x0 + j*dx j=0...N-1, with N=len(cj)
Edges are handled using mirror-symmetric boundary conditions.
Examples
--------
We can filter a signal to reduce and smooth out high-frequency noise with
a quadratic spline:
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.signal import qspline1d, qspline1d_eval
>>> rng = np.random.default_rng()
>>> sig = np.repeat([0., 1., 0.], 100)
>>> sig += rng.standard_normal(len(sig))*0.05 # add noise
>>> time = np.linspace(0, len(sig))
>>> filtered = qspline1d_eval(qspline1d(sig), time)
>>> plt.plot(sig, label="signal")
>>> plt.plot(time, filtered, label="filtered")
>>> plt.legend()
>>> plt.show()
"""
newx = (asarray(newx) - x0) / dx
res = zeros_like(newx)
if res.size == 0:
return res
N = len(cj)
cond1 = newx < 0
cond2 = newx > (N - 1)
cond3 = ~(cond1 | cond2)
# handle general mirror-symmetry
res[cond1] = qspline1d_eval(cj, -newx[cond1])
res[cond2] = qspline1d_eval(cj, 2 * (N - 1) - newx[cond2])
newx = newx[cond3]
if newx.size == 0:
return res
result = zeros_like(newx)
jlower = floor(newx - 1.5).astype(int) + 1
for i in range(3):
thisj = jlower + i
indj = thisj.clip(0, N - 1) # handle edge cases
result += cj[indj] * _quadratic(newx - thisj)
res[cond3] = result
return res