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671 lines
19 KiB
Python
671 lines
19 KiB
Python
from numpy import (logical_and, asarray, pi, zeros_like,
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piecewise, array, arctan2, tan, zeros, arange, floor)
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from numpy.core.umath import (sqrt, exp, greater, less, cos, add, sin,
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less_equal, greater_equal)
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# From splinemodule.c
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from .spline import cspline2d, sepfir2d
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from scipy.special import comb
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from scipy._lib._util import float_factorial
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__all__ = ['spline_filter', 'bspline', 'gauss_spline', 'cubic', 'quadratic',
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'cspline1d', 'qspline1d', 'cspline1d_eval', 'qspline1d_eval']
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def spline_filter(Iin, lmbda=5.0):
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"""Smoothing spline (cubic) filtering of a rank-2 array.
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Filter an input data set, `Iin`, using a (cubic) smoothing spline of
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fall-off `lmbda`.
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Parameters
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----------
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Iin : array like
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input data set
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lmbda : float, optional
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spline smooghing fall-off value, default is `5.0`.
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Returns
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-------
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res : ndarray
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filterd input data
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Examples
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--------
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We can filter an multi dimentional signal (ex: 2D image) using cubic
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B-spline filter:
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>>> from scipy.signal import spline_filter
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>>> import matplotlib.pyplot as plt
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>>> orig_img = np.eye(20) # create an image
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>>> orig_img[10, :] = 1.0
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>>> sp_filter = spline_filter(orig_img, lmbda=0.1)
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>>> f, ax = plt.subplots(1, 2, sharex=True)
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>>> for ind, data in enumerate([[orig_img, "original image"],
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... [sp_filter, "spline filter"]]):
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... ax[ind].imshow(data[0], cmap='gray_r')
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... ax[ind].set_title(data[1])
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>>> plt.tight_layout()
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>>> plt.show()
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"""
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intype = Iin.dtype.char
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hcol = array([1.0, 4.0, 1.0], 'f') / 6.0
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if intype in ['F', 'D']:
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Iin = Iin.astype('F')
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ckr = cspline2d(Iin.real, lmbda)
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cki = cspline2d(Iin.imag, lmbda)
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outr = sepfir2d(ckr, hcol, hcol)
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outi = sepfir2d(cki, hcol, hcol)
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out = (outr + 1j * outi).astype(intype)
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elif intype in ['f', 'd']:
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ckr = cspline2d(Iin, lmbda)
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out = sepfir2d(ckr, hcol, hcol)
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out = out.astype(intype)
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else:
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raise TypeError("Invalid data type for Iin")
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return out
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_splinefunc_cache = {}
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def _bspline_piecefunctions(order):
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"""Returns the function defined over the left-side pieces for a bspline of
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a given order.
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The 0th piece is the first one less than 0. The last piece is a function
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identical to 0 (returned as the constant 0). (There are order//2 + 2 total
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pieces).
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Also returns the condition functions that when evaluated return boolean
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arrays for use with `numpy.piecewise`.
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"""
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try:
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return _splinefunc_cache[order]
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except KeyError:
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pass
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def condfuncgen(num, val1, val2):
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if num == 0:
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return lambda x: logical_and(less_equal(x, val1),
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greater_equal(x, val2))
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elif num == 2:
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return lambda x: less_equal(x, val2)
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else:
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return lambda x: logical_and(less(x, val1),
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greater_equal(x, val2))
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last = order // 2 + 2
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if order % 2:
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startbound = -1.0
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else:
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startbound = -0.5
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condfuncs = [condfuncgen(0, 0, startbound)]
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bound = startbound
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for num in range(1, last - 1):
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condfuncs.append(condfuncgen(1, bound, bound - 1))
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bound = bound - 1
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condfuncs.append(condfuncgen(2, 0, -(order + 1) / 2.0))
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# final value of bound is used in piecefuncgen below
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# the functions to evaluate are taken from the left-hand side
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# in the general expression derived from the central difference
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# operator (because they involve fewer terms).
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fval = float_factorial(order)
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def piecefuncgen(num):
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Mk = order // 2 - num
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if (Mk < 0):
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return 0 # final function is 0
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coeffs = [(1 - 2 * (k % 2)) * float(comb(order + 1, k, exact=1)) / fval
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for k in range(Mk + 1)]
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shifts = [-bound - k for k in range(Mk + 1)]
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def thefunc(x):
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res = 0.0
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for k in range(Mk + 1):
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res += coeffs[k] * (x + shifts[k]) ** order
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return res
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return thefunc
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funclist = [piecefuncgen(k) for k in range(last)]
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_splinefunc_cache[order] = (funclist, condfuncs)
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return funclist, condfuncs
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def bspline(x, n):
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"""B-spline basis function of order n.
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Parameters
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----------
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x : array like
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a knot vector
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n : int
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The order of the spline. Must be non-negative, i.e., n >= 0
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Returns
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-------
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res : ndarray
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B-spline basis function values
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See Also
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--------
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cubic : A cubic B-spline.
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quadratic : A quadratic B-spline.
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Notes
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-----
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Uses numpy.piecewise and automatic function-generator.
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Examples
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--------
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We can calculate B-Spline basis function of several orders:
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>>> from scipy.signal import bspline, cubic, quadratic
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>>> bspline(0.0, 1)
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1
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>>> knots = [-1.0, 0.0, -1.0]
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>>> bspline(knots, 2)
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array([0.125, 0.75, 0.125])
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>>> np.array_equal(bspline(knots, 2), quadratic(knots))
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True
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>>> np.array_equal(bspline(knots, 3), cubic(knots))
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True
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"""
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ax = -abs(asarray(x))
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# number of pieces on the left-side is (n+1)/2
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funclist, condfuncs = _bspline_piecefunctions(n)
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condlist = [func(ax) for func in condfuncs]
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return piecewise(ax, condlist, funclist)
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def gauss_spline(x, n):
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r"""Gaussian approximation to B-spline basis function of order n.
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Parameters
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----------
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x : array
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a knot vector
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n : int
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The order of the spline. Must be non-negative, i.e., n >= 0
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Returns
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-------
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res : ndarray
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B-spline basis function values approximated by a zero-mean Gaussian
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function.
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Notes
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-----
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The B-spline basis function can be approximated well by a zero-mean
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Gaussian function with standard-deviation equal to :math:`\sigma=(n+1)/12`
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for large `n` :
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.. math:: \frac{1}{\sqrt {2\pi\sigma^2}}exp(-\frac{x^2}{2\sigma})
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References
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----------
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.. [1] Bouma H., Vilanova A., Bescos J.O., ter Haar Romeny B.M., Gerritsen
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F.A. (2007) Fast and Accurate Gaussian Derivatives Based on B-Splines. In:
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Sgallari F., Murli A., Paragios N. (eds) Scale Space and Variational
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Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer
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Science, vol 4485. Springer, Berlin, Heidelberg
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.. [2] http://folk.uio.no/inf3330/scripting/doc/python/SciPy/tutorial/old/node24.html
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Examples
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--------
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We can calculate B-Spline basis functions approximated by a gaussian
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distribution:
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>>> from scipy.signal import gauss_spline, bspline
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>>> knots = np.array([-1.0, 0.0, -1.0])
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>>> gauss_spline(knots, 3)
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array([0.15418033, 0.6909883, 0.15418033]) # may vary
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>>> bspline(knots, 3)
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array([0.16666667, 0.66666667, 0.16666667]) # may vary
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"""
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x = asarray(x)
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signsq = (n + 1) / 12.0
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return 1 / sqrt(2 * pi * signsq) * exp(-x ** 2 / 2 / signsq)
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def cubic(x):
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"""A cubic B-spline.
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This is a special case of `bspline`, and equivalent to ``bspline(x, 3)``.
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Parameters
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----------
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x : array like
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a knot vector
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Returns
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-------
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res : ndarray
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Cubic B-spline basis function values
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See Also
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--------
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bspline : B-spline basis function of order n
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quadratic : A quadratic B-spline.
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Examples
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--------
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We can calculate B-Spline basis function of several orders:
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>>> from scipy.signal import bspline, cubic, quadratic
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>>> bspline(0.0, 1)
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1
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>>> knots = [-1.0, 0.0, -1.0]
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>>> bspline(knots, 2)
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array([0.125, 0.75, 0.125])
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>>> np.array_equal(bspline(knots, 2), quadratic(knots))
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True
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>>> np.array_equal(bspline(knots, 3), cubic(knots))
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True
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"""
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ax = abs(asarray(x))
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res = zeros_like(ax)
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cond1 = less(ax, 1)
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if cond1.any():
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ax1 = ax[cond1]
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res[cond1] = 2.0 / 3 - 1.0 / 2 * ax1 ** 2 * (2 - ax1)
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cond2 = ~cond1 & less(ax, 2)
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if cond2.any():
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ax2 = ax[cond2]
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res[cond2] = 1.0 / 6 * (2 - ax2) ** 3
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return res
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def quadratic(x):
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"""A quadratic B-spline.
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This is a special case of `bspline`, and equivalent to ``bspline(x, 2)``.
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Parameters
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----------
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x : array like
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a knot vector
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Returns
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-------
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res : ndarray
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Quadratic B-spline basis function values
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See Also
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--------
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bspline : B-spline basis function of order n
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cubic : A cubic B-spline.
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Examples
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--------
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We can calculate B-Spline basis function of several orders:
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>>> from scipy.signal import bspline, cubic, quadratic
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>>> bspline(0.0, 1)
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1
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>>> knots = [-1.0, 0.0, -1.0]
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>>> bspline(knots, 2)
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array([0.125, 0.75, 0.125])
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>>> np.array_equal(bspline(knots, 2), quadratic(knots))
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True
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>>> np.array_equal(bspline(knots, 3), cubic(knots))
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True
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"""
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ax = abs(asarray(x))
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res = zeros_like(ax)
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cond1 = less(ax, 0.5)
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if cond1.any():
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ax1 = ax[cond1]
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res[cond1] = 0.75 - ax1 ** 2
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cond2 = ~cond1 & less(ax, 1.5)
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if cond2.any():
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ax2 = ax[cond2]
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res[cond2] = (ax2 - 1.5) ** 2 / 2.0
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return res
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def _coeff_smooth(lam):
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xi = 1 - 96 * lam + 24 * lam * sqrt(3 + 144 * lam)
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omeg = arctan2(sqrt(144 * lam - 1), sqrt(xi))
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rho = (24 * lam - 1 - sqrt(xi)) / (24 * lam)
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rho = rho * sqrt((48 * lam + 24 * lam * sqrt(3 + 144 * lam)) / xi)
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return rho, omeg
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def _hc(k, cs, rho, omega):
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return (cs / sin(omega) * (rho ** k) * sin(omega * (k + 1)) *
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greater(k, -1))
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def _hs(k, cs, rho, omega):
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c0 = (cs * cs * (1 + rho * rho) / (1 - rho * rho) /
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(1 - 2 * rho * rho * cos(2 * omega) + rho ** 4))
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gamma = (1 - rho * rho) / (1 + rho * rho) / tan(omega)
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ak = abs(k)
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return c0 * rho ** ak * (cos(omega * ak) + gamma * sin(omega * ak))
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def _cubic_smooth_coeff(signal, lamb):
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rho, omega = _coeff_smooth(lamb)
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cs = 1 - 2 * rho * cos(omega) + rho * rho
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K = len(signal)
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yp = zeros((K,), signal.dtype.char)
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k = arange(K)
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yp[0] = (_hc(0, cs, rho, omega) * signal[0] +
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add.reduce(_hc(k + 1, cs, rho, omega) * signal))
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yp[1] = (_hc(0, cs, rho, omega) * signal[0] +
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_hc(1, cs, rho, omega) * signal[1] +
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add.reduce(_hc(k + 2, cs, rho, omega) * signal))
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for n in range(2, K):
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yp[n] = (cs * signal[n] + 2 * rho * cos(omega) * yp[n - 1] -
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rho * rho * yp[n - 2])
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y = zeros((K,), signal.dtype.char)
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y[K - 1] = add.reduce((_hs(k, cs, rho, omega) +
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_hs(k + 1, cs, rho, omega)) * signal[::-1])
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y[K - 2] = add.reduce((_hs(k - 1, cs, rho, omega) +
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_hs(k + 2, cs, rho, omega)) * signal[::-1])
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for n in range(K - 3, -1, -1):
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y[n] = (cs * yp[n] + 2 * rho * cos(omega) * y[n + 1] -
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rho * rho * y[n + 2])
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return y
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def _cubic_coeff(signal):
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zi = -2 + sqrt(3)
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K = len(signal)
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yplus = zeros((K,), signal.dtype.char)
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powers = zi ** arange(K)
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yplus[0] = signal[0] + zi * add.reduce(powers * signal)
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for k in range(1, K):
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yplus[k] = signal[k] + zi * yplus[k - 1]
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output = zeros((K,), signal.dtype)
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output[K - 1] = zi / (zi - 1) * yplus[K - 1]
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for k in range(K - 2, -1, -1):
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output[k] = zi * (output[k + 1] - yplus[k])
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return output * 6.0
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def _quadratic_coeff(signal):
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zi = -3 + 2 * sqrt(2.0)
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K = len(signal)
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yplus = zeros((K,), signal.dtype.char)
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powers = zi ** arange(K)
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yplus[0] = signal[0] + zi * add.reduce(powers * signal)
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for k in range(1, K):
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yplus[k] = signal[k] + zi * yplus[k - 1]
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output = zeros((K,), signal.dtype.char)
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output[K - 1] = zi / (zi - 1) * yplus[K - 1]
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for k in range(K - 2, -1, -1):
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output[k] = zi * (output[k + 1] - yplus[k])
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return output * 8.0
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def cspline1d(signal, lamb=0.0):
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"""
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Compute cubic spline coefficients for rank-1 array.
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Find the cubic spline coefficients for a 1-D signal assuming
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mirror-symmetric boundary conditions. To obtain the signal back from the
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spline representation mirror-symmetric-convolve these coefficients with a
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length 3 FIR window [1.0, 4.0, 1.0]/ 6.0 .
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Parameters
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----------
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signal : ndarray
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A rank-1 array representing samples of a signal.
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lamb : float, optional
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Smoothing coefficient, default is 0.0.
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Returns
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-------
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c : ndarray
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Cubic spline coefficients.
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See Also
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--------
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cspline1d_eval : Evaluate a cubic spline at the new set of points.
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Examples
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--------
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We can filter a signal to reduce and smooth out high-frequency noise with
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a cubic spline:
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>>> import matplotlib.pyplot as plt
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>>> from scipy.signal import cspline1d, cspline1d_eval
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>>> sig = np.repeat([0., 1., 0.], 100)
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>>> sig += np.random.randn(len(sig))*0.05 # add noise
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>>> time = np.linspace(0, len(sig))
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>>> filtered = cspline1d_eval(cspline1d(sig), time)
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>>> plt.plot(sig, label="signal")
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>>> plt.plot(time, filtered, label="filtered")
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>>> plt.legend()
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>>> plt.show()
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"""
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if lamb != 0.0:
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return _cubic_smooth_coeff(signal, lamb)
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else:
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return _cubic_coeff(signal)
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def qspline1d(signal, lamb=0.0):
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"""Compute quadratic spline coefficients for rank-1 array.
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Parameters
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----------
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signal : ndarray
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A rank-1 array representing samples of a signal.
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lamb : float, optional
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Smoothing coefficient (must be zero for now).
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Returns
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-------
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c : ndarray
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Quadratic spline coefficients.
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See Also
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--------
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qspline1d_eval : Evaluate a quadratic spline at the new set of points.
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Notes
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-----
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Find the quadratic spline coefficients for a 1-D signal assuming
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mirror-symmetric boundary conditions. To obtain the signal back from the
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spline representation mirror-symmetric-convolve these coefficients with a
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length 3 FIR window [1.0, 6.0, 1.0]/ 8.0 .
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Examples
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--------
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We can filter a signal to reduce and smooth out high-frequency noise with
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a quadratic spline:
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>>> import matplotlib.pyplot as plt
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>>> from scipy.signal import qspline1d, qspline1d_eval
|
|
>>> sig = np.repeat([0., 1., 0.], 100)
|
|
>>> sig += np.random.randn(len(sig))*0.05 # add noise
|
|
>>> time = np.linspace(0, len(sig))
|
|
>>> filtered = qspline1d_eval(qspline1d(sig), time)
|
|
>>> plt.plot(sig, label="signal")
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|
>>> 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 matplotlib.pyplot as plt
|
|
>>> from scipy.signal import cspline1d, cspline1d_eval
|
|
>>> sig = np.repeat([0., 1., 0.], 100)
|
|
>>> sig += np.random.randn(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 matplotlib.pyplot as plt
|
|
>>> from scipy.signal import qspline1d, qspline1d_eval
|
|
>>> sig = np.repeat([0., 1., 0.], 100)
|
|
>>> sig += np.random.randn(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
|