You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
395 lines
12 KiB
Python
395 lines
12 KiB
Python
6 years ago
|
from __future__ import division, print_function, absolute_import
|
||
|
|
|
||
|
|
from scipy._lib.six import xrange
|
||
|
|
from numpy import (logical_and, asarray, pi, zeros_like,
|
||
|
|
piecewise, array, arctan2, tan, zeros, arange, floor)
|
||
|
|
from numpy.core.umath import (sqrt, exp, greater, less, cos, add, sin,
|
||
|
|
less_equal, greater_equal)
|
||
|
|
|
||
|
|
# From splinemodule.c
|
||
|
|
from .spline import cspline2d, sepfir2d
|
||
|
|
|
||
|
|
from scipy.special import comb, gamma
|
||
|
|
|
||
|
|
__all__ = ['spline_filter', 'bspline', 'gauss_spline', 'cubic', 'quadratic',
|
||
|
|
'cspline1d', 'qspline1d', 'cspline1d_eval', 'qspline1d_eval']
|
||
|
|
|
||
|
|
|
||
|
|
def factorial(n):
|
||
|
|
return gamma(n + 1)
|
||
|
|
|
||
|
|
|
||
|
|
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`.
|
||
|
|
"""
|
||
|
|
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 _bspline_piecefunctions(order):
|
||
|
|
"""Returns the function defined over the left-side pieces for a bspline of
|
||
|
|
a given order.
|
||
|
|
|
||
|
|
The 0th piece is the first one less than 0. The last piece is a function
|
||
|
|
identical to 0 (returned as the constant 0). (There are order//2 + 2 total
|
||
|
|
pieces).
|
||
|
|
|
||
|
|
Also returns the condition functions that when evaluated return boolean
|
||
|
|
arrays for use with `numpy.piecewise`.
|
||
|
|
"""
|
||
|
|
try:
|
||
|
|
return _splinefunc_cache[order]
|
||
|
|
except KeyError:
|
||
|
|
pass
|
||
|
|
|
||
|
|
def condfuncgen(num, val1, val2):
|
||
|
|
if num == 0:
|
||
|
|
return lambda x: logical_and(less_equal(x, val1),
|
||
|
|
greater_equal(x, val2))
|
||
|
|
elif num == 2:
|
||
|
|
return lambda x: less_equal(x, val2)
|
||
|
|
else:
|
||
|
|
return lambda x: logical_and(less(x, val1),
|
||
|
|
greater_equal(x, val2))
|
||
|
|
|
||
|
|
last = order // 2 + 2
|
||
|
|
if order % 2:
|
||
|
|
startbound = -1.0
|
||
|
|
else:
|
||
|
|
startbound = -0.5
|
||
|
|
condfuncs = [condfuncgen(0, 0, startbound)]
|
||
|
|
bound = startbound
|
||
|
|
for num in xrange(1, last - 1):
|
||
|
|
condfuncs.append(condfuncgen(1, bound, bound - 1))
|
||
|
|
bound = bound - 1
|
||
|
|
condfuncs.append(condfuncgen(2, 0, -(order + 1) / 2.0))
|
||
|
|
|
||
|
|
# final value of bound is used in piecefuncgen below
|
||
|
|
|
||
|
|
# the functions to evaluate are taken from the left-hand-side
|
||
|
|
# in the general expression derived from the central difference
|
||
|
|
# operator (because they involve fewer terms).
|
||
|
|
|
||
|
|
fval = factorial(order)
|
||
|
|
|
||
|
|
def piecefuncgen(num):
|
||
|
|
Mk = order // 2 - num
|
||
|
|
if (Mk < 0):
|
||
|
|
return 0 # final function is 0
|
||
|
|
coeffs = [(1 - 2 * (k % 2)) * float(comb(order + 1, k, exact=1)) / fval
|
||
|
|
for k in xrange(Mk + 1)]
|
||
|
|
shifts = [-bound - k for k in xrange(Mk + 1)]
|
||
|
|
|
||
|
|
def thefunc(x):
|
||
|
|
res = 0.0
|
||
|
|
for k in range(Mk + 1):
|
||
|
|
res += coeffs[k] * (x + shifts[k]) ** order
|
||
|
|
return res
|
||
|
|
return thefunc
|
||
|
|
|
||
|
|
funclist = [piecefuncgen(k) for k in xrange(last)]
|
||
|
|
|
||
|
|
_splinefunc_cache[order] = (funclist, condfuncs)
|
||
|
|
|
||
|
|
return funclist, condfuncs
|
||
|
|
|
||
|
|
|
||
|
|
def bspline(x, n):
|
||
|
|
"""B-spline basis function of order n.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Uses numpy.piecewise and automatic function-generator.
|
||
|
|
|
||
|
|
"""
|
||
|
|
ax = -abs(asarray(x))
|
||
|
|
# number of pieces on the left-side is (n+1)/2
|
||
|
|
funclist, condfuncs = _bspline_piecefunctions(n)
|
||
|
|
condlist = [func(ax) for func in condfuncs]
|
||
|
|
return piecewise(ax, condlist, funclist)
|
||
|
|
|
||
|
|
|
||
|
|
def gauss_spline(x, n):
|
||
|
|
"""Gaussian approximation to B-spline basis function of order n.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n : int
|
||
|
|
The order of the spline. Must be nonnegative, i.e. n >= 0
|
||
|
|
|
||
|
|
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
|
||
|
|
"""
|
||
|
|
signsq = (n + 1) / 12.0
|
||
|
|
return 1 / sqrt(2 * pi * signsq) * exp(-x ** 2 / 2 / signsq)
|
||
|
|
|
||
|
|
|
||
|
|
def cubic(x):
|
||
|
|
"""A cubic B-spline.
|
||
|
|
|
||
|
|
This is a special case of `bspline`, and equivalent to ``bspline(x, 3)``.
|
||
|
|
"""
|
||
|
|
ax = abs(asarray(x))
|
||
|
|
res = zeros_like(ax)
|
||
|
|
cond1 = less(ax, 1)
|
||
|
|
if cond1.any():
|
||
|
|
ax1 = ax[cond1]
|
||
|
|
res[cond1] = 2.0 / 3 - 1.0 / 2 * ax1 ** 2 * (2 - ax1)
|
||
|
|
cond2 = ~cond1 & less(ax, 2)
|
||
|
|
if cond2.any():
|
||
|
|
ax2 = ax[cond2]
|
||
|
|
res[cond2] = 1.0 / 6 * (2 - ax2) ** 3
|
||
|
|
return res
|
||
|
|
|
||
|
|
|
||
|
|
def quadratic(x):
|
||
|
|
"""A quadratic B-spline.
|
||
|
|
|
||
|
|
This is a special case of `bspline`, and equivalent to ``bspline(x, 2)``.
|
||
|
|
"""
|
||
|
|
ax = abs(asarray(x))
|
||
|
|
res = zeros_like(ax)
|
||
|
|
cond1 = less(ax, 0.5)
|
||
|
|
if cond1.any():
|
||
|
|
ax1 = ax[cond1]
|
||
|
|
res[cond1] = 0.75 - ax1 ** 2
|
||
|
|
cond2 = ~cond1 & less(ax, 1.5)
|
||
|
|
if cond2.any():
|
||
|
|
ax2 = ax[cond2]
|
||
|
|
res[cond2] = (ax2 - 1.5) ** 2 / 2.0
|
||
|
|
return res
|
||
|
|
|
||
|
|
|
||
|
|
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)
|
||
|
|
yp = zeros((K,), signal.dtype.char)
|
||
|
|
k = arange(K)
|
||
|
|
yp[0] = (_hc(0, cs, rho, omega) * signal[0] +
|
||
|
|
add.reduce(_hc(k + 1, cs, rho, omega) * signal))
|
||
|
|
|
||
|
|
yp[1] = (_hc(0, cs, rho, omega) * signal[0] +
|
||
|
|
_hc(1, cs, rho, omega) * signal[1] +
|
||
|
|
add.reduce(_hc(k + 2, cs, rho, omega) * signal))
|
||
|
|
|
||
|
|
for n in range(2, K):
|
||
|
|
yp[n] = (cs * signal[n] + 2 * rho * cos(omega) * yp[n - 1] -
|
||
|
|
rho * rho * yp[n - 2])
|
||
|
|
|
||
|
|
y = zeros((K,), signal.dtype.char)
|
||
|
|
|
||
|
|
y[K - 1] = add.reduce((_hs(k, cs, rho, omega) +
|
||
|
|
_hs(k + 1, cs, rho, omega)) * signal[::-1])
|
||
|
|
y[K - 2] = add.reduce((_hs(k - 1, cs, rho, omega) +
|
||
|
|
_hs(k + 2, cs, rho, omega)) * signal[::-1])
|
||
|
|
|
||
|
|
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])
|
||
|
|
|
||
|
|
return y
|
||
|
|
|
||
|
|
|
||
|
|
def _cubic_coeff(signal):
|
||
|
|
zi = -2 + sqrt(3)
|
||
|
|
K = len(signal)
|
||
|
|
yplus = zeros((K,), signal.dtype.char)
|
||
|
|
powers = zi ** arange(K)
|
||
|
|
yplus[0] = signal[0] + zi * add.reduce(powers * signal)
|
||
|
|
for k in range(1, K):
|
||
|
|
yplus[k] = signal[k] + zi * yplus[k - 1]
|
||
|
|
output = zeros((K,), signal.dtype)
|
||
|
|
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])
|
||
|
|
return output * 6.0
|
||
|
|
|
||
|
|
|
||
|
|
def _quadratic_coeff(signal):
|
||
|
|
zi = -3 + 2 * sqrt(2.0)
|
||
|
|
K = len(signal)
|
||
|
|
yplus = zeros((K,), signal.dtype.char)
|
||
|
|
powers = zi ** arange(K)
|
||
|
|
yplus[0] = signal[0] + zi * add.reduce(powers * signal)
|
||
|
|
for k in range(1, K):
|
||
|
|
yplus[k] = signal[k] + zi * yplus[k - 1]
|
||
|
|
output = zeros((K,), signal.dtype.char)
|
||
|
|
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])
|
||
|
|
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.
|
||
|
|
|
||
|
|
"""
|
||
|
|
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.
|
||
|
|
|
||
|
|
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 .
|
||
|
|
|
||
|
|
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
|
||
|
|
Cubic spline coefficients.
|
||
|
|
|
||
|
|
"""
|
||
|
|
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 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.
|
||
|
|
|
||
|
|
"""
|
||
|
|
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.
|
||
|
|
|
||
|
|
`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.
|
||
|
|
|
||
|
|
"""
|
||
|
|
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
|