def decimate(x, q, n=None, ftype='iir', axis=-1):
"""downsample the signal x by an integer factor q, using an order n filter
By default an order 8 Chebyshev type I filter is used or a 30 point FIR
filter with hamming window if ftype is 'fir'.
Parameters
----------
x : N-d array
the signal to be downsampled
q : int
the downsampling factor
n : int or None
the order of the filter (1 less than the length for 'fir')
ftype : {'iir' or 'fir'}
the type of the lowpass filter
axis : int
the axis along which to decimate
Returns
-------
y : N-d array
the down-sampled signal
See also: resample
"""
if not isinstance(q, int):
raise TypeError, "q must be an integer"
if n is None:
if ftype == 'fir':
n = 30
else:
n = 8
if ftype == 'fir':
b = firwin(n+1, 1./q, window='hamming')
a = 1.
else:
b, a = cheby1(n, 0.05, 0.8/q)
y = lfilter(b, a, x, axis=axis)
sl = [None]*y.ndim
sl[axis] = slice(None, None, q)
return y[sl]
def decimate(x, q, n=None, ftype='iir', axis=-1):
"""downsample the signal x by an integer factor q, using an order n filter
By default an order 8 Chebyshev type I filter is used or a 30 point FIR
filter with hamming window if ftype is 'fir'.
Parameters
----------
x : N-d array
the signal to be downsampled
q : int
the downsampling factor
n : int or None
the order of the filter (1 less than the length for 'fir')
ftype : {'iir' or 'fir'}
the type of the lowpass filter
axis : int
the axis along which to decimate
Returns
-------
y : N-d array
the down-sampled signal
See also: resample
"""
if not isinstance(q, int):
raise TypeError, "q must be an integer"
if n is None:
if ftype == 'fir':
n = 30
else:
n = 8
if ftype == 'fir':
b = firwin(n + 1, 1. / q, window='hamming')
a = 1.
else:
b, a = cheby1(n, 0.05, 0.8 / q)
y = lfilter(b, a, x, axis=axis)
sl = [None] * y.ndim
sl[axis] = slice(None, None, q)
return y[sl]
def __design(self):
order = self['filter_order']
ft = self['type']
if self['algorithm']==0: # remez
if ft == 0:#'lowpass':
dg = [1., 0.]
if ft == 1:#'highpass':
dg = [0., 1.]
fq = [0., self['f1'], self['f2'], .5]
out = signaltools.remez(order, fq, dg)
elif self['algorithm']==1: # windowing
from scipy.signal.filter_design import firwin
# N -- order of filter (number of taps)
# cutoff -- cutoff frequency of filter (normalized so that 1 corresponds
# to Nyquist or pi radians / sample)
out = firwin(order, self['f1'])
self.__den = out
def dfilters(fname, type):
""" DFILTERS Generate directional 2D filters
Input:
fname: Filter name. Available 'fname' are:
'haar': the Haar filters
'vk': McClellan transformed of the filter from the VK book
'ko': orthogonal filter in the Kovacevic's paper
'kos': smooth 'ko' filter
'lax': 17 x 17 by Lu, Antoniou and Xu
'sk': 9 x 9 by Shah and Kalker
'cd': 7 and 9 McClellan transformed by Cohen and Daubechies
'pkva': ladder filters by Phong et al.
'oqf_362': regular 3 x 6 filter
'dvmlp': regular linear phase biorthogonal filter with 3 dvm
'sinc': ideal filter (*NO perfect recontruction*)
'dmaxflat': diamond maxflat filters obtained from a three stage ladder
type: 'd' or 'r' for decomposition or reconstruction filters
Output:
h0, h1: diamond filter pair (lowpass and highpass)
To test those filters (for the PR condition for the FIR case),
verify that:
convolve(h0, modulate2(h1, 'b')) + convolve(modulate2(h0, 'b'), h1) = 2
(replace + with - for even size filters)
To test for orthogonal filter
convolve(h, reverse2(h)) + modulate2(convolve(h, reverse2(h)), 'b') = 2
"""
# The diamond-shaped filter pair
if fname == "haar":
if str.lower(type[0]) == 'd':
h0 = array([1, 1]) / sqrt(2)
h1 = array([-1, 1]) / sqrt(2)
else:
h0 = array([1, 1]) / sqrt(2)
h1 = array([1, -1]) / sqrt(2)
return h0, h1
elif fname == "vk":
if str.lower(type[0]) == 'd':
h0 = array([1, 2, 1]) / 4.0
h1 = array([-1, -2, 6, -2, -1]) / 4.0
else:
h0 = array([-1, 2, 6, 2, -1]) / 4.0
h1 = array([-1, 2, -1]) / 4.0
# McClellan transfrom to obtain 2D diamond filters
t = array([[0, 1, 0], [1, 0, 1], [0, 1, 0]]) / 4.0 # diamond kernel
h0 = mctrans(h0, t)
h1 = mctrans(h1, t)
return h0, h1
elif fname == "ko": # orthogonal filters in Kovacevic's thesis
a0, a1, a2 = 2, 0.5, 1
h0 = array([[0, -a1, -a0 * a1, 0],
[-a2, -a0 * a2, -a0, 1],
[0, a0 * a1 * a2, -a1 * a2, 0]])
# h1 = qmf2(h0);
h1 = array([[0, -a1 * a2, -a0 * a1 * a2, 0],
[1, a0, -a0 * a2, a2],
[0, -a0 * a1, a1, 0]])
# Normalize filter sum and norm;
norm = sqrt(2) / sum(h0)
h0 = h0 * norm
h1 = h1 * norm
if str.lower(type[0]) == 'r':
# Reverse filters for reconstruction
h0 = h0[::-1, ::-1]
h1 = h1[::-1, ::-1]
return h0, h1
elif fname == "kos": # Smooth orthogonal filters in Kovacevic's thesis
a0, a1, a2 = -sqrt(3), -sqrt(3), 2 + sqrt(3)
h0 = array([[0, -a1, -a0 * a1, 0],
[-a2, -a0 * a2, -a0, 1],
[0, a0 * a1 * a2, -a1 * a2, 0]])
# h1 = qmf2(h0);
h1 = array([[0, -a1 * a2, -a0 * a1 * a2, 0],
[1, a0, -a0 * a2, a2],
[0, -a0 * a1, a1, 0]])
# Normalize filter sum and norm;
norm = sqrt(2) / sum(h0)
h0 = h0 * norm
h1 = h1 * norm
if str.lower(type[0]) == 'r':
# Reverse filters for reconstruction
h0 = h0[::-1, ::-1]
h1 = h1[::-1, ::-1]
return h0, h1
elif fname == "lax": # by Lu, Antoniou and Xu
h = array([[-1.2972901e-5, 1.2316237e-4, -7.5212207e-5, 6.3686104e-5,
9.4800610e-5, -7.5862919e-5, 2.9586164e-4, -1.8430337e-4],
[1.2355540e-4, -1.2780882e-4, -1.9663685e-5, -4.5956538e-5,
-6.5195193e-4, -2.4722942e-4, -2.1538331e-5, -7.0882131e-4],
[-7.5319075e-5, -1.9350810e-5, -7.1947086e-4, 1.2295412e-3,
5.7411214e-4, 4.4705422e-4, 1.9623554e-3, 3.3596717e-4],
[6.3400249e-5, -2.4947178e-4, 4.4905711e-4, -4.1053629e-3,
-2.8588307e-3, 4.3782726e-3, -3.1690509e-3, -3.4371484e-3],
[9.6404973e-5, -4.6116254e-5, 1.2371871e-3, -1.1675575e-2,
1.6173911e-2, -4.1197559e-3, 4.4911165e-3, 1.1635130e-2],
[-7.6955555e-5, -6.5618379e-4, 5.7752252e-4, 1.6211426e-2,
2.1310378e-2, -2.8712621e-3, -4.8422645e-2, -5.9246338e-3],
[2.9802986e-4, -2.1365364e-5, 1.9701350e-3, 4.5047673e-3,
-4.8489158e-2, -3.1809526e-3, -2.9406153e-2, 1.8993868e-1],
[-1.8556637e-4, -7.1279432e-4, 3.3839195e-4, 1.1662001e-2,
-5.9398223e-3, -3.4467920e-3, 1.9006499e-1, 5.7235228e-1]])
h0 = sqrt(2) * vstack((hstack((h, h[:, len(h) - 2::-1])),
hstack((h[len(h) - 2::-1, :],
h[len(h) - 2::-1, len(h) - 2::-1]))))
h1 = modulate2(h0, 'b', None)
return h0, h1
elif fname == "sk": # by Shah and Kalker
h = array([[0.621729, 0.161889, -0.0126949, -0.00542504, 0.00124838],
[0.161889, -0.0353769, -0.0162751, -0.00499353, 0],
[-0.0126949, -0.0162751, 0.00749029, 0, 0],
[-0.00542504, 0.00499353, 0, 0, 0],
[0.00124838, 0, 0, 0, 0]])
h0 = sqrt(2) * vstack((hstack((h[len(h):0:-1, len(h):0:-1],
h[len(h):0:-1, :])),
hstack((h[:, len(h):0:-1], h))))
h1 = modulate2(h0, 'b', None)
return h0, h1
elif fname == "dvmlp":
q = sqrt(2)
b = 0.02
b1 = b * b
h = array([[b / q, 0, -2 * q * b, 0, 3 * q * b, 0, -2 * q * b, 0, b / q],
[0, -1 / (16 * q), 0, 9 / (16 * q), 1 / q, 9 / (16 * q), 0, -1 / (16 * q), 0],
[b / q, 0, -2 * q * b, 0, 3 * q * b, 0, -2 * q * b, 0, b / q]])
g0 = array([[-b1 / q, 0, 4 * b1 * q, 0, -14 * q * b1, 0, 28 * q * b1, 0, -35 * q * b1, 0,
28 * q * b1, 0, -14 * q * b1, 0, 4 * b1 * q, 0, -b1 / q],
[0, b / (8 * q), 0, -13 * b / (8 * q), b / q, 33 * b / (8 * q), -2 * q * b,
-21 * b / (8 * q), 3 * q * b, -21 * b / (8 * q), -2 * q * b, 33 * b / (8 * q),
b / q, -13 * b / (8 * q), 0, b / (8 * q), 0],
[-q * b1, 0, -1 / (256 * q) + 8 * q * b1, 0, 9 / (128 * q) - 28 * q * b1,
-1 / (q * 16), -63 / (256 * q) + 56 * q * b1, 9 / (16 * q),
87 / (64 * q) - 70 * q * b1, 9 / (16 * q), -63 / (256 * q) + 56 * q * b1,
-1 / (q * 16), 9 / (128 * q) - 28 * q * b1, 0, -1 / (256 * q) + 8 * q * b1,
0, -q * b1],
[0, b / (8 * q), 0, -13 * b / (8 * q), b / q, 33 * b / (8 * q), -2 * q * b,
-21 * b / (8 * q), 3 * q * b, -21 * b / (8 * q), -2 * q * b, 33 * b / (8 * q),
b / q, -13 * b / (8 * q), 0, b / (8 * q), 0],
[-b1 / q, 0, 4 * b1 * q, 0, -14 * q * b1, 0, 28 * q * b1, 0, -35 * q * b1,
0, 28 * q * b1, 0, -14 * q * b1, 0, 4 * b1 * q, 0, -b1 / q]])
h1 = modulate2(g0, 'b', None)
h0 = h.copy()
if str.lower(type[0]) == 'r':
h1 = modulate2(h, 'b', None)
h0 = g0.copy()
return h0, h1
elif fname == "cd" or fname == "7-9": # by Cohen and Daubechies
# 1D prototype filters: the '7-9' pair
h0 = array([0.026748757411, -0.016864118443, -0.078223266529,
0.266864118443, 0.602949018236, 0.266864118443,
-0.078223266529, -0.016864118443, 0.026748757411])
g0 = array([-0.045635881557, -0.028771763114, 0.295635881557,
0.557543526229, 0.295635881557, -0.028771763114,
-0.045635881557])
if str.lower(type[0]) == 'd':
h1 = modulate2(g0, 'c', None)
else:
h1 = modulate2(h0, 'c', None)
h0 = g0.copy()
# Use McClellan to obtain 2D filters
t = array([[0, 1, 0], [1, 0, 1], [0, 1, 0]]) / 4.0 # diamond kernel
h0 = sqrt(2) * mctrans(h0, t)
h1 = sqrt(2) * mctrans(h1, t)
return h0, h1
elif fname == "pkva" or fname == "ldtest":
# Filters from the ladder structure
# Allpass filter for the ladder structure network
beta = ldfilter(fname)
# Analysis filters
h0, h1 = ld2quin(beta)
# Normalize norm
h0 = sqrt(2) * h0
h1 = sqrt(2) * h1
# Synthesis filters
if str.lower(type[0]) == 'r':
f0 = modulate2(h1, 'b', None)
f1 = modulate2(h0, 'b', None)
h0 = f0.copy()
h1 = f1.copy()
return h0, h1
elif fname == "pkva-half4": # Filters from the ladder structure
# Allpass filter for the ladder structure network
beta = ldfilterhalf(4)
# Analysis filters
h0, h1 = ld2quin(beta)
# Normalize norm
h0 = sqrt(2) * h0
h1 = sqrt(2) * h1
# Synthesis filters
if str.lower(type[0]) == 'r':
f0 = modulate2(h1, 'b', None)
f1 = modulate2(h0, 'b', None)
h0 = f0
h1 = f1
return h0, h1
elif fname == "pkva-half6": # Filters from the ladder structure
# Allpass filter for the ladder structure network
beta = ldfilterhalf(6)
# Analysis filters
h0, h1 = ld2quin(beta)
# Normalize norm
h0 = sqrt(2) * h0
h1 = sqrt(2) * h1
# Synthesis filters
if srtring.lower(type[0]) == 'r':
f0 = modulate2(h1, 'b', None)
f1 = modulate2(h0, 'b', None)
h0 = f0
h1 = f1
return h0, h1
elif fname == "pkva-half8": # Filters from the ladder structure
# Allpass filter for the ladder structure network
beta = ldfilterhalf(8)
# Analysis filters
h0, h1 = ld2quin(beta)
# Normalize norm
h0 = sqrt(2) * h0
h1 = sqrt(2) * h1
# Synthesis filters
if str.lower(type[0]) == 'r':
f0 = modulate2(h1, 'b', None)
f1 = modulate2(h0, 'b', None)
h0 = f0
h1 = f1
return h0, h1
elif fname == "sinc": # The "sinc" case, NO Perfect Reconstruction
# Ideal low and high pass filters
flength = 30
h0 = firwin(flength + 1, 0.5)
h1 = modulate2(h0, 'c', None)
# Use McClellan to obtain 2D filters
t = array([[0, 1, 0], [1, 0, 1], [0, 1, 0]]) / 4.0 # diamond kernel
h0 = sqrt(2) * mctrans(h0, t)
h1 = sqrt(2) * mctrans(h1, t)
return h0, h1
elif fname == "oqf_362": # Some "home-made" filters!
h0 = sqrt(2) / 64 * array([[sqrt(15), -3, 0],
[0, 5, -sqrt(15)],
[-2 * sqrt(15), 30, 0],
[0, 30, 2 * sqrt(15)],
[sqrt(15), 5, 0],
[0, -3, -sqrt(15)]]).conj().T
h1 = -reverse2(modulate2(h0, 'b', None))
if str.lower(type[0]) == 'r':
# Reverse filters for reconstruction
h0 = h0[::-1, ::-1]
h1 = h1[::-1, ::-1]
return h0, h1
elif fname == "test": # Only for the shape, not for PR
h0 = array([[0, 1, 0], [1, 4, 1], [0, 1, 0]])
h1 = array([[0, -1, 0], [-1, 4, -1], [0, -1, 0]])
return h0, h1
elif fname == "testDVM": # Only for directional vanishing moment
h0 = array([[1, 1], [1, 1]]) / sqrt(2)
h1 = array([[-1, 1], [1, -1]]) / sqrt(2)
return h0, h1
elif fname == "qmf": # by Lu, Antoniou and Xu
# ideal response
# window
m, n = 2, 2
w = empty([5, 5])
w1d = kaiser(4 * m + 1, 2.6)
for n1 in xrange(-m, m + 1):
for n2 in xrange(-n, n + 1):
w[n1 + m, n2 + n] = w1d[2 * m + n1 + n2] * w1d[2 * m + n1 - n2]
h = empty([5, 5])
for n1 in xrange(-m, m + 1):
for n2 in xrange(-n, n + 1):
h[n1 + m, n2 + n] = .5 * sinc((n1 + n2) / 2.0) * .5 * sinc((n1 - n2) / 2.0)
c = sum(h)
h = sqrt(2) * h / c
h0 = h * w
h1 = modulate2(h0, 'b', None)
return h0, h1
#h0 = modulate2(h,'r');
#h1 = modulate2(h,'b');
elif fname == "qmf2": # by Lu, Antoniou and Xu
# ideal response
# window
h = array([[-.001104, .002494, -0.001744, 0.004895,
-0.000048, -.000311],
[0.008918, -0.002844, -0.025197, -0.017135,
0.003905, -0.000081],
[-0.007587, -0.065904, 0.100431, -0.055878,
0.007023, 0.001504],
[0.001725, 0.184162, 0.632115, 0.099414,
-0.027006, -0.001110],
[-0.017935, -0.000491, 0.191397, -0.001787,
-0.010587, 0.002060],
[.001353, 0.005635, -0.001231, -0.009052,
-0.002668, 0.000596]])
h0 = h / sum(h)
h1 = modulate2(h0, 'b', None)
return h0, h1
#h0 = modulate2(h,'r');
#h1 = modulate2(h,'b');
elif fname == "dmaxflat4":
M1 = 1 / sqrt(2)
M2 = M1
k1 = 1 - sqrt(2)
k3 = k1
k2 = M1
h = array([.25 * k2 * k3, .5 * k2, 1 + .5 * k2 * k3]) * M1
h = hstack((h, h[len(h) - 2::-1]))
g = array([-.125 * k1 * k2 * k3, 0.25 * k1 * k2, (-0.5 * k1 - 0.5 * k3 - 0.375 * k1 * k2 * k3),
1 + .5 * k1 * k2]) * M2
g = hstack((g, g[len(g) - 2::-1]))
B = dmaxflat(4, 0)
h0 = mctrans(h, B)
g0 = mctrans(g, B)
h0 = sqrt(2) * h0 / sum(h0)
g0 = sqrt(2) * g0 / sum(g0)
h1 = modulate2(g0, 'b', None)
if str.lower(type[0]) == 'r':
h1 = modulate2(h0, 'b', None)
h0 = g0.copy()
return h0, h1
elif fname == "dmaxflat5":
M1 = 1 / sqrt(2)
M2 = M1
k1 = 1 - sqrt(2)
k3 = k1
k2 = M1
h = array([.25 * k2 * k3, .5 * k2, 1 + .5 * k2 * k3]) * M1
h = hstack((h, h[len(h) - 2::-1]))
g = array([-.125 * k1 * k2 * k3, 0.25 * k1 * k2,
(-0.5 * k1 - 0.5 * k3 - 0.375 * k1 * k2 * k3), 1 + .5 * k1 * k2]) * M2
g = hstack((g, g[len(g) - 2::-1]))
B = dmaxflat(5, 0)
h0 = mctrans(h, B)
g0 = mctrans(g, B)
h0 = sqrt(2) * h0 / sum(h0)
g0 = sqrt(2) * g0 / sum(g0)
h1 = modulate2(g0, 'b', None)
if str.lower(type[0]) == 'r':
h1 = modulate2(h0, 'b', None)
h0 = g0.copy()
return h0, h1
elif fname == "dmaxflat6":
M1 = 1 / sqrt(2)
M2 = M1
k1 = 1 - sqrt(2)
k3 = k1
k2 = M1
h = array([.25 * k2 * k3, .5 * k2, 1 + .5 * k2 * k3]) * M1
h = hstack((h, h[len(h) - 2::-1]))
g = array([-.125 * k1 * k2 * k3, 0.25 * k1 * k2,
(-0.5 * k1 - 0.5 * k3 - 0.375 * k1 * k2 * k3), 1 + .5 * k1 * k2]) * M2
g = hstack((g, g[len(g) - 2::-1]))
B = dmaxflat(6, 0)
h0 = mctrans(h, B)
g0 = mctrans(g, B)
h0 = sqrt(2) * h0 / sum(h0)
g0 = sqrt(2) * g0 / sum(g0)
h1 = modulate2(g0, 'b', None)
if str.lower(type[0]) == 'r':
h1 = modulate2(h0, 'b', None)
h0 = g0.copy()
return h0, h1
elif fname == "dmaxflat7":
M1 = 1 / sqrt(2)
M2 = M1
k1 = 1 - sqrt(2)
k3 = k1
k2 = M1
h = array([.25 * k2 * k3, .5 * k2, 1 + .5 * k2 * k3]) * M1
h = hstack((h, h[len(h) - 2::-1]))
g = array([-.125 * k1 * k2 * k3, 0.25 * k1 * k2,
(-0.5 * k1 - 0.5 * k3 - 0.375 * k1 * k2 * k3), 1 + .5 * k1 * k2]) * M2
g = hstack((g, g[len(g) - 2::-1]))
B = dmaxflat(7, 0)
h0 = mctrans(h, B)
g0 = mctrans(g, B)
h0 = sqrt(2) * h0 / sum(h0)
g0 = sqrt(2) * g0 / sum(g0)
h1 = modulate2(g0, 'b', None)
if str.lower(type[0]) == 'r':
h1 = modulate2(h0, 'b', None)
h0 = g0.copy()
return h0, h1
