def plot_filters(bank=None, filt_opt_bank = None):
if bank is None:
filt_opt_bank, bank = test_filter_bank(return_bank=True)
for k in range(2):
print('Bank {}'.format(k))
center_psi, bw_psi, bw_phi = morlet_freq_1d(filt_opt_bank[k])
sigma0 = bank[k].meta.sigma0
sigma_psi = sigma0*np.pi/2./bw_psi
sigma_phi = sigma0*np.pi/2./bw_phi
J = bank[k].meta.J
P = bank[k].meta.P
print('J = {}, P = {}'.format(J, P))
for j in range(J)[::J//5]:
f = bank[k].psi.filter[j]
# f = np.concatenate([f[:len(f)//2][::-1], f[:len(f)//2]])
ifft_f = mkl_fft.ifft(f)
ifft_f = np.real(ifft_f)
fig, ax = plt.subplots(1, 2, figsize = (24, 4))
ax[0].plot(np.concatenate([ifft_f[-int(3*sigma_psi[-1]):],
ifft_f[:int(3*sigma_psi[-1])]]))
# ax[0].plot(ifft_f)
ax[1].plot(f)
# ax[1].plot(np.concatenate([f[:len(f)//2][::-1], f[:len(f)//2]]))
fig.suptitle('Psi filter, j{} = {}'.format(k+1, j), size = 'xx-large')
ax[0].set_title('Time domain')
ax[1].set_title('Frequency domain')
plt.show()
if P > 0:
f = bank[k].phi.filter[0]
ifft_f = mkl_fft.ifft(f)
ifft_f = np.real(ifft_f)
fig, ax = plt.subplots(1, 2, figsize = (24, 4))
ax[0].plot(np.concatenate([ifft_f[-int(3*sigma_psi[-1]):],
ifft_f[:int(3*sigma_psi[-1])]]))
# ax[0].plot(ifft_f)
ax[1].plot(f)
fig.suptitle('Phi filter', size = 'xx-large')
ax[0].set_title('Time domain')
ax[1].set_title('Frequency domain')
plt.show()
def FFT_t_2(self, A):
if MKL_AVAILABLE:
if global_variables.PRE_FFTSHIFT:
return mkl_fft.ifft(A)
else:
return ifftshift(mkl_fft.ifft(fftshift(A)))
else:
if global_variables.PRE_FFTSHIFT:
return scipy.fftpack.ifft(A)
else:
return ifftshift(scipy.fftpack.ifft(fftshift(A)))
def mkl_ifft1d(a):
n,m = a.shape
b = np.zeros((n,m),dtype=np.complex128)
for i in xrange(m):
b[:,i] = mkl_fft.ifft(a[:,i]+0j)/n
return b
def ifft(a, overwrite_x = False):
"""Computes ifft on a matrix of shape (..., n).
This is identical to np.ifft2(a)
Parameters
----------
a : array_like
Input array (must be complex).
overwrite_x : bool
Specifies whether original array can be destroyed (for inplace transform)
Returns
-------
out : complex ndarray
Result of the transformation along the (-4,-3) axes.
"""
a = np.asarray(a, dtype = CDTYPE)
libname = DTMMConfig["fftlib"]
if libname == "mkl_fft":
return mkl_fft.ifft(a, overwrite_x = overwrite_x)
elif libname == "scipy":
return spfft.ifft(a, overwrite_x = overwrite_x)
elif libname == "numpy":
return npfft.ifft(a)
elif libname == "pyfftw":
return pyfftw.interfaces.scipy_fft.ifft(a, overwrite_x = overwrite_x)
else: #default implementation is numpy
return npfft.ifft(a)
def mkl_ifft1d(a):
n,m = a.shape
b = np.zeros((n,m),dtype=np.complex128)
for i in xrange(m):
b[:,i] = mkl_fft.ifft(a[:,i]+0j)/n
return b
def test_vector3(self):
"fft(ifft(x)) is identity"
f1 = mkl_fft.ifft(self.xz1)
f2 = mkl_fft.fft(f1)
assert_(np.allclose(self.xz1,f2))
f1 = mkl_fft.ifft(self.xc1)
f2 = mkl_fft.fft(f1)
assert_( np.allclose(self.xc1,f2))
f1 = mkl_fft.ifft(self.xd1)
f2 = mkl_fft.fft(f1)
assert_( np.allclose(self.xd1,f2))
f1 = mkl_fft.ifft(self.xf1)
f2 = mkl_fft.fft(f1)
assert_( np.allclose(self.xf1, f2, atol = 2.0e-7))
def mifft(fh):
M = fh.shape[0]
NS = fh.shape[1]
N = NS + 1
N2 = int(N / 2)
temp = np.empty((M, N), dtype=complex)
temp[:, :N2] = fh[:, :N2]
temp[:, N2] = 0.0
temp[:, N2 + 1:] = fh[:, N2:]
return ifft(temp)
def _ifft(a, overwrite_x=False):
libname = CDDMConfig["fftlib"]
if libname == "mkl_fft":
return mkl_fft.ifft(a, overwrite_x=overwrite_x)
elif libname == "scipy":
return spfft.ifft(a, overwrite_x=overwrite_x)
elif libname == "pyfftw":
return fftw.scipy_fftpack.ifft(a, overwrite_x=overwrite_x)
elif libname == "numpy":
return np.fft.ifft(a)
def test_scale_1d_array(self):
X = np.ones((8, 4, 4,), dtype='d')
f1 = mkl_fft.fft(X, axis=1, forward_scale=0.25)
f2 = mkl_fft.fft(X, axis=1)
r_tol, a_tol = _get_rtol_atol(X)
assert_allclose(4*f1, f2, rtol=r_tol, atol=a_tol)
X1 = mkl_fft.ifft(f1, axis=1, forward_scale=0.25)
assert_allclose(X, X1, rtol=r_tol, atol=a_tol)
f3 = mkl_fft.rfft(X, axis=0, forward_scale=0.5)
X2 = mkl_fft.irfft(f3, axis=0, forward_scale=0.5)
assert_allclose(X, X2, rtol=r_tol, atol=a_tol)
def test_scale_1d_vector(self):
X = np.ones(128, dtype='d')
f1 = mkl_fft.fft(X, forward_scale=0.25)
f2 = mkl_fft.fft(X)
r_tol, a_tol = _get_rtol_atol(X)
assert_allclose(4*f1, f2, rtol=r_tol, atol=a_tol)
X1 = mkl_fft.ifft(f1, forward_scale=0.25)
assert_allclose(X, X1, rtol=r_tol, atol=a_tol)
f3 = mkl_fft.rfft(X, forward_scale=0.5)
X2 = mkl_fft.irfft(f3, forward_scale=0.5)
assert_allclose(X, X2, rtol=r_tol, atol=a_tol)
def hilbert(signal: np.ndarray) -> np.ndarray:
"""
hilbert(signal)
calculate the analytic signal using the `Hilbert transform` (same with scipy.signal.hilbert)
.. math::
S_a = S + iS_H
where S_H is the hilbert transform of the signal
Parameters
----------
signal : np.ndarray
signal
Returns
-------
np.ndarray
:math:'s + s_{H} i'
"""
@njit
def _modifyfft(x: np.ndarray) -> np.ndarray:
"""used by the function hilbert
Parameters
----------
x : np.ndarray
[description]
Returns
-------
np.ndarray
[description]
"""
N = x.size
if N % 2 == 0:
x[1:N // 2] = x[1:N // 2] * 2
x[N // 2 + 1:] = 0
else:
x[1:(N + 1) // 2] = x[1:(N + 1) // 2] * 2
x[(N + 1) // 2:] = 0
return x
Xf = fft(signal)
Xf = _modifyfft(Xf)
x = ifft(Xf)
return x
def test_fft(self):
n_tests = 1000
n_max = 2
d_max = 100
norm = 'ortho'
inplace = True
scrambled = True
passed = True
for i in range(n_tests):
ndim = np.random.randint(1, high=n_max + 1)
axis = np.random.randint(0, high=ndim)
dims = np.random.randint(1, high=d_max + 1, size=(ndim))
x = np.random.normal(size=dims) + 1j * np.random.normal(size=dims)
if inplace:
X = x.copy()
fft(X, axis=axis, norm=norm, out=X, scrambled=scrambled)
else:
X = fft(x, axis=axis, norm=norm, scrambled=scrambled)
Y = np.fft.fft(x, axis=axis, norm=norm)
x_ = ifft(X, axis=axis, norm=norm)
if not np.allclose(X, Y) and not scrambled:
print(
' Failed forward with ndim=%d axis=%d dims=' %
(ndim, axis), dims)
passed = False
if not np.allclose(x, x_):
print(
' Failed backward with ndim=%d axis=%d dims=' %
(ndim, axis), dims)
passed = False
self.assertTrue(passed)
def test_fft_truncation(self):
n_tests = 1000
n_max = 2
d_max = 100
norm = None
inplace = False
scrambled = False
passed = True
for i in range(n_tests):
ndim = np.random.randint(1, high=n_max + 1)
axis = np.random.randint(0, high=ndim)
dims = np.random.randint(2, high=d_max + 1, size=(ndim))
x = np.random.normal(size=dims) + 1j * np.random.normal(size=dims)
x_trunc = np.swapaxes(x, axis, -1)[..., :dims[axis] // 2]
x_trunc = np.swapaxes(x_trunc, -1, axis)
X = fft(x, axis=axis, n=dims[axis] // 2)
Y = np.fft.fft(x, axis=axis, n=dims[axis] // 2)
x_ = ifft(X, axis=axis, norm=norm)
if not np.allclose(X, Y) and not scrambled:
print(
' Failed forward with ndim=%d axis=%d dims=' %
(ndim, axis), dims)
passed = False
if not np.allclose(x_trunc, x_):
print(
' Failed backward with ndim=%d axis=%d dims=' %
(ndim, axis), dims)
passed = False
self.assertTrue(passed)
def test_fft(self):
n_tests = 1000
n_max = 2
d_max = 100
norm = 'ortho'
inplace = True
scrambled = True
passed = True
for i in range(n_tests):
ndim = np.random.randint(1,high=n_max+1)
axis = np.random.randint(0,high=ndim)
dims = np.random.randint(1,high=d_max+1,size=(ndim))
x = np.random.normal(size=dims) + 1j*np.random.normal(size=dims)
if inplace:
X = x.copy()
fft(X, axis=axis, norm=norm, out=X, scrambled=scrambled)
else:
X = fft(x, axis=axis, norm=norm, scrambled=scrambled)
Y = np.fft.fft(x, axis=axis, norm=norm)
x_ = ifft(X, axis=axis, norm=norm)
if not np.allclose(X,Y) and not scrambled:
print(' Failed forward with ndim=%d axis=%d dims=' % (ndim, axis), dims)
passed = False
if not np.allclose(x, x_):
print(' Failed backward with ndim=%d axis=%d dims=' % (ndim, axis), dims)
passed = False
self.assertTrue(passed)
def ifft(x, n=None, axis=-1, overwrite_input=False, extra_info=None):
return mkl_fft.ifft(x, n=n, axis=axis, overwrite_x=overwrite_input)
def ifft(x, n=None, axis=-1, overwrite_input=False, extra_info=None):
return mkl_fft.ifft(x, n=n, axis=axis, overwrite_x=overwrite_input)
def FFT_t_shift(self, A):
if MKL_AVAILABLE:
return ifftshift(mkl_fft.ifft(fftshift(A)))
else:
return ifftshift(scipy.fftpack.ifft(fftshift(A)))
# Ensure Image can be split in blocks by 8 and split it
height = height - height % block
width = width - width % block
hmax = height // block
wmax = width // block
CompIMG = np.zeros((height, width, 3))
# Process every component
for kk in range(3):
I = img[:height, :width, kk]
MM = np.array(I, dtype=np.float32)
CC = np.zeros((height, width))
for i in range(hmax):
for j in range(wmax):
# Splint int 8x8 blocks
M = MM[block * i: block * (i + 1), block * j: block * (j + 1)]
# Execute fft
D = np.real(mkl_fft.fft(M)).reshape((-1, 8))
# Compress dividing by quantization matrix (most coef are zeros)
C = np.round(D * Q)
CC[block * i: block * (i + 1), block * j: block * (j + 1)] = C.copy()
for i in range(hmax):
for j in range(wmax):
# Restore image with ifft
CompIMG[block * i: block * (i + 1), block * j: block * (j + 1), kk] = np.real(
mkl_fft.ifft(CC[block * i: block * (i + 1), block * j: block * (j + 1)])).reshape(-1, 8)
# Convert image back to RGB
t.saveImage(CompIMG, file + "_compressedFFT")
t.showResults(file, file + "_compressedFFT")
def test_array4(self):
x = self.ad3
for ax in range(x.ndim):
f1 = mkl_fft.ifft(x, axis = ax)
f2 = mkl_fft.fft(f1, axis = ax)
assert_allclose(f2, x, atol=2e-15)
def test_matrix6(self):
x = self.ad2;
f1 = mkl_fft.ifft(x)
f2 = mkl_fft.fft(f1)
assert_allclose(x, f2, atol=1e-10)
def test_vector8(self):
"ifft of real array is the same as fft of its complex cast"
x = self.xd1[3:17:2]
f1 = mkl_fft.ifft(x)
f2 = mkl_fft.ifft(x.astype(np.complex128))
assert_(np.allclose(f1,f2))
def pfourier_multiply(fh, m):
pos = int(fh.shape[1] // 2 + 1)
fh[:, pos] = 0.0
oh = fft(m * ifft(fh))
oh[:, pos] = 0.0
return oh
def pifftr(fh):
pos = int(fh.shape[1] // 2 + 1)
fh[:, pos] = 0.0
return ifft(fh).real
def ffourier_multiply(fh, m):
return fft(m * ifft(fh))
def remove_stripe(img, level, wname='db5', sigma=1.5):
"""
Suppress horizontal stripe in a sinogram using the Fourier-Wavelet based
method by Munch et al. [2]_.
Parameters
----------
img : 2d array
The two-dimensional array representig the image or the sinogram to de-stripe.
level : int
The highest decomposition level.
wname : str, optional
The wavelet type. Default value is ``db5``
sigma : float, optional
The damping factor in the Fourier space. Default value is ``1.5``
Returns
-------
out : 2d array
The resulting filtered image.
References
----------
.. [2] B. Munch, P. Trtik, F. Marone, M. Stampanoni, Stripe and ring artifact removal with
combined wavelet-Fourier filtering, Optics Express 17(10):8567-8591, 2009.
"""
nrow, ncol = img.shape
# wavelet decomposition.
cH = []
cV = []
cD = []
for i in range(0, level):
img, (cHi, cVi, cDi) = pywt.dwt2(img, wname)
cH.append(cHi)
cV.append(cVi)
cD.append(cDi)
# FFT transform of horizontal frequency bands
for i in range(0, level):
# FFT
fcV = fftshift(fft(cV[i], axis=0))
my, mx = fcV.shape
# damping of vertical stripe information
yy2 = (np.arange(-np.floor(my / 2), -np.floor(my / 2) + my))**2
damp = -np.expm1(-yy2 / (2.0 * (sigma**2)))
fcV = fcV * np.tile(damp.reshape(damp.size, 1), (1, mx))
#inverse FFT
cV[i] = np.real(ifft(ifftshift(fcV), axis=0))
# wavelet reconstruction
for i in range(level - 1, -1, -1):
img = img[0:cH[i].shape[0], 0:cH[i].shape[1]]
img = pywt.idwt2((img, (cH[i], cV[i], cD[i])), wname)
return img[0:nrow, 0:ncol]
def conv_sub_1d(x_fft, f, ds):
'''
Input
x_fft: The Fourier transform of the signals to be convolved, 2-d array N x K,
where K is number of signals.
f: The filter in the frequency domain (= Fourier transform of the filter).
ds: The downsampling factor as a power of 2 with respect to x_fft
Output
y_ds: the filtered, downsampled signalm in the time domain
'''
# periodize filter so that it has the correct resolution
# j0 = int(np.log2(f.shape[0]/x_fft.shape[0]))
# f = f.reshape((f.shape[0]//2**j0, 2**j0)).sum(axis = 1)
sig_length = x_fft.shape[0]
if type(f) != dict:
end = len(f)
f = np.concatenate([
f[:sig_length // 2],
[f[sig_length // 2] / 2 + f[end - sig_length // 2] / 2],
f[end - sig_length // 2 + 1:]
])
# represent filter in matrix form
f = f[:, np.newaxis] + 0 * 1j
# print('Inside conv_sub_1d: filter')
# print(f.shape)
# print(f[:10,0])
# Calculate Fourier coefficients
y_fft = f * x_fft
#print('Inside conv_sub_1d: y_fft')
#print(y_fft.shape)
# print(y_fft[:10,0])
elif f['type'] == 'fourier_truncated':
coefft, start = f['coefft'], f['start']
if len(coefft) > sig_length:
# filter is larger than signal, lowpass filter & periodize the former
# create lowpass filter
start0 = np.mod(start, f['N'])
nCoeffts0 = len(coefft)
if start0 + nCoeffts0 - 1 <= f['N']:
rng = np.arange(start0, nCoeffts0)
else:
rng = np.concatenate([
np.arange(start0, f['N']),
np.arange(nCoeffts0 + start0 - f['N'])
])
lowpass = np.zeros(shape=coefft.shape)
lowpass[rng < sig_length // 2] = 1
lowpass[rng == sig_length // 2] = 0.5
lowpass[rng == f['N'] - sig_length // 2] = 0.5
lowpass[rng > f['N'] - sig_length // 2] = 1
coefft = (coefft * lowpass).reshape(
(sig_length, len(coefft) // sig_length), order='F').sum(axis=1)
nCoeffts = len(coefft)
j = int(np.log2(nCoeffts / sig_length))
start = np.mod(start, x_fft.shape[0])
if start + nCoeffts <= x_fft.shape[0]:
# filter support contained in one period, no wrap-around
y_fft = coefft[:, np.newaxis] * x_fft[start:(nCoeffts + start), :]
else:
# filter support wraps around, extract both parts
y_fft = coefft[:, np.newaxis] * np.concatenate([
x_fft[start:, :], x_fft[:nCoeffts + start - x_fft.shape[0], :]
])
# Calculate downsampling factor with respect to y_fft
# Пока в имплементации у нас всегда y_fft.shape[0]=x_fft.shape[0]
ds_factor = ds + np.log2(y_fft.shape[0] / sig_length)
#print('Inside conv_sub_1d: dsj: ', ds_factor)
if ds_factor > 0:
y_fft = downsample_filter(y_fft, ds_factor=ds_factor)
elif ds_factor < 0:
# upsample, so zero-pad in Fourier
# note that this only happens for fourier_truncated filters, since otherwise
# filter sizes are always the same as the signal size
# also, we have to do one-sided padding since otherwise we might break
# continuity of Fourier transform
y_fft = np.concatenate([
y_fft,
np.zeros(((2**(-ds_factor) - 1) * y_fft.shape[0], y_fft.shape[1]))
])
#print('Inside conv_sub_1d; downsampled y_fft: ')
#print(y_fft.shape)
# print(y_fft[:4,0])
if (type(f) == dict) and (f['type']
== 'fourier_truncated') and f['recenter']:
# result has been shifted in frequency so that the zero fre-
# quency is actually at -filter.start+1
y_fft = np.roll(y_fft, f['start'])
# Calculate inverse Fourier transform
# Use .T to apply transform along columns (coefficients of 1-d signals)
# Divede to the 2**(ds_factor/2) to get the same signal magnitude in time domain
# after downsampling in frequency domain (CHECK TAHT WE SHOULD DEVIDE ifft NOT fft)
x_filtered_ds = mkl_fft.ifft(y_fft.T).T / (2**(ds_factor / 2))
# print('Inside conv_sub_1d; x_filtered_ds: ')
# print(x_filtered_ds.shape)
# print(x_filtered_ds[:4,0])
return x_filtered_ds