def test_shape_argument(self):
small_x = [[1, 2, 3], [4, 5, 6]]
large_x1 = [[1, 2, 3, 0], [4, 5, 6, 0], [0, 0, 0, 0], [0, 0, 0, 0]]
y = fftn(small_x, shape=(4, 4))
assert_array_almost_equal(y, fftn(large_x1))
y = fftn(small_x, shape=(3, 4))
assert_array_almost_equal(y, fftn(large_x1[:-1]))
def test_definition(self):
x = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
y = fftn(x)
assert_array_almost_equal(y, direct_dftn(x))
x = random((20, 26))
assert_array_almost_equal(fftn(x), direct_dftn(x))
x = random((5, 4, 3, 20))
assert_array_almost_equal(fftn(x), direct_dftn(x))
def test_definition(self):
x = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
y = fftn(np.array(x, np.float32))
if not y.dtype == np.complex64:
raise ValueError("double precision output with single precision")
y_r = np.array(fftn(x), np.complex64)
assert_array_almost_equal_nulp(y, y_r)
def test_shape_axes_argument2(self):
# Change shape of the last axis
x = numpy.random.random((10, 5, 3, 7))
y = fftn(x, axes=(-1, ), shape=(8, ))
assert_array_almost_equal(y, fft(x, axis=-1, n=8))
# Change shape of an arbitrary axis which is not the last one
x = numpy.random.random((10, 5, 3, 7))
y = fftn(x, axes=(-2, ), shape=(8, ))
assert_array_almost_equal(y, fft(x, axis=-2, n=8))
# Change shape of axes: cf #244, where shape and axes were mixed up
x = numpy.random.random((4, 4, 2))
y = fftn(x, axes=(-3, -2), shape=(8, 8))
assert_array_almost_equal(y, numpy.fft.fftn(x, axes=(-3, -2),
s=(8, 8)))
def test_shape_axes_argument(self):
small_x = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
large_x1 = array([[1, 2, 3, 0], [4, 5, 6, 0], [7, 8, 9, 0],
[0, 0, 0, 0]])
# Disable tests with shape and axes of different lengths
# y = fftn(small_x,shape=(4,4),axes=(-1,))
# for i in range(4):
# assert_array_almost_equal (y[i],fft(large_x1[i]))
# y = fftn(small_x,shape=(4,4),axes=(-2,))
# for i in range(4):
# assert_array_almost_equal (y[:,i],fft(large_x1[:,i]))
y = fftn(small_x, shape=(4, 4), axes=(-2, -1))
assert_array_almost_equal(y, fftn(large_x1))
y = fftn(small_x, shape=(4, 4), axes=(-1, -2))
assert_array_almost_equal(
y, swapaxes(fftn(swapaxes(large_x1, -1, -2)), -1, -2))
def test_float16_input(self):
for size in SMALL_COMPOSITE_SIZES + SMALL_PRIME_SIZES:
np.random.seed(1234)
x = np.random.rand(size, size) + 1j * np.random.rand(size, size)
y1 = fftn(x.real.astype(np.float16))
y2 = fftn(x.real.astype(np.float64)).astype(np.complex64)
assert_equal(y1.dtype, np.complex64)
assert_array_almost_equal_nulp(y1, y2, 5e5)
for size in LARGE_COMPOSITE_SIZES + LARGE_PRIME_SIZES:
np.random.seed(1234)
x = np.random.rand(size, 3) + 1j * np.random.rand(size, 3)
y1 = fftn(x.real.astype(np.float16))
y2 = fftn(x.real.astype(np.float64)).astype(np.complex64)
assert_equal(y1.dtype, np.complex64)
assert_array_almost_equal_nulp(y1, y2, 2e6)
def fftconvolve(in1, in2, mode="full"):
"""Convolve two N-dimensional arrays using FFT.
Convolve `in1` and `in2` using the fast Fourier transform method, with
the output size determined by the `mode` argument.
This is generally much faster than `convolve` for large arrays (n > ~500),
but can be slower when only a few output values are needed, and can only
output float arrays (int or object array inputs will be cast to float).
As of v0.19, `convolve` automatically chooses this method or the direct
method based on an estimation of which is faster.
Parameters
----------
in1 : array_like
First input.
in2 : array_like
Second input. Should have the same number of dimensions as `in1`.
If operating in 'valid' mode, either `in1` or `in2` must be
at least as large as the other in every dimension.
mode : str {'full', 'valid', 'same'}, optional
A string indicating the size of the output:
``full``
The output is the full discrete linear convolution
of the inputs. (Default)
``valid``
The output consists only of those elements that do not
rely on the zero-padding.
``same``
The output is the same size as `in1`, centered
with respect to the 'full' output.
Returns
-------
out : array
An N-dimensional array containing a subset of the discrete linear
convolution of `in1` with `in2`.
Examples
--------
Autocorrelation of white noise is an impulse.
>>> from scipy import signal
>>> sig = np.random.randn(1000)
>>> autocorr = signal.fftconvolve(sig, sig[::-1], mode='full')
>>> import matplotlib.pyplot as plt
>>> fig, (ax_orig, ax_mag) = plt.subplots(2, 1)
>>> ax_orig.plot(sig)
>>> ax_orig.set_title('White noise')
>>> ax_mag.plot(np.arange(-len(sig)+1,len(sig)), autocorr)
>>> ax_mag.set_title('Autocorrelation')
>>> fig.tight_layout()
>>> fig.show()
Gaussian blur implemented using FFT convolution. Notice the dark borders
around the image, due to the zero-padding beyond its boundaries.
The `convolve2d` function allows for other types of image boundaries,
but is far slower.
>>> from scipy import misc
>>> face = misc.face(gray=True)
>>> kernel = np.outer(signal.gaussian(70, 8), signal.gaussian(70, 8))
>>> blurred = signal.fftconvolve(face, kernel, mode='same')
>>> fig, (ax_orig, ax_kernel, ax_blurred) = plt.subplots(3, 1,
... figsize=(6, 15))
>>> ax_orig.imshow(face, cmap='gray')
>>> ax_orig.set_title('Original')
>>> ax_orig.set_axis_off()
>>> ax_kernel.imshow(kernel, cmap='gray')
>>> ax_kernel.set_title('Gaussian kernel')
>>> ax_kernel.set_axis_off()
>>> ax_blurred.imshow(blurred, cmap='gray')
>>> ax_blurred.set_title('Blurred')
>>> ax_blurred.set_axis_off()
>>> fig.show()
"""
in1 = asarray(in1)
in2 = asarray(in2)
if in1.ndim == in2.ndim == 0: # scalar inputs
return in1 * in2
elif not in1.ndim == in2.ndim:
raise ValueError("in1 and in2 should have the same dimensionality")
elif in1.size == 0 or in2.size == 0: # empty arrays
return array([])
s1 = array(in1.shape)
s2 = array(in2.shape)
complex_result = (np.issubdtype(in1.dtype, np.complexfloating)
or np.issubdtype(in2.dtype, np.complexfloating))
shape = s1 + s2 - 1
# Check that input sizes are compatible with 'valid' mode
if _inputs_swap_needed(mode, s1, s2):
# Convolution is commutative; order doesn't have any effect on output
in1, s1, in2, s2 = in2, s2, in1, s1
# Speed up FFT by padding to optimal size for FFTPACK
fshape = [fftpack.helper.next_fast_len(int(d)) for d in shape]
fslice = tuple([slice(0, int(sz)) for sz in shape])
# Pre-1.9 NumPy FFT routines are not threadsafe. For older NumPys, make
# sure we only call rfftn/irfftn from one thread at a time.
if not complex_result and (_rfft_mt_safe or _rfft_lock.acquire(False)):
try:
sp1 = np.fft.rfftn(in1, fshape)
sp2 = np.fft.rfftn(in2, fshape)
ret = (np.fft.irfftn(sp1 * sp2, fshape)[fslice].copy())
finally:
if not _rfft_mt_safe:
_rfft_lock.release()
else:
# If we're here, it's either because we need a complex result, or we
# failed to acquire _rfft_lock (meaning rfftn isn't threadsafe and
# is already in use by another thread). In either case, use the
# (threadsafe but slower) SciPy complex-FFT routines instead.
sp1 = fftpack.fftn(in1, fshape)
sp2 = fftpack.fftn(in2, fshape)
ret = fftpack.ifftn(sp1 * sp2)[fslice].copy()
if not complex_result:
ret = ret.real
if mode == "full":
return ret
elif mode == "same":
return _centered(ret, s1)
elif mode == "valid":
return _centered(ret, s1 - s2 + 1)
else:
raise ValueError("Acceptable mode flags are 'valid',"
" 'same', or 'full'.")
def test_random_complex(self):
for size in [1, 2, 51, 32, 64, 92]:
x = random([size, size]) + 1j * random([size, size])
assert_array_almost_equal_nulp(ifftn(fftn(x)), x, self.maxnlp)
assert_array_almost_equal_nulp(fftn(ifftn(x)), x, self.maxnlp)
def test_axes_argument(self):
# plane == ji_plane, x== kji_space
plane1 = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
plane2 = [[10, 11, 12], [13, 14, 15], [16, 17, 18]]
plane3 = [[19, 20, 21], [22, 23, 24], [25, 26, 27]]
ki_plane1 = [[1, 2, 3], [10, 11, 12], [19, 20, 21]]
ki_plane2 = [[4, 5, 6], [13, 14, 15], [22, 23, 24]]
ki_plane3 = [[7, 8, 9], [16, 17, 18], [25, 26, 27]]
jk_plane1 = [[1, 10, 19], [4, 13, 22], [7, 16, 25]]
jk_plane2 = [[2, 11, 20], [5, 14, 23], [8, 17, 26]]
jk_plane3 = [[3, 12, 21], [6, 15, 24], [9, 18, 27]]
kj_plane1 = [[1, 4, 7], [10, 13, 16], [19, 22, 25]]
kj_plane2 = [[2, 5, 8], [11, 14, 17], [20, 23, 26]]
kj_plane3 = [[3, 6, 9], [12, 15, 18], [21, 24, 27]]
ij_plane1 = [[1, 4, 7], [2, 5, 8], [3, 6, 9]]
ij_plane2 = [[10, 13, 16], [11, 14, 17], [12, 15, 18]]
ij_plane3 = [[19, 22, 25], [20, 23, 26], [21, 24, 27]]
ik_plane1 = [[1, 10, 19], [2, 11, 20], [3, 12, 21]]
ik_plane2 = [[4, 13, 22], [5, 14, 23], [6, 15, 24]]
ik_plane3 = [[7, 16, 25], [8, 17, 26], [9, 18, 27]]
ijk_space = [jk_plane1, jk_plane2, jk_plane3]
ikj_space = [kj_plane1, kj_plane2, kj_plane3]
jik_space = [ik_plane1, ik_plane2, ik_plane3]
jki_space = [ki_plane1, ki_plane2, ki_plane3]
kij_space = [ij_plane1, ij_plane2, ij_plane3]
x = array([plane1, plane2, plane3])
assert_array_almost_equal(fftn(x),
fftn(x, axes=(-3, -2, -1))) # kji_space
assert_array_almost_equal(fftn(x), fftn(x, axes=(0, 1, 2)))
assert_array_almost_equal(fftn(x, axes=(0, 2)), fftn(x, axes=(0, -1)))
y = fftn(x, axes=(2, 1, 0)) # ijk_space
assert_array_almost_equal(swapaxes(y, -1, -3), fftn(ijk_space))
y = fftn(x, axes=(2, 0, 1)) # ikj_space
assert_array_almost_equal(swapaxes(swapaxes(y, -1, -3), -1, -2),
fftn(ikj_space))
y = fftn(x, axes=(1, 2, 0)) # jik_space
assert_array_almost_equal(swapaxes(swapaxes(y, -1, -3), -3, -2),
fftn(jik_space))
y = fftn(x, axes=(1, 0, 2)) # jki_space
assert_array_almost_equal(swapaxes(y, -2, -3), fftn(jki_space))
y = fftn(x, axes=(0, 2, 1)) # kij_space
assert_array_almost_equal(swapaxes(y, -2, -1), fftn(kij_space))
y = fftn(x, axes=(-2, -1)) # ji_plane
assert_array_almost_equal(fftn(plane1), y[0])
assert_array_almost_equal(fftn(plane2), y[1])
assert_array_almost_equal(fftn(plane3), y[2])
y = fftn(x, axes=(1, 2)) # ji_plane
assert_array_almost_equal(fftn(plane1), y[0])
assert_array_almost_equal(fftn(plane2), y[1])
assert_array_almost_equal(fftn(plane3), y[2])
y = fftn(x, axes=(-3, -2)) # kj_plane
assert_array_almost_equal(fftn(x[:, :, 0]), y[:, :, 0])
assert_array_almost_equal(fftn(x[:, :, 1]), y[:, :, 1])
assert_array_almost_equal(fftn(x[:, :, 2]), y[:, :, 2])
y = fftn(x, axes=(-3, -1)) # ki_plane
assert_array_almost_equal(fftn(x[:, 0, :]), y[:, 0, :])
assert_array_almost_equal(fftn(x[:, 1, :]), y[:, 1, :])
assert_array_almost_equal(fftn(x[:, 2, :]), y[:, 2, :])
y = fftn(x, axes=(-1, -2)) # ij_plane
assert_array_almost_equal(fftn(ij_plane1), swapaxes(y[0], -2, -1))
assert_array_almost_equal(fftn(ij_plane2), swapaxes(y[1], -2, -1))
assert_array_almost_equal(fftn(ij_plane3), swapaxes(y[2], -2, -1))
y = fftn(x, axes=(-1, -3)) # ik_plane
assert_array_almost_equal(fftn(ik_plane1),
swapaxes(y[:, 0, :], -1, -2))
assert_array_almost_equal(fftn(ik_plane2),
swapaxes(y[:, 1, :], -1, -2))
assert_array_almost_equal(fftn(ik_plane3),
swapaxes(y[:, 2, :], -1, -2))
y = fftn(x, axes=(-2, -3)) # jk_plane
assert_array_almost_equal(fftn(jk_plane1),
swapaxes(y[:, :, 0], -1, -2))
assert_array_almost_equal(fftn(jk_plane2),
swapaxes(y[:, :, 1], -1, -2))
assert_array_almost_equal(fftn(jk_plane3),
swapaxes(y[:, :, 2], -1, -2))
y = fftn(x, axes=(-1, )) # i_line
for i in range(3):
for j in range(3):
assert_array_almost_equal(fft(x[i, j, :]), y[i, j, :])
y = fftn(x, axes=(-2, )) # j_line
for i in range(3):
for j in range(3):
assert_array_almost_equal(fft(x[i, :, j]), y[i, :, j])
y = fftn(x, axes=(0, )) # k_line
for i in range(3):
for j in range(3):
assert_array_almost_equal(fft(x[:, i, j]), y[:, i, j])
y = fftn(x, axes=()) # point
assert_array_almost_equal(y, x)
def test_definition_float16(self):
x = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
y = fftn(np.array(x, np.float16))
assert_equal(y.dtype, np.complex64)
y_r = np.array(fftn(x), np.complex64)
assert_array_almost_equal_nulp(y, y_r)