def test_fourier_trafo_completely():
# Complete explicit test of all FT components on two small examples
# Discretization with 4 points
discr = odl.uniform_discr(-2, 2, 4, dtype="complex")
# Interval boundaries -2, -1, 0, 1, 2
assert np.allclose(discr.partition.cell_boundary_vecs[0], [-2, -1, 0, 1, 2])
# Grid points -1.5, -0.5, 0.5, 1.5
assert np.allclose(discr.grid.coord_vectors[0], [-1.5, -0.5, 0.5, 1.5])
# First test function, symmetric. Can be represented exactly in the
# discretization.
def f(x):
return (x >= -1) & (x <= 1)
def fhat(x):
return np.sqrt(2 / np.pi) * sinc(x)
# Discretize f, check values
f_discr = discr.element(f)
assert np.allclose(f_discr, [0, 1, 1, 0])
# "s" = shifted, "n" = not shifted
# Reciprocal grids
recip_s = reciprocal_grid(discr.grid, shift=True)
recip_n = reciprocal_grid(discr.grid, shift=False)
assert np.allclose(recip_s.coord_vectors[0], np.linspace(-np.pi, np.pi / 2, 4))
assert np.allclose(recip_n.coord_vectors[0], np.linspace(-3 * np.pi / 4, 3 * np.pi / 4, 4))
# Range
range_part_s = odl.uniform_partition_fromgrid(recip_s)
range_s = odl.uniform_discr_frompartition(range_part_s, dtype="complex")
range_part_n = odl.uniform_partition_fromgrid(recip_n)
range_n = odl.uniform_discr_frompartition(range_part_n, dtype="complex")
# Pre-processing
preproc_s = [1, -1, 1, -1]
preproc_n = [np.exp(1j * 3 / 4 * np.pi * k) for k in range(4)]
fpre_s = dft_preprocess_data(f_discr, shift=True)
fpre_n = dft_preprocess_data(f_discr, shift=False)
assert np.allclose(fpre_s, f_discr * discr.element(preproc_s))
assert np.allclose(fpre_n, f_discr * discr.element(preproc_n))
# FFT step, replicating the _call_numpy method
fft_s = np.fft.fftn(fpre_s, s=discr.shape, axes=[0])
fft_n = np.fft.fftn(fpre_n, s=discr.shape, axes=[0])
assert np.allclose(fft_s, [0, -1 + 1j, 2, -1 - 1j])
assert np.allclose(
fft_n, [np.exp(1j * np.pi * (3 - 2 * k) / 4) + np.exp(1j * np.pi * (3 - 2 * k) / 2) for k in range(4)]
)
# Interpolation kernel FT
interp_s = np.sinc(np.linspace(-1 / 2, 1 / 4, 4)) / np.sqrt(2 * np.pi)
interp_n = np.sinc(np.linspace(-3 / 8, 3 / 8, 4)) / np.sqrt(2 * np.pi)
assert np.allclose(interp_s, _interp_kernel_ft(np.linspace(-1 / 2, 1 / 4, 4), interp="nearest"))
assert np.allclose(interp_n, _interp_kernel_ft(np.linspace(-3 / 8, 3 / 8, 4), interp="nearest"))
# Post-processing
postproc_s = np.exp(1j * np.pi * np.linspace(-3 / 2, 3 / 4, 4))
postproc_n = np.exp(1j * np.pi * np.linspace(-9 / 8, 9 / 8, 4))
fpost_s = dft_postprocess_data(
range_s.element(fft_s), real_grid=discr.grid, recip_grid=recip_s, shift=[True], axes=(0,), interp="nearest"
)
fpost_n = dft_postprocess_data(
range_n.element(fft_n), real_grid=discr.grid, recip_grid=recip_n, shift=[False], axes=(0,), interp="nearest"
)
assert np.allclose(fpost_s, fft_s * postproc_s * interp_s)
assert np.allclose(fpost_n, fft_n * postproc_n * interp_n)
# Comparing to the known result sqrt(2/pi) * sinc(x)
assert np.allclose(fpost_s, fhat(recip_s.coord_vectors[0]))
assert np.allclose(fpost_n, fhat(recip_n.coord_vectors[0]))
# Doing the exact same with direct application of the FT operator
ft_op_s = FourierTransform(discr, shift=True)
ft_op_n = FourierTransform(discr, shift=False)
assert ft_op_s.range.grid == recip_s
assert ft_op_n.range.grid == recip_n
ft_f_s = ft_op_s(f)
ft_f_n = ft_op_n(f)
assert np.allclose(ft_f_s, fhat(recip_s.coord_vectors[0]))
assert np.allclose(ft_f_n, fhat(recip_n.coord_vectors[0]))
# Second test function, asymmetric. Can also be represented exactly in the
# discretization.
def f(x):
return (x >= 0) & (x <= 1)
def fhat(x):
return np.exp(-1j * x / 2) * sinc(x / 2) / np.sqrt(2 * np.pi)
# Discretize f, check values
f_discr = discr.element(f)
assert np.allclose(f_discr, [0, 0, 1, 0])
# Pre-processing
fpre_s = dft_preprocess_data(f_discr, shift=True)
fpre_n = dft_preprocess_data(f_discr, shift=False)
assert np.allclose(fpre_s, [0, 0, 1, 0])
assert np.allclose(fpre_n, [0, 0, -1j, 0])
# FFT step
fft_s = np.fft.fftn(fpre_s, s=discr.shape, axes=[0])
fft_n = np.fft.fftn(fpre_n, s=discr.shape, axes=[0])
assert np.allclose(fft_s, [1, -1, 1, -1])
assert np.allclose(fft_n, [-1j, 1j, -1j, 1j])
fpost_s = dft_postprocess_data(
range_s.element(fft_s), real_grid=discr.grid, recip_grid=recip_s, shift=[True], axes=(0,), interp="nearest"
)
fpost_n = dft_postprocess_data(
range_n.element(fft_n), real_grid=discr.grid, recip_grid=recip_n, shift=[False], axes=(0,), interp="nearest"
)
assert np.allclose(fpost_s, fft_s * postproc_s * interp_s)
assert np.allclose(fpost_n, fft_n * postproc_n * interp_n)
# Comparing to the known result exp(-1j*x/2) * sinc(x/2) / sqrt(2*pi)
assert np.allclose(fpost_s, fhat(recip_s.coord_vectors[0]))
assert np.allclose(fpost_n, fhat(recip_n.coord_vectors[0]))
# Doing the exact same with direct application of the FT operator
ft_f_s = ft_op_s(f)
ft_f_n = ft_op_n(f)
assert np.allclose(ft_f_s, fhat(recip_s.coord_vectors[0]))
assert np.allclose(ft_f_n, fhat(recip_n.coord_vectors[0]))
def test_fourier_trafo_completely():
# Complete explicit test of all FT components on two small examples
# Discretization with 4 points
discr = odl.uniform_discr(-2, 2, 4, dtype='complex')
# Interval boundaries -2, -1, 0, 1, 2
assert np.allclose(discr.partition.cell_boundary_vecs[0],
[-2, -1, 0, 1, 2])
# Grid points -1.5, -0.5, 0.5, 1.5
assert np.allclose(discr.grid.coord_vectors[0], [-1.5, -0.5, 0.5, 1.5])
# First test function, symmetric. Can be represented exactly in the
# discretization.
def f(x):
return (x >= -1) & (x <= 1)
def fhat(x):
return np.sqrt(2 / np.pi) * sinc(x)
# Discretize f, check values
f_discr = discr.element(f)
assert np.allclose(f_discr, [0, 1, 1, 0])
# "s" = shifted, "n" = not shifted
# Reciprocal grids
recip_s = reciprocal_grid(discr.grid, shift=True)
recip_n = reciprocal_grid(discr.grid, shift=False)
assert np.allclose(recip_s.coord_vectors[0],
np.linspace(-np.pi, np.pi / 2, 4))
assert np.allclose(recip_n.coord_vectors[0],
np.linspace(-3 * np.pi / 4, 3 * np.pi / 4, 4))
# Range
range_part_s = odl.uniform_partition_fromgrid(recip_s)
range_s = odl.uniform_discr_frompartition(range_part_s, dtype='complex')
range_part_n = odl.uniform_partition_fromgrid(recip_n)
range_n = odl.uniform_discr_frompartition(range_part_n, dtype='complex')
# Pre-processing
preproc_s = [1, -1, 1, -1]
preproc_n = [np.exp(1j * 3 / 4 * np.pi * k) for k in range(4)]
fpre_s = dft_preprocess_data(f_discr, shift=True)
fpre_n = dft_preprocess_data(f_discr, shift=False)
assert np.allclose(fpre_s, f_discr * discr.element(preproc_s))
assert np.allclose(fpre_n, f_discr * discr.element(preproc_n))
# FFT step, replicating the _call_numpy method
fft_s = np.fft.fftn(fpre_s, s=discr.shape, axes=[0])
fft_n = np.fft.fftn(fpre_n, s=discr.shape, axes=[0])
assert np.allclose(fft_s, [0, -1 + 1j, 2, -1 - 1j])
assert np.allclose(fft_n, [
np.exp(1j * np.pi * (3 - 2 * k) / 4) + np.exp(1j * np.pi *
(3 - 2 * k) / 2)
for k in range(4)
])
# Interpolation kernel FT
interp_s = np.sinc(np.linspace(-1 / 2, 1 / 4, 4)) / np.sqrt(2 * np.pi)
interp_n = np.sinc(np.linspace(-3 / 8, 3 / 8, 4)) / np.sqrt(2 * np.pi)
assert np.allclose(
interp_s,
_interp_kernel_ft(np.linspace(-1 / 2, 1 / 4, 4), interp='nearest'))
assert np.allclose(
interp_n,
_interp_kernel_ft(np.linspace(-3 / 8, 3 / 8, 4), interp='nearest'))
# Post-processing
postproc_s = np.exp(1j * np.pi * np.linspace(-3 / 2, 3 / 4, 4))
postproc_n = np.exp(1j * np.pi * np.linspace(-9 / 8, 9 / 8, 4))
fpost_s = dft_postprocess_data(range_s.element(fft_s),
real_grid=discr.grid,
recip_grid=recip_s,
shift=[True],
axes=(0, ),
interp='nearest')
fpost_n = dft_postprocess_data(range_n.element(fft_n),
real_grid=discr.grid,
recip_grid=recip_n,
shift=[False],
axes=(0, ),
interp='nearest')
assert np.allclose(fpost_s, fft_s * postproc_s * interp_s)
assert np.allclose(fpost_n, fft_n * postproc_n * interp_n)
# Comparing to the known result sqrt(2/pi) * sinc(x)
assert np.allclose(fpost_s, fhat(recip_s.coord_vectors[0]))
assert np.allclose(fpost_n, fhat(recip_n.coord_vectors[0]))
# Doing the exact same with direct application of the FT operator
ft_op_s = FourierTransform(discr, shift=True)
ft_op_n = FourierTransform(discr, shift=False)
assert ft_op_s.range.grid == recip_s
assert ft_op_n.range.grid == recip_n
ft_f_s = ft_op_s(f)
ft_f_n = ft_op_n(f)
assert np.allclose(ft_f_s, fhat(recip_s.coord_vectors[0]))
assert np.allclose(ft_f_n, fhat(recip_n.coord_vectors[0]))
# Second test function, asymmetric. Can also be represented exactly in the
# discretization.
def f(x):
return (x >= 0) & (x <= 1)
def fhat(x):
return np.exp(-1j * x / 2) * sinc(x / 2) / np.sqrt(2 * np.pi)
# Discretize f, check values
f_discr = discr.element(f)
assert np.allclose(f_discr, [0, 0, 1, 0])
# Pre-processing
fpre_s = dft_preprocess_data(f_discr, shift=True)
fpre_n = dft_preprocess_data(f_discr, shift=False)
assert np.allclose(fpre_s, [0, 0, 1, 0])
assert np.allclose(fpre_n, [0, 0, -1j, 0])
# FFT step
fft_s = np.fft.fftn(fpre_s, s=discr.shape, axes=[0])
fft_n = np.fft.fftn(fpre_n, s=discr.shape, axes=[0])
assert np.allclose(fft_s, [1, -1, 1, -1])
assert np.allclose(fft_n, [-1j, 1j, -1j, 1j])
fpost_s = dft_postprocess_data(range_s.element(fft_s),
real_grid=discr.grid,
recip_grid=recip_s,
shift=[True],
axes=(0, ),
interp='nearest')
fpost_n = dft_postprocess_data(range_n.element(fft_n),
real_grid=discr.grid,
recip_grid=recip_n,
shift=[False],
axes=(0, ),
interp='nearest')
assert np.allclose(fpost_s, fft_s * postproc_s * interp_s)
assert np.allclose(fpost_n, fft_n * postproc_n * interp_n)
# Comparing to the known result exp(-1j*x/2) * sinc(x/2) / sqrt(2*pi)
assert np.allclose(fpost_s, fhat(recip_s.coord_vectors[0]))
assert np.allclose(fpost_n, fhat(recip_n.coord_vectors[0]))
# Doing the exact same with direct application of the FT operator
ft_f_s = ft_op_s(f)
ft_f_n = ft_op_n(f)
assert np.allclose(ft_f_s, fhat(recip_s.coord_vectors[0]))
assert np.allclose(ft_f_n, fhat(recip_n.coord_vectors[0]))
