def test_fourier_trafo_create_temp():
shape = 10
space_discr = odl.uniform_discr(0, 1, shape, dtype="complex64")
ft = FourierTransform(space_discr)
ft.create_temporaries()
assert ft._tmp_r is not None
assert ft._tmp_f is not None
ift = ft.inverse
assert ift._tmp_r is not None
assert ift._tmp_f is not None
ft.clear_temporaries()
assert ft._tmp_r is None
assert ft._tmp_f is None
def test_fourier_trafo_call(impl, floating_dtype):
# Test if all variants can be called without error
# Not supported, skip
if floating_dtype == np.dtype("float16") and impl == "pyfftw":
return
shape = 10
halfcomplex, _ = _params_from_dtype(floating_dtype)
space_discr = odl.uniform_discr(0, 1, shape, dtype=floating_dtype)
ft = FourierTransform(space_discr, impl=impl, halfcomplex=halfcomplex)
ift = ft.inverse
one = space_discr.one()
assert np.allclose(ift(ft(one)), one)
# With temporaries
ft.create_temporaries()
ift = ft.inverse # shares temporaries
one = space_discr.one()
assert np.allclose(ift(ft(one)), one)
def test_fourier_trafo_scaling():
# Test if the FT scales correctly
# Characteristic function of [0, 1], its Fourier transform is
# given by exp(-1j * y / 2) * sinc(y/2)
def char_interval(x):
return (x >= 0) & (x <= 1)
def char_interval_ft(x):
return np.exp(-1j * x / 2) * sinc(x / 2) / np.sqrt(2 * np.pi)
fspace = odl.FunctionSpace(odl.IntervalProd(-2, 2),
field=odl.ComplexNumbers())
discr = odl.uniform_discr_fromspace(fspace, 40, impl='numpy')
dft = FourierTransform(discr)
for factor in (2, 1j, -2.5j, 1 - 4j):
func_true_ft = factor * dft.range.element(char_interval_ft)
func_dft = dft(factor * fspace.element(char_interval))
assert (func_dft - func_true_ft).norm() < 1e-6
def test_fourier_trafo_hat_1d():
# Hat function as used in linear interpolation. It is not so
# well discretized by nearest neighbor interpolation, so a larger
# error is to be expected.
def hat_func(x):
out = np.where(x < 0, 1 + x, 1 - x)
out[x < -1] = 0
out[x > 1] = 0
return out
def hat_func_ft(x):
return sinc(x / 2)**2 / np.sqrt(2 * np.pi)
# Using a single-precision implementation, should be as good
# With linear interpolation in the discretization, should be better?
discr = odl.uniform_discr(-2, 2, 101, impl='numpy', dtype='float32')
dft = FourierTransform(discr)
func_true_ft = dft.range.element(hat_func_ft)
func_dft = dft(hat_func)
assert (func_dft - func_true_ft).norm() < 0.001
def test_fourier_trafo_create_temp():
shape = 10
space_discr = odl.uniform_discr(0, 1, shape, dtype='complex64')
ft = FourierTransform(space_discr)
ft.create_temporaries()
assert ft._tmp_r is not None
assert ft._tmp_f is not None
ift = ft.inverse
assert ift._tmp_r is not None
assert ift._tmp_f is not None
ft.clear_temporaries()
assert ft._tmp_r is None
assert ft._tmp_f is None
def test_dft_with_known_pairs_2d():
# Frequency-shifted product of characteristic functions
def fshift_char_rect(x):
# Characteristic function of the cuboid
# [-1, 1] x [1, 2]
return (x[0] >= -1) & (x[0] <= 1) & (x[1] >= 1) & (x[1] <= 2)
def fshift_char_rect_ft(x):
# FT is a product of shifted and frequency-shifted sinc functions
# 1st comp.: 2 * sinc(y)
# 2nd comp.: exp(-1j * y * 3/2) * sinc(y/2)
# Overall factor: (2 * pi)^(-1)
return (2 * sinc(x[0]) * np.exp(-1j * x[1] * 3 / 2) * sinc(x[1] / 2) /
(2 * np.pi))
discr = odl.uniform_discr([-2] * 2, [2] * 2, (100, 400),
impl='numpy',
dtype='complex64')
dft = FourierTransform(discr)
func_true_ft = dft.range.element(fshift_char_rect_ft)
func_dft = dft(fshift_char_rect)
assert (func_dft - func_true_ft).norm() < 0.001
def test_fourier_trafo_inverse(impl, sign):
# Test if the inverse really is the inverse
def char_interval(x):
return (x >= 0) & (x <= 1)
# Complex-to-complex
discr = odl.uniform_discr(-2, 2, 40, impl="numpy", dtype="complex64")
discr_char = discr.element(char_interval)
ft = FourierTransform(discr, sign=sign, impl=impl)
assert all_almost_equal(ft.inverse(ft(char_interval)), discr_char)
assert all_almost_equal(ft.adjoint(ft(char_interval)), discr_char)
# Half-complex
discr = odl.uniform_discr(-2, 2, 40, impl="numpy", dtype="float32")
ft = FourierTransform(discr, impl=impl, halfcomplex=True)
assert all_almost_equal(ft.inverse(ft(char_interval)), discr_char)
def char_rect(x):
return (x[0] >= 0) & (x[0] <= 1) & (x[1] >= 0) & (x[1] <= 1)
# 2D with axes, C2C
discr = odl.uniform_discr([-2, -2], [2, 2], (20, 10), impl="numpy", dtype="complex64")
discr_rect = discr.element(char_rect)
for axes in [(0,), 1]:
ft = FourierTransform(discr, sign=sign, impl=impl, axes=axes)
assert all_almost_equal(ft.inverse(ft(char_rect)), discr_rect)
assert all_almost_equal(ft.adjoint(ft(char_rect)), discr_rect)
# 2D with axes, halfcomplex
discr = odl.uniform_discr([-2, -2], [2, 2], (20, 10), impl="numpy", dtype="float32")
discr_rect = discr.element(char_rect)
for halfcomplex in [False, True]:
if halfcomplex and sign == "+":
continue # cannot mix halfcomplex with sign
for axes in [(0,), (1,)]:
ft = FourierTransform(discr, sign=sign, impl=impl, axes=axes, halfcomplex=halfcomplex)
assert all_almost_equal(ft.inverse(ft(char_rect)), discr_rect)
assert all_almost_equal(ft.adjoint(ft(char_rect)), discr_rect)
def test_fourier_trafo_init_plan(impl, floating_dtype):
# Not supported, skip
if floating_dtype == np.dtype("float16") and impl == "pyfftw":
return
shape = 10
halfcomplex, _ = _params_from_dtype(floating_dtype)
space_discr = odl.uniform_discr(0, 1, shape, dtype=floating_dtype)
ft = FourierTransform(space_discr, impl=impl, halfcomplex=halfcomplex)
if impl != "pyfftw":
with pytest.raises(ValueError):
ft.init_fftw_plan()
with pytest.raises(ValueError):
ft.clear_fftw_plan()
else:
ft.init_fftw_plan()
# Make sure plan can be used
ft._fftw_plan(ft.domain.element().asarray(), ft.range.element().asarray())
ft.clear_fftw_plan()
assert ft._fftw_plan is None
# With temporaries
ft.create_temporaries(r=True, f=False)
if impl != "pyfftw":
with pytest.raises(ValueError):
ft.init_fftw_plan()
with pytest.raises(ValueError):
ft.clear_fftw_plan()
else:
ft.init_fftw_plan()
# Make sure plan can be used
ft._fftw_plan(ft.domain.element().asarray(), ft.range.element().asarray())
ft.clear_fftw_plan()
assert ft._fftw_plan is None
ft.create_temporaries(r=False, f=True)
if impl != "pyfftw":
with pytest.raises(ValueError):
ft.init_fftw_plan()
with pytest.raises(ValueError):
ft.clear_fftw_plan()
else:
ft.init_fftw_plan()
# Make sure plan can be used
ft._fftw_plan(ft.domain.element().asarray(), ft.range.element().asarray())
ft.clear_fftw_plan()
assert ft._fftw_plan is None
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_inverse(impl, sign):
# Test if the inverse really is the inverse
def char_interval(x):
return (x >= 0) & (x <= 1)
# Complex-to-complex
discr = odl.uniform_discr(-2, 2, 40, impl='numpy', dtype='complex64')
discr_char = discr.element(char_interval)
ft = FourierTransform(discr, sign=sign, impl=impl)
assert all_almost_equal(ft.inverse(ft(char_interval)), discr_char)
assert all_almost_equal(ft.adjoint(ft(char_interval)), discr_char)
# Half-complex
discr = odl.uniform_discr(-2, 2, 40, impl='numpy', dtype='float32')
ft = FourierTransform(discr, impl=impl, halfcomplex=True)
assert all_almost_equal(ft.inverse(ft(char_interval)), discr_char)
def char_rect(x):
return (x[0] >= 0) & (x[0] <= 1) & (x[1] >= 0) & (x[1] <= 1)
# 2D with axes, C2C
discr = odl.uniform_discr([-2, -2], [2, 2], (20, 10),
impl='numpy',
dtype='complex64')
discr_rect = discr.element(char_rect)
for axes in [(0, ), 1]:
ft = FourierTransform(discr, sign=sign, impl=impl, axes=axes)
assert all_almost_equal(ft.inverse(ft(char_rect)), discr_rect)
assert all_almost_equal(ft.adjoint(ft(char_rect)), discr_rect)
# 2D with axes, halfcomplex
discr = odl.uniform_discr([-2, -2], [2, 2], (20, 10),
impl='numpy',
dtype='float32')
discr_rect = discr.element(char_rect)
for halfcomplex in [False, True]:
if halfcomplex and sign == '+':
continue # cannot mix halfcomplex with sign
for axes in [(0, ), (1, )]:
ft = FourierTransform(discr,
sign=sign,
impl=impl,
axes=axes,
halfcomplex=halfcomplex)
assert all_almost_equal(ft.inverse(ft(char_rect)), discr_rect)
assert all_almost_equal(ft.adjoint(ft(char_rect)), discr_rect)
def test_fourier_trafo_init_plan(impl, floating_dtype):
# Not supported, skip
if floating_dtype == np.dtype('float16') and impl == 'pyfftw':
return
shape = 10
halfcomplex, _ = _params_from_dtype(floating_dtype)
space_discr = odl.uniform_discr(0, 1, shape, dtype=floating_dtype)
ft = FourierTransform(space_discr, impl=impl, halfcomplex=halfcomplex)
if impl != 'pyfftw':
with pytest.raises(ValueError):
ft.init_fftw_plan()
with pytest.raises(ValueError):
ft.clear_fftw_plan()
else:
ft.init_fftw_plan()
# Make sure plan can be used
ft._fftw_plan(ft.domain.element().asarray(),
ft.range.element().asarray())
ft.clear_fftw_plan()
assert ft._fftw_plan is None
# With temporaries
ft.create_temporaries(r=True, f=False)
if impl != 'pyfftw':
with pytest.raises(ValueError):
ft.init_fftw_plan()
with pytest.raises(ValueError):
ft.clear_fftw_plan()
else:
ft.init_fftw_plan()
# Make sure plan can be used
ft._fftw_plan(ft.domain.element().asarray(),
ft.range.element().asarray())
ft.clear_fftw_plan()
assert ft._fftw_plan is None
ft.create_temporaries(r=False, f=True)
if impl != 'pyfftw':
with pytest.raises(ValueError):
ft.init_fftw_plan()
with pytest.raises(ValueError):
ft.clear_fftw_plan()
else:
ft.init_fftw_plan()
# Make sure plan can be used
ft._fftw_plan(ft.domain.element().asarray(),
ft.range.element().asarray())
ft.clear_fftw_plan()
assert ft._fftw_plan is None
