def test_defining_spd_operator_by_taking_real_part(self):
with self.test_session() as sess:
# S is real and positive.
s = linear_operator_test_util.random_uniform(
shape=(10, 2, 3, 4), dtype=dtypes.float32, minval=1., maxval=2.)
# Let S = S1 + S2, the Hermitian and anti-hermitian parts.
# S1 = 0.5 * (S + S^H), S2 = 0.5 * (S - S^H),
# where ^H is the Hermitian transpose of the function:
# f(n0, n1, n2)^H := ComplexConjugate[f(N0-n0, N1-n1, N2-n2)].
# We want to isolate S1, since
# S1 is Hermitian by construction
# S1 is real since S is
# S1 is positive since it is the sum of two positive kernels
# IDFT[S] = IDFT[S1] + IDFT[S2]
# = H1 + H2
# where H1 is real since it is Hermitian,
# and H2 is imaginary since it is anti-Hermitian.
ifft_s = math_ops.ifft3d(math_ops.cast(s, dtypes.complex64))
# Throw away H2, keep H1.
real_ifft_s = math_ops.real(ifft_s)
# This is the perfect spectrum!
# spectrum = DFT[H1]
# = S1,
fft_real_ifft_s = math_ops.fft3d(
math_ops.cast(real_ifft_s, dtypes.complex64))
# S1 is Hermitian ==> operator is real.
# S1 is real ==> operator is self-adjoint.
# S1 is positive ==> operator is positive-definite.
operator = linalg.LinearOperatorCirculant3D(fft_real_ifft_s)
# Allow for complex output so we can check operator has zero imag part.
self.assertEqual(operator.dtype, dtypes.complex64)
matrix, matrix_t = sess.run([
operator.to_dense(),
array_ops.matrix_transpose(operator.to_dense())
])
operator.assert_positive_definite().run() # Should not fail.
np.testing.assert_allclose(0, np.imag(matrix), atol=1e-6)
self.assertAllClose(matrix, matrix_t)
# Just to test the theory, get S2 as well.
# This should create an imaginary operator.
# S2 is anti-Hermitian ==> operator is imaginary.
# S2 is real ==> operator is self-adjoint.
imag_ifft_s = math_ops.imag(ifft_s)
fft_imag_ifft_s = math_ops.fft3d(
1j * math_ops.cast(imag_ifft_s, dtypes.complex64))
operator_imag = linalg.LinearOperatorCirculant3D(fft_imag_ifft_s)
matrix, matrix_h = sess.run([
operator_imag.to_dense(),
array_ops.matrix_transpose(math_ops.conj(operator_imag.to_dense()))
])
self.assertAllClose(matrix, matrix_h)
np.testing.assert_allclose(0, np.real(matrix), atol=1e-7)
def test_defining_operator_using_real_convolution_kernel(self):
with self.cached_session():
convolution_kernel = linear_operator_test_util.random_normal(
shape=(2, 2, 3, 5), dtype=dtypes.float32)
# Convolution kernel is real ==> spectrum is Hermitian.
spectrum = math_ops.fft3d(
math_ops.cast(convolution_kernel, dtypes.complex64))
# spectrum is Hermitian ==> operator is real.
operator = linalg.LinearOperatorCirculant3D(spectrum)
self.assertAllEqual((2, 2 * 3 * 5, 2 * 3 * 5), operator.shape)
# Allow for complex output so we can make sure it has zero imag part.
self.assertEqual(operator.dtype, dtypes.complex64)
matrix = operator.to_dense().eval()
self.assertAllEqual((2, 2 * 3 * 5, 2 * 3 * 5), matrix.shape)
np.testing.assert_allclose(0, np.imag(matrix), atol=1e-6)
def test_defining_operator_using_real_convolution_kernel(self):
with self.test_session():
convolution_kernel = linear_operator_test_util.random_normal(
shape=(2, 2, 3, 5), dtype=dtypes.float32)
# Convolution kernel is real ==> spectrum is Hermitian.
spectrum = math_ops.fft3d(
math_ops.cast(convolution_kernel, dtypes.complex64))
# spectrum is Hermitian ==> operator is real.
operator = linalg.LinearOperatorCirculant3D(spectrum)
self.assertAllEqual((2, 2 * 3 * 5, 2 * 3 * 5), operator.shape)
# Allow for complex output so we can make sure it has zero imag part.
self.assertEqual(operator.dtype, dtypes.complex64)
matrix = operator.to_dense().eval()
self.assertAllEqual((2, 2 * 3 * 5, 2 * 3 * 5), matrix.shape)
np.testing.assert_allclose(0, np.imag(matrix), atol=1e-6)
def _IFFT3DGrad(_, grad):
rsize = 1. / math_ops.cast(array_ops.size(grad), dtypes.float32)
return math_ops.fft3d(grad) * math_ops.complex(rsize, 0.)
def _IFFT3DGrad(_, grad): rsize = 1. / math_ops.cast(_FFTSizeForGrad(grad, 3), dtypes.float32) return math_ops.fft3d(grad) * math_ops.complex(rsize, 0.)
def _IFFT3DGrad(_, grad): rsize = 1. / math_ops.cast(array_ops.size(grad), dtypes.float32) return math_ops.fft3d(grad) * math_ops.complex(rsize, 0.)
def test_defining_spd_operator_by_taking_real_part(self):
with self.cached_session() as sess:
# S is real and positive.
s = linear_operator_test_util.random_uniform(shape=(10, 2, 3, 4),
dtype=dtypes.float32,
minval=1.,
maxval=2.)
# Let S = S1 + S2, the Hermitian and anti-hermitian parts.
# S1 = 0.5 * (S + S^H), S2 = 0.5 * (S - S^H),
# where ^H is the Hermitian transpose of the function:
# f(n0, n1, n2)^H := ComplexConjugate[f(N0-n0, N1-n1, N2-n2)].
# We want to isolate S1, since
# S1 is Hermitian by construction
# S1 is real since S is
# S1 is positive since it is the sum of two positive kernels
# IDFT[S] = IDFT[S1] + IDFT[S2]
# = H1 + H2
# where H1 is real since it is Hermitian,
# and H2 is imaginary since it is anti-Hermitian.
ifft_s = math_ops.ifft3d(math_ops.cast(s, dtypes.complex64))
# Throw away H2, keep H1.
real_ifft_s = math_ops.real(ifft_s)
# This is the perfect spectrum!
# spectrum = DFT[H1]
# = S1,
fft_real_ifft_s = math_ops.fft3d(
math_ops.cast(real_ifft_s, dtypes.complex64))
# S1 is Hermitian ==> operator is real.
# S1 is real ==> operator is self-adjoint.
# S1 is positive ==> operator is positive-definite.
operator = linalg.LinearOperatorCirculant3D(fft_real_ifft_s)
# Allow for complex output so we can check operator has zero imag part.
self.assertEqual(operator.dtype, dtypes.complex64)
matrix, matrix_t = sess.run([
operator.to_dense(),
array_ops.matrix_transpose(operator.to_dense())
])
operator.assert_positive_definite().run() # Should not fail.
np.testing.assert_allclose(0, np.imag(matrix), atol=1e-6)
self.assertAllClose(matrix, matrix_t)
# Just to test the theory, get S2 as well.
# This should create an imaginary operator.
# S2 is anti-Hermitian ==> operator is imaginary.
# S2 is real ==> operator is self-adjoint.
imag_ifft_s = math_ops.imag(ifft_s)
fft_imag_ifft_s = math_ops.fft3d(
1j * math_ops.cast(imag_ifft_s, dtypes.complex64))
operator_imag = linalg.LinearOperatorCirculant3D(fft_imag_ifft_s)
matrix, matrix_h = sess.run([
operator_imag.to_dense(),
array_ops.matrix_transpose(
math_ops.conj(operator_imag.to_dense()))
])
self.assertAllClose(matrix, matrix_h)
np.testing.assert_allclose(0, np.real(matrix), atol=1e-7)
def _IFFT3DGrad(_, grad):
rsize = 1. / math_ops.cast(_FFTSizeForGrad(grad, 3), dtypes.float32)
return math_ops.fft3d(grad) * math_ops.complex(rsize, 0.)