def test_finite_diff_periodic_padding():
"""Finite difference using periodic padding."""
diff_forward = finite_diff(DATA_1D,
axis=0,
method='forward',
pad_mode='periodic')
assert diff_forward[0] == DATA_1D[1] - DATA_1D[0]
assert diff_forward[-1] == DATA_1D[0] - DATA_1D[-1]
diff_backward = finite_diff(DATA_1D,
axis=0,
method='backward',
pad_mode='periodic')
assert diff_backward[0] == DATA_1D[0] - DATA_1D[-1]
assert diff_backward[-1] == DATA_1D[-1] - DATA_1D[-2]
diff_central = finite_diff(DATA_1D,
axis=0,
method='central',
pad_mode='periodic')
assert diff_central[0] == (DATA_1D[1] - DATA_1D[-1]) / 2
assert diff_central[-1] == (DATA_1D[0] - DATA_1D[-2]) / 2
def test_discrete_divergence_cuda():
"""Discretized spatial divergence operator using CUDA."""
# Check result of operator with explicit summation
# DiscreteLp
space = odl.uniform_discr([0, 0], [1.5, 10], DATA_2D.shape, impl='cuda')
# operator instance
div = Divergence(range=space)
# apply operator
dom_vec = div.domain.element([DATA_2D, DATA_2D])
div_dom_vec = div(dom_vec)
# computation of divergence with helper function
dx0, dx1 = space.cell_sides
diff_0 = finite_diff(dom_vec[0].asarray(), axis=0, dx=dx0,
padding_method='constant')
diff_1 = finite_diff(dom_vec[1].asarray(), axis=1, dx=dx1,
padding_method='constant')
assert all_equal(diff_0 + diff_1, div_dom_vec.asarray())
# Adjoint operator
adj_div = div.adjoint
ran_vec = div.range.element(DATA_2D ** 2)
adj_div_ran_vec = adj_div(ran_vec)
# Adjoint condition
lhs = ran_vec.inner(div_dom_vec)
rhs = dom_vec.inner(adj_div_ran_vec)
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs)
def test_gradient_cuda():
"""Discretized spatial gradient operator using CUDA."""
# DiscreteLp Vector
discr_space = odl.uniform_discr([0, 0], [6, 2.5], DATA_2D.shape,
impl='cuda')
dom_vec = discr_space.element(DATA_2D)
# computation of gradient components with helper function
dx0, dx1 = discr_space.cell_sides
diff_0 = finite_diff(DATA_2D, axis=0, dx=dx0, padding_method='constant')
diff_1 = finite_diff(DATA_2D, axis=1, dx=dx1, padding_method='constant')
# gradient
grad = Gradient(discr_space)
grad_vec = grad(dom_vec)
assert len(grad_vec) == DATA_2D.ndim
assert all_equal(grad_vec[0].asarray(), diff_0)
assert all_equal(grad_vec[1].asarray(), diff_1)
# adjoint operator
ran_vec = grad.range.element([DATA_2D, DATA_2D ** 2])
adj_vec = grad.adjoint(ran_vec)
lhs = ran_vec.inner(grad_vec)
rhs = dom_vec.inner(adj_vec)
assert lhs != 0
assert rhs != 0
assert lhs == rhs
def test_finite_diff_constant_padding():
"""Finite difference using constant padding."""
for pad_const in [-1, 0, 1]:
diff_forward = finite_diff(DATA_1D,
axis=0,
method='forward',
pad_mode='constant',
pad_const=pad_const)
assert diff_forward[0] == DATA_1D[1] - DATA_1D[0]
assert diff_forward[-1] == pad_const - DATA_1D[-1]
diff_backward = finite_diff(DATA_1D,
axis=0,
method='backward',
pad_mode='constant',
pad_const=pad_const)
assert diff_backward[0] == DATA_1D[0] - pad_const
assert diff_backward[-1] == DATA_1D[-1] - DATA_1D[-2]
diff_central = finite_diff(DATA_1D,
axis=0,
method='central',
pad_mode='constant',
pad_const=pad_const)
assert diff_central[0] == (DATA_1D[1] - pad_const) / 2
assert diff_central[-1] == (pad_const - DATA_1D[-2]) / 2
def test_gradient(method, impl, padding):
"""Discretized spatial gradient operator."""
with pytest.raises(TypeError):
Gradient(odl.Rn(1), method=method)
if isinstance(padding, tuple):
padding_method, padding_value = padding
else:
padding_method, padding_value = padding, None
# DiscreteLp Vector
discr_space = odl.uniform_discr([0, 0], [1, 1], DATA_2D.shape, impl=impl)
dom_vec = discr_space.element(DATA_2D)
# computation of gradient components with helper function
dx0, dx1 = discr_space.cell_sides
diff_0 = finite_diff(DATA_2D, axis=0, dx=dx0, method=method,
padding_method=padding_method,
padding_value=padding_value)
diff_1 = finite_diff(DATA_2D, axis=1, dx=dx1, method=method,
padding_method=padding_method,
padding_value=padding_value)
# gradient
grad = Gradient(discr_space, method=method,
padding_method=padding_method,
padding_value=padding_value)
grad_vec = grad(dom_vec)
assert len(grad_vec) == DATA_2D.ndim
assert all_almost_equal(grad_vec[0].asarray(), diff_0)
assert all_almost_equal(grad_vec[1].asarray(), diff_1)
# Test adjoint operator
derivative = grad.derivative()
ran_vec = derivative.range.element([DATA_2D, DATA_2D ** 2])
deriv_grad_vec = derivative(dom_vec)
adj_grad_vec = derivative.adjoint(ran_vec)
lhs = ran_vec.inner(deriv_grad_vec)
rhs = dom_vec.inner(adj_grad_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs)
# higher dimensional arrays
lin_size = 3
for ndim in [1, 3, 6]:
# DiscreteLp Vector
space = odl.uniform_discr([0.] * ndim, [1.] * ndim, [lin_size] * ndim)
dom_vec = odl.phantom.cuboid(space, [0.2] * ndim, [0.8] * ndim)
# gradient
grad = Gradient(space, method=method,
padding_method=padding_method,
padding_value=padding_value)
grad(dom_vec)
def test_laplacian(space, padding):
"""Discretized spatial laplacian operator."""
# Invalid space
with pytest.raises(TypeError):
Laplacian(range=odl.rn(1))
if isinstance(padding, tuple):
pad_mode, pad_const = padding
else:
pad_mode, pad_const = padding, 0
if pad_mode in ('order1', 'order2'):
return # these pad modes not supported for laplacian
# Operator instance
lap = Laplacian(space, pad_mode=pad_mode, pad_const=pad_const)
# Apply operator
dom_vec = noise_element(space)
div_dom_vec = lap(dom_vec)
# computation of divergence with helper function
expected_result = np.zeros(space.shape)
for axis, dx in enumerate(space.cell_sides):
diff_f = finite_diff(dom_vec.asarray(),
axis=axis,
dx=dx**2,
method='forward',
pad_mode=pad_mode,
pad_const=pad_const)
diff_b = finite_diff(dom_vec.asarray(),
axis=axis,
dx=dx**2,
method='backward',
pad_mode=pad_mode,
pad_const=pad_const)
expected_result += diff_f - diff_b
assert all_almost_equal(expected_result, div_dom_vec.asarray())
# Adjoint operator
derivative = lap.derivative()
deriv_lap_dom_vec = derivative(dom_vec)
ran_vec = noise_element(lap.range)
adj_lap_ran_vec = derivative.adjoint(ran_vec)
# Adjoint condition
lhs = ran_vec.inner(deriv_lap_dom_vec)
rhs = dom_vec.inner(adj_lap_ran_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs, places=4)
def test_laplacian(impl, padding):
"""Discretized spatial laplacian operator."""
# Invalid space
with pytest.raises(TypeError):
Divergence(range=odl.Rn(1))
if isinstance(padding, tuple):
padding_method, padding_value = padding
else:
padding_method, padding_value = padding, None
# DiscreteLp
space = odl.uniform_discr([0, 0], [1, 1], DATA_2D.shape, impl=impl)
# Operator instance
lap = Laplacian(space,
padding_method=padding_method,
padding_value=padding_value)
# Apply operator
dom_vec = lap.domain.element(DATA_2D)
div_dom_vec = lap(dom_vec)
# computation of divergence with helper function
dx0, dx1 = space.cell_sides
expected_result = np.zeros(space.shape)
for axis, dx in enumerate(space.cell_sides):
diff_f = finite_diff(dom_vec.asarray(), axis=axis, dx=dx ** 2,
method='forward',
padding_method=padding_method,
padding_value=padding_value)
diff_b = finite_diff(dom_vec.asarray(), axis=axis, dx=dx ** 2,
method='backward',
padding_method=padding_method,
padding_value=padding_value)
expected_result += diff_f - diff_b
assert all_almost_equal(expected_result, div_dom_vec.asarray())
# Adjoint operator
derivative = lap.derivative()
deriv_lap_dom_vec = derivative(dom_vec)
ran_vec = lap.range.element(DATA_2D ** 2)
adj_lap_ran_vec = derivative.adjoint(ran_vec)
# Adjoint condition
lhs = ran_vec.inner(deriv_lap_dom_vec)
rhs = dom_vec.inner(adj_lap_ran_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs)
def test_divergence_cpu():
"""Discretized spatial divergence operator."""
# Invalid space
with pytest.raises(TypeError):
Divergence(range=odl.Rn(1))
# DiscreteLp
# space = odl.uniform_discr([0, 0], [6, 2.5], DATA.shape)
space = odl.uniform_discr([0, 0], [3, 5], DATA_2D.shape)
# Operator instance
div = Divergence(range=space, method='forward')
# Apply operator
# dom_vec = div.domain.element([DATA / 2, DATA ** 3])
dom_vec = div.domain.element([DATA_2D, DATA_2D])
div_dom_vec = div(dom_vec)
# computation of divergence with helper function
dx0, dx1 = space.cell_sides
diff_0 = finite_diff(dom_vec[0].asarray(), axis=0, dx=dx0,
padding_method='constant')
diff_1 = finite_diff(dom_vec[1].asarray(), axis=1, dx=dx1,
padding_method='constant')
assert all_equal(diff_0 + diff_1, div_dom_vec.asarray())
# Adjoint operator
adj_div = div.adjoint
ran_vec = div.range.element(DATA_2D ** 2)
adj_div_ran_vec = adj_div(ran_vec)
# Adjoint condition
lhs = ran_vec.inner(div_dom_vec)
rhs = dom_vec.inner(adj_div_ran_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs)
# Higher dimensional arrays
for ndim in range(1, 6):
# DiscreteLp Vector
lin_size = 3
space = odl.uniform_discr([0.] * ndim, [lin_size] * ndim,
[lin_size] * ndim)
# Divergence
div = Divergence(range=space)
dom_vec = div.domain.element([ndvolume(lin_size, ndim)] * ndim)
div(dom_vec)
def test_finite_diff_symmetric_padding():
"""Finite difference using replicate padding."""
# Using replicate padding forward and backward differences have zero
# derivative at the upper or lower endpoint, respectively
assert finite_diff(DATA_1D, method='forward',
padding_method='symmetric')[-1] == 0
assert finite_diff(DATA_1D, method='backward',
padding_method='symmetric')[0] == 0
diff = finite_diff(DATA_1D, method='central', padding_method='symmetric')
assert diff[0] == (DATA_1D[1] - DATA_1D[0]) / 2
assert diff[-1] == (DATA_1D[-1] - DATA_1D[-2]) / 2
def test_laplacian(space, padding):
"""Discretized spatial laplacian operator."""
# Invalid space
with pytest.raises(TypeError):
Laplacian(range=odl.rn(1))
if isinstance(padding, tuple):
pad_mode, pad_const = padding
else:
pad_mode, pad_const = padding, 0
if pad_mode in ('order1', 'order2'):
return # these pad modes not supported for laplacian
# Operator instance
lap = Laplacian(space, pad_mode=pad_mode, pad_const=pad_const)
# Apply operator
dom_vec = noise_element(space)
div_dom_vec = lap(dom_vec)
# computation of divergence with helper function
expected_result = np.zeros(space.shape)
for axis, dx in enumerate(space.cell_sides):
diff_f = finite_diff(dom_vec.asarray(), axis=axis, dx=dx ** 2,
method='forward', pad_mode=pad_mode,
pad_const=pad_const)
diff_b = finite_diff(dom_vec.asarray(), axis=axis, dx=dx ** 2,
method='backward', pad_mode=pad_mode,
pad_const=pad_const)
expected_result += diff_f - diff_b
assert all_almost_equal(expected_result, div_dom_vec.asarray())
# Adjoint operator
derivative = lap.derivative()
deriv_lap_dom_vec = derivative(dom_vec)
ran_vec = noise_element(lap.range)
adj_lap_ran_vec = derivative.adjoint(ran_vec)
# Adjoint condition
lhs = ran_vec.inner(deriv_lap_dom_vec)
rhs = dom_vec.inner(adj_lap_ran_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs, places=4)
def test_gradient(space, method, padding):
"""Discretized spatial gradient operator."""
places = 2 if space.dtype == np.float32 else 4
with pytest.raises(TypeError):
Gradient(odl.rn(1), method=method)
if isinstance(padding, tuple):
pad_mode, pad_const = padding
else:
pad_mode, pad_const = padding, 0
# DiscreteLp Vector
dom_vec = noise_element(space)
dom_vec_arr = dom_vec.asarray()
# gradient
grad = Gradient(space,
method=method,
pad_mode=pad_mode,
pad_const=pad_const)
grad_vec = grad(dom_vec)
assert len(grad_vec) == space.ndim
# computation of gradient components with helper function
for axis, dx in enumerate(space.cell_sides):
diff = finite_diff(dom_vec_arr,
axis=axis,
dx=dx,
method=method,
pad_mode=pad_mode,
pad_const=pad_const)
assert all_almost_equal(grad_vec[axis].asarray(), diff)
# Test adjoint operator
derivative = grad.derivative()
ran_vec = noise_element(derivative.range)
deriv_grad_vec = derivative(dom_vec)
adj_grad_vec = derivative.adjoint(ran_vec)
lhs = ran_vec.inner(deriv_grad_vec)
rhs = dom_vec.inner(adj_grad_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs, places=places)
# Higher-dimensional arrays
lin_size = 3
for ndim in [1, 3, 6]:
space = odl.uniform_discr([0.] * ndim, [1.] * ndim, [lin_size] * ndim)
dom_vec = odl.phantom.cuboid(space, [0.2] * ndim, [0.8] * ndim)
grad = Gradient(space,
method=method,
pad_mode=pad_mode,
pad_const=pad_const)
grad(dom_vec)
def test_forward_diff():
"""Forward finite differences."""
arr = np.array([0., 3., 5., 6.])
findiff_op = finite_diff(arr, padding_method='constant', method='forward')
assert all_equal(findiff_op, [3., 2., 1., -6.])
def test_gradient_cpu():
"""Discretized spatial gradient operator."""
with pytest.raises(TypeError):
Gradient(odl.Rn(1))
# DiscreteLp Vector
discr_space = odl.uniform_discr([0, 0], [6, 2.5], DATA_2D.shape)
dom_vec = discr_space.element(DATA_2D)
# computation of gradient components with helper function
dx0, dx1 = discr_space.cell_sides
diff_0 = finite_diff(DATA_2D, axis=0, dx=dx0, method='forward',
padding_method='constant')
diff_1 = finite_diff(DATA_2D, axis=1, dx=dx1, method='forward',
padding_method='constant')
# gradient
grad = Gradient(discr_space)
grad_vec = grad(dom_vec)
assert len(grad_vec) == DATA_2D.ndim
assert all_equal(grad_vec[0].asarray(), diff_0)
assert all_equal(grad_vec[1].asarray(), diff_1)
# Test adjoint operator
ran_vec = grad.range.element([DATA_2D, DATA_2D ** 2])
adj_vec = grad.adjoint(ran_vec)
lhs = ran_vec.inner(grad_vec)
rhs = dom_vec.inner(adj_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert lhs == rhs
# higher dimensional arrays
lin_size = 3
for ndim in range(1, 6):
# DiscreteLp Vector
discr_space = odl.uniform_discr([0.] * ndim, [lin_size] * ndim,
[lin_size] * ndim)
dom_vec = discr_space.element(ndvolume(lin_size, ndim))
# gradient
grad = Gradient(discr_space)
grad(dom_vec)
def test_gradient(space, method, padding):
"""Discretized spatial gradient operator."""
places = 2 if space.dtype == np.float32 else 4
with pytest.raises(TypeError):
Gradient(odl.rn(1), method=method)
if isinstance(padding, tuple):
pad_mode, pad_const = padding
else:
pad_mode, pad_const = padding, 0
# DiscreteLp Vector
dom_vec = noise_element(space)
dom_vec_arr = dom_vec.asarray()
# gradient
grad = Gradient(space, method=method,
pad_mode=pad_mode,
pad_const=pad_const)
grad_vec = grad(dom_vec)
assert len(grad_vec) == space.ndim
# computation of gradient components with helper function
for axis, dx in enumerate(space.cell_sides):
diff = finite_diff(dom_vec_arr, axis=axis, dx=dx, method=method,
pad_mode=pad_mode,
pad_const=pad_const)
assert all_almost_equal(grad_vec[axis].asarray(), diff)
# Test adjoint operator
derivative = grad.derivative()
ran_vec = noise_element(derivative.range)
deriv_grad_vec = derivative(dom_vec)
adj_grad_vec = derivative.adjoint(ran_vec)
lhs = ran_vec.inner(deriv_grad_vec)
rhs = dom_vec.inner(adj_grad_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs, places=places)
# higher dimensional arrays
lin_size = 3
for ndim in [1, 3, 6]:
# DiscreteLpElement
space = odl.uniform_discr([0.] * ndim, [1.] * ndim, [lin_size] * ndim)
dom_vec = odl.phantom.cuboid(space, [0.2] * ndim, [0.8] * ndim)
# gradient
grad = Gradient(space, method=method,
pad_mode=pad_mode,
pad_const=pad_const)
grad(dom_vec)
def test_finite_diff_periodic_padding():
"""Finite difference using periodic padding."""
diff_forward = finite_diff(DATA_1D, method='forward',
padding_method='periodic')
assert diff_forward[0] == DATA_1D[1] - DATA_1D[0]
assert diff_forward[-1] == DATA_1D[0] - DATA_1D[-1]
diff_backward = finite_diff(DATA_1D, method='backward',
padding_method='periodic')
assert diff_backward[0] == DATA_1D[0] - DATA_1D[-1]
assert diff_backward[-1] == DATA_1D[-1] - DATA_1D[-2]
diff_central = finite_diff(DATA_1D, method='central',
padding_method='periodic')
assert diff_central[0] == (DATA_1D[1] - DATA_1D[-1]) / 2
assert diff_central[-1] == (DATA_1D[0] - DATA_1D[-2]) / 2
def test_divergence(space, method, padding):
"""Discretized spatial divergence operator."""
# Invalid space
with pytest.raises(TypeError):
Divergence(range=odl.rn(1), method=method)
if isinstance(padding, tuple):
pad_mode, pad_const = padding
else:
pad_mode, pad_const = padding, 0
# Operator instance
div = Divergence(range=space,
method=method,
pad_mode=pad_mode,
pad_const=pad_const)
# Apply operator
dom_vec = noise_element(div.domain)
div_dom_vec = div(dom_vec)
# computation of divergence with helper function
expected_result = np.zeros(space.shape)
for axis, dx in enumerate(space.cell_sides):
expected_result += finite_diff(dom_vec[axis],
axis=axis,
dx=dx,
method=method,
pad_mode=pad_mode,
pad_const=pad_const)
assert all_almost_equal(expected_result, div_dom_vec.asarray())
# Adjoint operator
derivative = div.derivative()
deriv_div_dom_vec = derivative(dom_vec)
ran_vec = noise_element(div.range)
adj_div_ran_vec = derivative.adjoint(ran_vec)
# Adjoint condition
lhs = ran_vec.inner(deriv_div_dom_vec)
rhs = dom_vec.inner(adj_div_ran_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs, places=4)
# Higher dimensional arrays
for ndim in range(1, 6):
# DiscreteLpElement
lin_size = 3
space = odl.uniform_discr([0.] * ndim, [1.] * ndim, [lin_size] * ndim)
def test_divergence(space, method, padding):
"""Discretized spatial divergence operator."""
# Invalid space
with pytest.raises(TypeError):
Divergence(range=odl.rn(1), method=method)
if isinstance(padding, tuple):
pad_mode, pad_const = padding
else:
pad_mode, pad_const = padding, 0
# Operator instance
div = Divergence(range=space, method=method,
pad_mode=pad_mode,
pad_const=pad_const)
# Apply operator
dom_vec = noise_element(div.domain)
div_dom_vec = div(dom_vec)
# computation of divergence with helper function
expected_result = np.zeros(space.shape)
for axis, dx in enumerate(space.cell_sides):
expected_result += finite_diff(dom_vec[axis], axis=axis, dx=dx,
method=method, pad_mode=pad_mode,
pad_const=pad_const)
assert all_almost_equal(expected_result, div_dom_vec.asarray())
# Adjoint operator
derivative = div.derivative()
deriv_div_dom_vec = derivative(dom_vec)
ran_vec = noise_element(div.range)
adj_div_ran_vec = derivative.adjoint(ran_vec)
# Adjoint condition
lhs = ran_vec.inner(deriv_div_dom_vec)
rhs = dom_vec.inner(adj_div_ran_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs, places=4)
# Higher dimensional arrays
for ndim in range(1, 6):
# DiscreteLpElement
lin_size = 3
space = odl.uniform_discr([0.] * ndim, [1.] * ndim, [lin_size] * ndim)
def test_finite_diff_constant_padding():
"""Finite difference using constant padding."""
for pad_const in [-1, 0, 1]:
diff_forward = finite_diff(DATA_1D, axis=0, method='forward',
pad_mode='constant',
pad_const=pad_const)
assert diff_forward[0] == DATA_1D[1] - DATA_1D[0]
assert diff_forward[-1] == pad_const - DATA_1D[-1]
diff_backward = finite_diff(DATA_1D, axis=0, method='backward',
pad_mode='constant',
pad_const=pad_const)
assert diff_backward[0] == DATA_1D[0] - pad_const
assert diff_backward[-1] == DATA_1D[-1] - DATA_1D[-2]
diff_central = finite_diff(DATA_1D, axis=0, method='central',
pad_mode='constant',
pad_const=pad_const)
assert diff_central[0] == (DATA_1D[1] - pad_const) / 2
assert diff_central[-1] == (pad_const - DATA_1D[-2]) / 2
def test_finite_diff_constant_padding():
"""Finite difference using constant padding."""
for padding_value in [-1, 0, 1]:
diff_forward = finite_diff(DATA_1D, method='forward',
padding_method='constant',
padding_value=padding_value)
assert diff_forward[0] == DATA_1D[1] - DATA_1D[0]
assert diff_forward[-1] == padding_value - DATA_1D[-1]
diff_backward = finite_diff(DATA_1D, method='backward',
padding_method='constant',
padding_value=padding_value)
assert diff_backward[0] == DATA_1D[0] - padding_value
assert diff_backward[-1] == DATA_1D[-1] - DATA_1D[-2]
diff_central = finite_diff(DATA_1D, method='central',
padding_method='constant',
padding_value=padding_value)
assert diff_central[0] == (DATA_1D[1] - padding_value) / 2
assert diff_central[-1] == (padding_value - DATA_1D[-2]) / 2
def test_part_deriv(space, method, padding):
"""Discretized partial derivative."""
with pytest.raises(TypeError):
PartialDerivative(odl.rn(1))
if isinstance(padding, tuple):
pad_mode, pad_const = padding
else:
pad_mode, pad_const = padding, 0
# discretized space
dom_vec = noise_element(space)
dom_vec_arr = dom_vec.asarray()
# operator
for axis in range(space.ndim):
partial = PartialDerivative(space,
axis=axis,
method=method,
pad_mode=pad_mode,
pad_const=pad_const)
# Compare to helper function
dx = space.cell_sides[axis]
diff = finite_diff(dom_vec_arr,
axis=axis,
dx=dx,
method=method,
pad_mode=pad_mode,
pad_const=pad_const)
partial_vec = partial(dom_vec)
assert all_almost_equal(partial_vec.asarray(), diff)
# Test adjoint operator
derivative = partial.derivative()
ran_vec = noise_element(space)
deriv_vec = derivative(dom_vec)
adj_vec = derivative.adjoint(ran_vec)
lhs = ran_vec.inner(deriv_vec)
rhs = dom_vec.inner(adj_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs, places=4)
def test_part_deriv(impl, method, padding):
"""Discretized partial derivative."""
with pytest.raises(TypeError):
PartialDerivative(odl.Rn(1))
if isinstance(padding, tuple):
padding_method, padding_value = padding
else:
padding_method, padding_value = padding, None
# discretized space
space = odl.uniform_discr([0, 0], [2, 1], DATA_2D.shape, impl=impl)
dom_vec = space.element(DATA_2D)
# operator
for axis in range(space.ndim):
partial = PartialDerivative(space, axis=axis, method=method,
padding_method=padding_method,
padding_value=padding_value)
# Compare to helper function
dx = space.cell_sides[axis]
diff = finite_diff(DATA_2D, axis=axis, dx=dx, method=method,
padding_method=padding_method,
padding_value=padding_value)
partial_vec = partial(dom_vec)
assert all_almost_equal(partial_vec.asarray(), diff)
# Test adjoint operator
derivative = partial.derivative()
ran_vec = derivative.range.element(DATA_2D ** 2)
deriv_vec = derivative(dom_vec)
adj_vec = derivative.adjoint(ran_vec)
lhs = ran_vec.inner(deriv_vec)
rhs = dom_vec.inner(adj_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs)
def test_part_deriv(space, method, padding):
"""Discretized partial derivative."""
with pytest.raises(TypeError):
PartialDerivative(odl.rn(1))
if isinstance(padding, tuple):
pad_mode, pad_const = padding
else:
pad_mode, pad_const = padding, 0
# discretized space
dom_vec = noise_element(space)
dom_vec_arr = dom_vec.asarray()
# operator
for axis in range(space.ndim):
partial = PartialDerivative(space, axis=axis, method=method,
pad_mode=pad_mode,
pad_const=pad_const)
# Compare to helper function
dx = space.cell_sides[axis]
diff = finite_diff(dom_vec_arr, axis=axis, dx=dx, method=method,
pad_mode=pad_mode,
pad_const=pad_const)
partial_vec = partial(dom_vec)
assert all_almost_equal(partial_vec.asarray(), diff)
# Test adjoint operator
derivative = partial.derivative()
ran_vec = noise_element(space)
deriv_vec = derivative(dom_vec)
adj_vec = derivative.adjoint(ran_vec)
lhs = ran_vec.inner(deriv_vec)
rhs = dom_vec.inner(adj_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs, places=4)
def test_part_deriv(space, method, padding):
"""Discretized partial derivative."""
if isinstance(padding, tuple):
pad_mode, pad_const = padding
else:
pad_mode, pad_const = padding, 0
dom_vec = noise_element(space)
dom_vec_arr = dom_vec.asarray()
for axis in range(space.ndim):
partial = PartialDerivative(space,
axis=axis,
method=method,
pad_mode=pad_mode,
pad_const=pad_const)
# Compare to helper function
dx = space.cell_sides[axis]
diff = finite_diff(dom_vec_arr,
axis=axis,
dx=dx,
method=method,
pad_mode=pad_mode,
pad_const=pad_const)
partial_vec = partial(dom_vec)
assert all_almost_equal(partial_vec, diff)
# Test adjoint operator
derivative = partial.derivative()
ran_vec = noise_element(space)
deriv_vec = derivative(dom_vec)
adj_vec = derivative.adjoint(ran_vec)
lhs = ran_vec.inner(deriv_vec)
rhs = dom_vec.inner(adj_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert lhs == pytest.approx(rhs, rel=dtype_tol(space.dtype))
def test_finite_diff_explicit():
"""Compare finite differences function to explicit computation."""
# phantom data
arr = DATA_1D
# explicitly calculated finite difference
diff_ex = np.zeros_like(arr)
# interior: second-order accurate differences
diff_ex[1:-1] = (arr[2:] - arr[:-2]) / 2.0
# default: out=None, axis=0, dx=1.0, zero_padding=None, method='forward'
diff = finite_diff(arr, axis=0, dx=1.0, out=None, pad_mode='constant')
assert all_equal(diff, finite_diff(arr, axis=0))
# boundary: one-sided second-order accurate forward/backward difference
diff = finite_diff(arr,
axis=0,
dx=1.0,
out=None,
method='central',
pad_mode='order2')
diff_ex[0] = -(3 * arr[0] - 4 * arr[1] + arr[2]) / 2.0
diff_ex[-1] = (3 * arr[-1] - 4 * arr[-2] + arr[-3]) / 2.0
assert all_equal(diff, diff_ex)
# non-unit step length
dx = 0.5
diff = finite_diff(arr,
axis=0,
dx=dx,
method='central',
out=None,
pad_mode='order2')
assert all_equal(diff, diff_ex / dx)
# boundary: second-order accurate central differences with zero padding
diff = finite_diff(arr,
axis=0,
method='central',
pad_mode='constant',
pad_const=0)
diff_ex[0] = arr[1] / 2.0
diff_ex[-1] = -arr[-2] / 2.0
assert all_equal(diff, diff_ex)
# boundary: one-sided first-order forward/backward difference without zero
# padding
diff = finite_diff(arr, axis=0, method='central', pad_mode='order1')
diff_ex[0] = arr[1] - arr[0] # 1st-order accurate forward difference
diff_ex[-1] = arr[-1] - arr[-2] # 1st-order accurate backward diff.
assert all_equal(diff, diff_ex)
# different edge order really differ
df1 = finite_diff(arr, axis=0, method='central', pad_mode='order1')
df2 = finite_diff(arr, axis=0, method='central', pad_mode='order2')
assert all_equal(df1[1:-1], diff_ex[1:-1])
assert all_equal(df2[1:-1], diff_ex[1:-1])
assert df1[0] != df2[0]
assert df1[-1] != df2[-1]
# in-place evaluation
out = np.zeros_like(arr)
assert out is finite_diff(arr, axis=0, out=out)
assert all_equal(out, finite_diff(arr, axis=0))
assert out is not finite_diff(arr, axis=0)
# axis
arr = np.array([[0., 2., 4., 6., 8.], [1., 3., 5., 7., 9.]])
df0 = finite_diff(arr, axis=0, pad_mode='order1')
darr0 = 1 * np.ones(arr.shape)
assert all_equal(df0, darr0)
darr1 = 2 * np.ones(arr.shape)
df1 = finite_diff(arr, axis=1, pad_mode='order1')
assert all_equal(df1, darr1)
# complex arrays
arr = np.array([0., 1., 2., 3., 4.]) + 1j * np.array([10., 9., 8., 7., 6.])
diff = finite_diff(arr, axis=0, pad_mode='order1')
assert all(diff.real == 1)
assert all(diff.imag == -1)
def test_finite_diff_invalid_args():
"""Test finite difference function for invalid arguments."""
# Test that old "edge order" argument fails.
with pytest.raises(ValueError):
finite_diff(DATA_1D, axis=0, edge_order=0)
with pytest.raises(ValueError):
finite_diff(DATA_1D, axis=0, edge_order=3)
# at least a two-element array is required
with pytest.raises(ValueError):
finite_diff(np.array([0.0]), axis=0)
# axis
with pytest.raises(IndexError):
finite_diff(DATA_1D, axis=2)
# in-place argument
out = np.zeros(DATA_1D.size + 1)
with pytest.raises(ValueError):
finite_diff(DATA_1D, axis=0, out=out)
with pytest.raises(ValueError):
finite_diff(DATA_1D, axis=0, dx=0)
# wrong method
with pytest.raises(ValueError):
finite_diff(DATA_1D, axis=0, method='non-method')
def test_finite_diff_invalid_args():
"""Test finite difference function for invalid arguments."""
# Test that old "edge order" argument fails.
with pytest.raises(ValueError):
finite_diff(DATA_1D, axis=0, edge_order=0)
with pytest.raises(ValueError):
finite_diff(DATA_1D, axis=0, edge_order=3)
# at least a two-element array is required
with pytest.raises(ValueError):
finite_diff(np.array([0.0]), axis=0)
# axis
with pytest.raises(IndexError):
finite_diff(DATA_1D, axis=2)
# in-place argument
out = np.zeros(DATA_1D.size + 1)
with pytest.raises(ValueError):
finite_diff(DATA_1D, axis=0, out=out)
with pytest.raises(ValueError):
finite_diff(DATA_1D, axis=0, dx=0)
# wrong method
with pytest.raises(ValueError):
finite_diff(DATA_1D, axis=0, method='non-method')
def test_finite_diff_explicit():
"""Compare finite differences function to explicit computation."""
# phantom data
arr = DATA_1D
# explicitly calculated finite difference
diff_ex = np.zeros_like(arr)
# interior: second-order accurate differences
diff_ex[1:-1] = (arr[2:] - arr[:-2]) / 2.0
# default: out=None, axis=0, dx=1.0, zero_padding=None, method='forward'
diff = finite_diff(arr, out=None, axis=0, dx=1.0, padding_method=None)
assert all_equal(diff, finite_diff(arr))
# boundary: one-sided second-order accurate forward/backward difference
diff = finite_diff(arr, out=None, axis=0, dx=1.0, method='central',
padding_method=None)
diff_ex[0] = -(3 * arr[0] - 4 * arr[1] + arr[2]) / 2.0
diff_ex[-1] = (3 * arr[-1] - 4 * arr[-2] + arr[-3]) / 2.0
assert all_equal(diff, diff_ex)
# non-unit step length
dx = 0.5
diff = finite_diff(arr, dx=dx, method='central')
assert all_equal(diff, diff_ex / dx)
# boundary: second-order accurate central differences with zero padding
diff = finite_diff(arr, method='central', padding_method='constant',
padding_value=0)
diff_ex[0] = arr[1] / 2.0
diff_ex[-1] = -arr[-2] / 2.0
assert all_equal(diff, diff_ex)
# boundary: one-sided first-order forward/backward difference without zero
# padding
diff = finite_diff(arr, method='central', edge_order=1)
diff_ex[0] = arr[1] - arr[0] # 1st-order accurate forward difference
diff_ex[-1] = arr[-1] - arr[-2] # 1st-order accurate backward diff.
assert all_equal(diff, diff_ex)
# different edge order really differ
df1 = finite_diff(arr, method='central', edge_order=1)
df2 = finite_diff(arr, method='central', edge_order=2)
assert all_equal(df1[1:-1], diff_ex[1:-1])
assert all_equal(df2[1:-1], diff_ex[1:-1])
assert df1[0] != df2[0]
assert df1[-1] != df2[-1]
# in-place evaluation
out = np.zeros_like(arr)
assert out is finite_diff(arr, out=out)
assert all_equal(out, finite_diff(arr))
assert out is not finite_diff(arr)
# axis
arr = np.array([[0., 1., 2., 3., 4.],
[1., 2., 3., 4., 5.]])
df0 = finite_diff(arr, axis=0)
df1 = finite_diff(arr, axis=1)
assert all_equal(df0, df1)
# complex arrays
arr = np.array([0., 1., 2., 3., 4.]) + 1j * np.array([10., 9., 8., 7.,
6.])
diff = finite_diff(arr)
assert all(diff.real == 1)
assert all(diff.imag == -1)
def test_divergence(method, impl, padding):
"""Discretized spatial divergence operator."""
# Invalid space
with pytest.raises(TypeError):
Divergence(range=odl.Rn(1), method=method)
if isinstance(padding, tuple):
padding_method, padding_value = padding
else:
padding_method, padding_value = padding, None
# DiscreteLp
space = odl.uniform_discr([0, 0], [1, 1], DATA_2D.shape, impl=impl)
# Operator instance
div = Divergence(range=space, method=method,
padding_method=padding_method,
padding_value=padding_value)
# Apply operator
dom_vec = div.domain.element([DATA_2D, DATA_2D])
div_dom_vec = div(dom_vec)
# computation of divergence with helper function
dx0, dx1 = space.cell_sides
diff_0 = finite_diff(dom_vec[0].asarray(), axis=0, dx=dx0, method=method,
padding_method=padding_method,
padding_value=padding_value)
diff_1 = finite_diff(dom_vec[1].asarray(), axis=1, dx=dx1, method=method,
padding_method=padding_method,
padding_value=padding_value)
assert all_almost_equal(diff_0 + diff_1, div_dom_vec.asarray())
# Adjoint operator
derivative = div.derivative()
deriv_div_dom_vec = derivative(dom_vec)
ran_vec = div.range.element(DATA_2D ** 2)
adj_div_ran_vec = derivative.adjoint(ran_vec)
# Adjoint condition
lhs = ran_vec.inner(deriv_div_dom_vec)
rhs = dom_vec.inner(adj_div_ran_vec)
# Check not to use trivial data
assert lhs != 0
assert rhs != 0
assert almost_equal(lhs, rhs)
# Higher dimensional arrays
for ndim in range(1, 6):
# DiscreteLp Vector
lin_size = 3
space = odl.uniform_discr([0.] * ndim, [1.] * ndim, [lin_size] * ndim)
# Divergence
div = Divergence(range=space, method=method,
padding_method=padding_method,
padding_value=padding_value)
dom_vec = odl.phantom.cuboid(space, [0.2] * ndim, [0.8] * ndim)
div([dom_vec] * ndim)
def test_finite_diff_invalid_args():
"""Test finite difference function for invalid arguments."""
# edge order in {1,2}
with pytest.raises(ValueError):
finite_diff(DATA_1D, edge_order=0)
with pytest.raises(ValueError):
finite_diff(DATA_1D, edge_order=3)
# central differences and zero padding use second-order accurate edges
with pytest.raises(ValueError):
finite_diff(DATA_1D, method='central', padding_method=0, edge_order=1)
with pytest.raises(ValueError):
finite_diff(DATA_1D, method='forward', padding_method=0, edge_order=2)
# at least a two-element array is required
with pytest.raises(ValueError):
finite_diff(np.array([0.0]))
# axis
with pytest.raises(IndexError):
finite_diff(DATA_1D, axis=2)
# in-place argument
out = np.zeros(DATA_1D.size + 1)
with pytest.raises(ValueError):
finite_diff(DATA_1D, out=out)
with pytest.raises(ValueError):
finite_diff(DATA_1D, dx=0)
# wrong method
with pytest.raises(ValueError):
finite_diff(DATA_1D, method='non-method')