def test_element_setitem_single():
H = odl.ProductSpace(odl.rn(1), odl.rn(2))
x1 = H[0].element([0])
x2 = H[1].element([1, 2])
x = H.element([x1, x2])
x1_1 = H[0].element([1])
x[-2] = x1_1
assert x[-2] is x1_1
x2_1 = H[1].element([3, 4])
x[-1] = x2_1
assert x[-1] is x2_1
x1_2 = H[0].element([5])
x[0] = x1_2
x2_2 = H[1].element([3, 4])
x[1] = x2_2
assert x[1] is x2_2
with pytest.raises(IndexError):
x[-3] = x2
x[2] = x1
def test_fixed_disp_init():
"""Verify that the init method and checks work properly."""
space = odl.uniform_discr(0, 1, 5)
disp_field = space.tangent_bundle.element(
disp_field_factory(space.ndim))
# Valid input
print(LinDeformFixedDisp(disp_field))
print(LinDeformFixedDisp(disp_field, templ_space=space))
# Non-valid input
with pytest.raises(TypeError): # displacement not ProductSpaceElement
LinDeformFixedDisp(space.one())
with pytest.raises(TypeError): # templ_space not DiscreteLp
LinDeformFixedDisp(disp_field, space.tangent_bundle)
with pytest.raises(TypeError): # templ_space not a power space
bad_pspace = odl.ProductSpace(space, odl.rn(3))
LinDeformFixedDisp(disp_field, bad_pspace)
with pytest.raises(TypeError): # templ_space not based on DiscreteLp
bad_pspace = odl.ProductSpace(odl.rn(2), 1)
LinDeformFixedDisp(disp_field, bad_pspace)
with pytest.raises(TypeError): # wrong dtype on templ_space
wrong_dtype = odl.ProductSpace(space.astype(complex), 1)
LinDeformFixedDisp(disp_field, wrong_dtype)
with pytest.raises(ValueError): # vector field spaces don't match
bad_space = odl.uniform_discr(0, 1, 10)
LinDeformFixedDisp(disp_field, bad_space)
def test_ufuncs():
# Cannot use fixture due to bug in pytest
H = odl.ProductSpace(odl.rn(1), odl.rn(2))
# one arg
x = H.element([[-1], [-2, -3]])
z = x.ufunc.absolute()
assert all_almost_equal(z, [[1], [2, 3]])
# one arg with out
x = H.element([[-1], [-2, -3]])
y = H.element()
z = x.ufunc.absolute(out=y)
assert y is z
assert all_almost_equal(z, [[1], [2, 3]])
# Two args
x = H.element([[1], [2, 3]])
y = H.element([[4], [5, 6]])
w = H.element()
z = x.ufunc.add(y)
assert all_almost_equal(z, [[5], [7, 9]])
# Two args with out
x = H.element([[1], [2, 3]])
y = H.element([[4], [5, 6]])
w = H.element()
z = x.ufunc.add(y, out=w)
assert w is z
assert all_almost_equal(z, [[5], [7, 9]])
def test_element_setitem_fancy():
"""Test assignment of pspace parts with lists."""
pspace = odl.ProductSpace(odl.rn(1), odl.rn(2), odl.rn(3))
x0 = pspace[0].element([0])
x1 = pspace[1].element([1, 2])
x2 = pspace[2].element([3, 4, 5])
x = pspace.element([x0, x1, x2])
old_x0 = x[0]
old_x2 = x[2]
# Check that values are set, but identity is preserved
new_x0 = pspace[0].element([6])
new_x2 = pspace[2].element([7, 8, 9])
x[[0, 2]] = pspace[[0, 2]].element([new_x0, new_x2])
assert x[[0, 2]][0] is old_x0
assert x[[0, 2]][0] == new_x0
assert x[[0, 2]][1] is old_x2
assert x[[0, 2]][1] == new_x2
# Set values with sequences of scalars
x[[0, 2]] = [-1, -2]
assert x[[0, 2]][0] is old_x0
assert all_equal(x[[0, 2]][0], [-1])
assert x[[0, 2]][1] is old_x2
assert all_equal(x[[0, 2]][1], [-2, -2, -2])
def test_init(exponent):
# Validate that the different init patterns work and do not crash.
space = odl.FunctionSpace(odl.IntervalProd(0, 1))
part = odl.uniform_partition_fromintv(space.domain, 10)
rn = odl.rn(10, exponent=exponent)
odl.DiscreteLp(space, part, rn, exponent=exponent)
odl.DiscreteLp(space, part, rn, exponent=exponent, interp='linear')
# Normal discretization of unit interval with complex
complex_space = odl.FunctionSpace(odl.IntervalProd(0, 1),
field=odl.ComplexNumbers())
cn = odl.cn(10, exponent=exponent)
odl.DiscreteLp(complex_space, part, cn, exponent=exponent)
space = odl.FunctionSpace(odl.IntervalProd([0, 0], [1, 1]))
part = odl.uniform_partition_fromintv(space.domain, (10, 10))
rn = odl.rn(100, exponent=exponent)
odl.DiscreteLp(space, part, rn, exponent=exponent,
interp=['nearest', 'linear'])
# Real space should not work with complex
with pytest.raises(ValueError):
odl.DiscreteLp(space, part, cn)
# Complex space should not work with reals
with pytest.raises(ValueError):
odl.DiscreteLp(complex_space, part, rn)
# Wrong size of underlying space
rn_wrong_size = odl.rn(20)
with pytest.raises(ValueError):
odl.DiscreteLp(space, part, rn_wrong_size)
def test_element_setitem_single():
"""Test assignment of pspace parts with single indices."""
pspace = odl.ProductSpace(odl.rn(1), odl.rn(2))
x0 = pspace[0].element([0])
x1 = pspace[1].element([1, 2])
x = pspace.element([x0, x1])
old_x0 = x[0]
old_x1 = x[1]
# Check that values are set, but identity is preserved
new_x0 = pspace[0].element([1])
x[-2] = new_x0
assert x[-2] == new_x0
assert x[-2] is old_x0
new_x1 = pspace[1].element([3, 4])
x[-1] = new_x1
assert x[-1] == new_x1
assert x[-1] is old_x1
# Set values with scalars
x[1] = -1
assert all_equal(x[1], [-1, -1])
assert x[1] is old_x1
# Check that out-of-bounds indices raise IndexError
with pytest.raises(IndexError):
x[-3] = x1
with pytest.raises(IndexError):
x[2] = x0
def test_reductions():
H = odl.ProductSpace(odl.rn(1), odl.rn(2))
x = H.element([[1], [2, 3]])
assert x.ufunc.sum() == 6.0
assert x.ufunc.prod() == 6.0
assert x.ufunc.min() == 1.0
assert x.ufunc.max() == 3.0
def test_custom_funcs():
# Checking the standard 1-norm and standard inner product, just to
# see that the functions are handled correctly.
r2 = odl.rn(2)
r2x = r2.element([1, -1])
r2y = r2.element([-2, 3])
# inner = -5, dist = 5, norms = (sqrt(2), sqrt(13))
r3 = odl.rn(3)
r3x = r3.element([3, 4, 4])
r3y = r3.element([1, -2, 1])
# inner = -1, dist = 7, norms = (sqrt(41), sqrt(6))
pspace_2 = odl.ProductSpace(r2, r3, exponent=2.0)
x = pspace_2.element((r2x, r3x))
y = pspace_2.element((r2y, r3y))
pspace_custom = odl.ProductSpace(r2, r3, inner=custom_inner)
xc = pspace_custom.element((r2x, r3x))
yc = pspace_custom.element((r2y, r3y))
assert almost_equal(x.inner(y), xc.inner(yc))
pspace_1 = odl.ProductSpace(r2, r3, exponent=1.0)
x = pspace_1.element((r2x, r3x))
y = pspace_1.element((r2y, r3y))
pspace_custom = odl.ProductSpace(r2, r3, norm=custom_norm)
xc = pspace_custom.element((r2x, r3x))
assert almost_equal(x.norm(), xc.norm())
pspace_custom = odl.ProductSpace(r2, r3, dist=custom_dist)
xc = pspace_custom.element((r2x, r3x))
yc = pspace_custom.element((r2y, r3y))
assert almost_equal(x.dist(y), xc.dist(yc))
with pytest.raises(TypeError):
odl.ProductSpace(r2, r3, a=1) # extra keyword argument
with pytest.raises(ValueError):
odl.ProductSpace(r2, r3, norm=custom_norm, inner=custom_inner)
with pytest.raises(ValueError):
odl.ProductSpace(r2, r3, dist=custom_dist, inner=custom_inner)
with pytest.raises(ValueError):
odl.ProductSpace(r2, r3, norm=custom_norm, dist=custom_dist)
with pytest.raises(ValueError):
odl.ProductSpace(r2, r3, norm=custom_norm, exponent=1.0)
with pytest.raises(ValueError):
odl.ProductSpace(r2, r3, norm=custom_norm, weight=2.0)
with pytest.raises(ValueError):
odl.ProductSpace(r2, r3, dist=custom_dist, weight=2.0)
with pytest.raises(ValueError):
odl.ProductSpace(r2, r3, inner=custom_inner, weight=2.0)
def __init__(self, matrix, domain=None, range=None):
dom = (odl.rn(matrix.shape[1])
if domain is None else domain)
ran = (odl.rn(matrix.shape[0])
if range is None else range)
super().__init__(dom, ran)
self.matrix = matrix
def test_getitem_fancy():
r1 = odl.rn(1)
r2 = odl.rn(2)
r3 = odl.rn(3)
H = odl.ProductSpace(r1, r2, r3)
assert H[[0, 2]] == odl.ProductSpace(r1, r3)
assert H[[0, 2]][0] is r1
assert H[[0, 2]][1] is r3
def test_getitem_slice():
r1 = odl.rn(1)
r2 = odl.rn(2)
r3 = odl.rn(3)
H = odl.ProductSpace(r1, r2, r3)
assert H[:2] == odl.ProductSpace(r1, r2)
assert H[:2][0] is r1
assert H[:2][1] is r2
assert H[3:] == odl.ProductSpace(field=r1.field)
def test_element_getitem_fancy():
H = odl.ProductSpace(odl.rn(1), odl.rn(2), odl.rn(3))
x1 = H[0].element([0])
x2 = H[1].element([1, 2])
x3 = H[2].element([3, 4, 5])
x = H.element([x1, x2, x3])
assert x[[0, 2]].space == H[[0, 2]]
assert x[[0, 2]][0] is x1
assert x[[0, 2]][1] is x3
def test_element_getitem_slice():
H = odl.ProductSpace(odl.rn(1), odl.rn(2), odl.rn(3))
x0 = H[0].element([0])
x1 = H[1].element([1, 2])
x2 = H[2].element([3, 4, 5])
x = H.element([x0, x1, x2])
assert x[:2].space == H[:2]
assert x[:2][0] is x0
assert x[:2][1] is x1
def test_getitem_single():
r1 = odl.rn(1)
r2 = odl.rn(2)
H = odl.ProductSpace(r1, r2)
assert H[-2] is r1
assert H[-1] is r2
assert H[0] is r1
assert H[1] is r2
with pytest.raises(IndexError):
H[-3]
H[2]
def test_mixed_space():
"""Verify that a mixed productspace is handled properly."""
r2_1 = odl.rn(2, dtype='float64')
r2_2 = odl.rn(2, dtype='float32')
pspace = odl.ProductSpace(r2_1, r2_2)
assert not pspace.is_power_space
assert pspace.spaces[0] is r2_1
assert pspace.spaces[1] is r2_2
# dtype not well defined for this space
with pytest.raises(AttributeError):
pspace.dtype
def test_element_setitem_slice():
H = odl.ProductSpace(odl.rn(1), odl.rn(2), odl.rn(3))
x1 = H[0].element([0])
x2 = H[1].element([1, 2])
x3 = H[2].element([3, 4, 5])
x = H.element([x1, x2, x3])
x1_new = H[0].element([6])
x2_new = H[1].element([7, 8])
x[:2] = H[:2].element([x1_new, x2_new])
assert x[:2][0] is x1_new
assert x[:2][1] is x2_new
def test_element_setitem_fancy():
H = odl.ProductSpace(odl.rn(1), odl.rn(2), odl.rn(3))
x1 = H[0].element([0])
x2 = H[1].element([1, 2])
x3 = H[2].element([3, 4, 5])
x = H.element([x1, x2, x3])
x1_new = H[0].element([6])
x3_new = H[2].element([7, 8, 9])
x[[0, 2]] = H[[0, 2]].element([x1_new, x3_new])
assert x[[0, 2]][0] is x1_new
assert x[[0, 2]][1] is x3_new
def test_element_getitem_single():
H = odl.ProductSpace(odl.rn(1), odl.rn(2))
x1 = H[0].element([0])
x2 = H[1].element([1, 2])
x = H.element([x1, x2])
assert x[-2] is x1
assert x[-1] is x2
assert x[0] is x1
assert x[1] is x2
with pytest.raises(IndexError):
x[-3]
x[2]
def test_element_equals():
H = odl.ProductSpace(odl.rn(1), odl.rn(2))
x = H.element([[0], [1, 2]])
assert x != 0 # test == not always true
assert x == x
x_2 = H.element([[0], [1, 2]])
assert x == x_2
x_3 = H.element([[3], [1, 2]])
assert x != x_3
x_4 = H.element([[0], [1, 3]])
assert x != x_4
def test_nonlinear_functional():
r3 = odl.rn(3)
x = r3.element([1, 2, 3])
op = SumSquaredFunctional(r3)
assert almost_equal(op(x), np.sum(x**2))
def test_functional():
r3 = odl.rn(3)
x = r3.element([1, 2, 3])
op = SumFunctional(r3)
assert op(x) == 6
def test_arithmetic():
"""Test that all standard arithmetic works."""
space = odl.rn(3)
# Create elements needed for later
functional = odl.solvers.L2Norm(space).translated([1, 2, 3])
functional2 = odl.solvers.L2NormSquared(space)
operator = odl.IdentityOperator(space) - space.element([4, 5, 6])
x = noise_element(functional.domain)
y = noise_element(functional.domain)
scalar = np.pi
# Simple tests here, more in depth comes later
assert functional(x) == functional(x)
assert functional(x) != functional2(x)
assert (scalar * functional)(x) == scalar * functional(x)
assert (scalar * (scalar * functional))(x) == scalar**2 * functional(x)
assert (functional * scalar)(x) == functional(scalar * x)
assert ((functional * scalar) * scalar)(x) == functional(scalar**2 * x)
assert (functional + functional2)(x) == functional(x) + functional2(x)
assert (functional - functional2)(x) == functional(x) - functional2(x)
assert (functional * operator)(x) == functional(operator(x))
assert all_almost_equal((y * functional)(x), y * functional(x))
assert all_almost_equal((y * (y * functional))(x), (y * y) * functional(x))
assert all_almost_equal((functional * y)(x), functional(y * x))
assert all_almost_equal(((functional * y) * y)(x), functional((y * y) * x))
def test_primal_dual_with_li():
"""Test for the forward-backward solver with infimal convolution.
The test is done by minimizing the functional ``(g @ l)(x)``, where
``(g @ l)(x) = inf_y { g(y) + l(x - y) }``,
g is the indicator function on [-3, -1], and l(x) = 1/2||x||_2^2.
The optimal solution to this problem is given by x in [-3, -1].
"""
# Parameter values for the box constraint
upper_lim = -1
lower_lim = -3
space = odl.rn(1)
lin_ops = [odl.IdentityOperator(space)]
g = [odl.solvers.IndicatorBox(space, lower=lower_lim, upper=upper_lim)]
f = odl.solvers.ZeroFunctional(space)
l = [odl.solvers.L2NormSquared(space)]
# Centering around a point further away from [-3,-1].
x = space.element(10)
douglas_rachford_pd(x, f, g, lin_ops, tau=0.5, sigma=[1.0], niter=20, l=l)
assert lower_lim - 10 ** -LOW_ACCURACY <= float(x)
assert float(x) <= upper_lim + 10 ** -LOW_ACCURACY
def test_power_method_opnorm_exceptions():
# Test the exceptions
space = odl.rn(2)
op = odl.IdentityOperator(space)
with pytest.raises(ValueError):
# Too small number of iterates
power_method_opnorm(op, maxiter=0)
with pytest.raises(ValueError):
# Negative number of iterates
power_method_opnorm(op, maxiter=-5)
with pytest.raises(ValueError):
# Input vector is zero
power_method_opnorm(op, maxiter=2, xstart=space.zero())
with pytest.raises(ValueError):
# Input vector in the nullspace
op = odl.MatrixOperator([[0., 1.], [0., 0.]])
power_method_opnorm(op, maxiter=2, xstart=op.domain.one())
with pytest.raises(ValueError):
# Uneven number of iterates for non square operator
op = odl.MatrixOperator([[1., 2., 3.], [4., 5., 6.]])
power_method_opnorm(op, maxiter=1, xstart=op.domain.one())
def test_functional_scale():
r3 = odl.rn(3)
Aop = SumFunctional(r3)
x = r3.element([1, 2, 3])
y = 1
# Test a range of scalars (scalar multiplication could implement
# optimizations for (-1, 0, 1).
scalars = [-1.432, -1, 0, 1, 3.14]
for scale in scalars:
C = OperatorRightScalarMult(Aop, scale)
assert C.is_linear
assert C.adjoint.is_linear
assert C(x) == scale * np.sum(x)
assert all_almost_equal(C.adjoint(y), scale * y * np.ones(3))
assert all_almost_equal(C.adjoint.adjoint(x), C(x))
# Using operator overloading
assert (scale * Aop)(x) == scale * np.sum(x)
assert (Aop * scale)(x) == scale * np.sum(x)
assert all_almost_equal((scale * Aop).adjoint(y),
scale * y * np.ones(3))
assert all_almost_equal((Aop * scale).adjoint(y),
scale * y * np.ones(3))
def test_element_1d(exponent):
discr = odl.uniform_discr(0, 1, 3, impl='numpy', exponent=exponent)
weight = 1.0 if exponent == float('inf') else discr.cell_volume
dspace = odl.rn(3, exponent=exponent, weighting=weight)
elem = discr.element()
assert isinstance(elem, odl.DiscreteLpElement)
assert elem.ntuple in dspace
def test_metric():
H = odl.rn(2)
v11 = H.element([1, 2])
v12 = H.element([5, 3])
v21 = H.element([1, 2])
v22 = H.element([8, 9])
# 1-norm
HxH = odl.ProductSpace(H, H, exponent=1.0)
w1 = HxH.element([v11, v12])
w2 = HxH.element([v21, v22])
assert almost_equal(HxH.dist(w1, w2),
H.dist(v11, v21) + H.dist(v12, v22))
# 2-norm
HxH = odl.ProductSpace(H, H, exponent=2.0)
w1 = HxH.element([v11, v12])
w2 = HxH.element([v21, v22])
assert almost_equal(
HxH.dist(w1, w2),
(H.dist(v11, v21) ** 2 + H.dist(v12, v22) ** 2) ** (1 / 2.0))
# inf norm
HxH = odl.ProductSpace(H, H, exponent=float('inf'))
w1 = HxH.element([v11, v12])
w2 = HxH.element([v21, v22])
assert almost_equal(
HxH.dist(w1, w2),
max(H.dist(v11, v21), H.dist(v12, v22)))
def test_pnorm(exponent):
for fn in (odl.rn(3, exponent=exponent), odl.cn(3, exponent=exponent)):
xarr, x = noise_elements(fn)
correct_norm = np.linalg.norm(xarr, ord=exponent)
assert almost_equal(fn.norm(x), correct_norm)
assert almost_equal(x.norm(), correct_norm)
def test_getitem_single():
r1 = odl.rn(1)
r2 = odl.rn(2)
H = odl.ProductSpace(r1, r2)
assert H[-2] is r1
assert H[-1] is r2
assert H[0] is r1
assert H[1] is r2
with pytest.raises(IndexError):
H[-3]
H[2]
assert H[(1, )] == r2
with pytest.raises(IndexError):
H[0, 1]
def test_element_getitem_multi():
"""Test element access with multiple indices."""
pspace = odl.ProductSpace(odl.rn(1), odl.rn(2))
pspace2 = odl.ProductSpace(pspace, 3)
pspace3 = odl.ProductSpace(pspace2, 2)
z = pspace3.element([[[[1], [2, 3]], [[4], [5, 6]], [[7], [8, 9]]],
[[[10], [12, 13]], [[14], [15, 16]], [[17], [18,
19]]]])
assert pspace3.shape == (2, 3, 2)
assert z[0, 0, 0, 0] == 1
assert all_equal(z[0, 0, 1], [2, 3])
assert all_equal(z[0, 0], [[1], [2, 3]])
assert all_equal(z[0, 1:], [[[4], [5, 6]], [[7], [8, 9]]])
assert all_equal(z[0, 1:, 1], [[5, 6], [8, 9]])
assert all_equal(z[0, 1:, :, 0], [[[4], [5]], [[7], [8]]])
def test_gradient_init():
"""Check initialization of ``Gradient``."""
space = odl.uniform_discr([0, 0], [1, 1], (4, 5))
vspace = space ** 2
op = Gradient(space)
assert repr(op) != ''
op = Gradient(range=vspace)
assert repr(op) != ''
op = Gradient(space, range=space.astype('float32') ** 2)
assert repr(op) != ''
op = Gradient(space, method='central')
assert repr(op) != ''
op = Gradient(space, pad_const=1)
assert repr(op) != ''
op = Gradient(space, pad_mode='order1')
assert repr(op) != ''
with pytest.raises(TypeError):
Gradient(odl.rn(1))
with pytest.raises(TypeError):
Gradient(space, range=space)
with pytest.raises(ValueError):
Gradient(space, range=space ** 3)
def test_divergence_init():
"""Check initialization of ``Divergence``."""
space = odl.uniform_discr([0, 0], [1, 1], (4, 5))
vspace = space ** 2
op = Divergence(vspace)
assert repr(op) != ''
op = Divergence(range=space)
assert repr(op) != ''
op = Divergence(vspace, range=space.astype('float32'))
assert repr(op) != ''
op = Divergence(vspace, method='central')
assert repr(op) != ''
op = Divergence(vspace, pad_const=1)
assert repr(op) != ''
op = Divergence(vspace, pad_mode='order1')
assert repr(op) != ''
with pytest.raises(TypeError):
Divergence(odl.rn(1) ** 2)
with pytest.raises(TypeError):
Divergence(vspace, range=vspace)
with pytest.raises(ValueError):
Divergence(space ** 3, range=space)
def space(request, tspace_impl):
name = request.param.strip()
if name == 'r10':
return odl.rn(10, impl=tspace_impl)
elif name == 'uniform_discr':
return odl.uniform_discr(0, 1, 7, impl=tspace_impl)
def test_operators(arithmetic_op):
# Test of the operators `+`, `-`, etc work as expected by numpy
space = odl.rn(3)
pspace = odl.ProductSpace(space, 2)
# Interactions with scalars
for scalar in [-31.2, -1, 0, 1, 2.13]:
# Left op
x_arr, x = noise_elements(pspace)
if scalar == 0 and arithmetic_op in [operator.truediv,
operator.itruediv]:
# Check for correct zero division behaviour
with pytest.raises(ZeroDivisionError):
y = arithmetic_op(x, scalar)
else:
y_arr = arithmetic_op(x_arr, scalar)
y = arithmetic_op(x, scalar)
assert all_almost_equal([x, y], [x_arr, y_arr])
# Right op
x_arr, x = noise_elements(pspace)
y_arr = arithmetic_op(scalar, x_arr)
y = arithmetic_op(scalar, x)
assert all_almost_equal([x, y], [x_arr, y_arr])
# Verify that the statement z=op(x, y) gives equivalent results to NumPy
x_arr, x = noise_elements(space, 1)
y_arr, y = noise_elements(pspace, 1)
# non-aliased left
if arithmetic_op in [operator.iadd,
operator.isub,
operator.itruediv,
operator.imul]:
# Check for correct error since in-place op is not possible here
with pytest.raises(TypeError):
z = arithmetic_op(x, y)
else:
z_arr = arithmetic_op(x_arr, y_arr)
z = arithmetic_op(x, y)
assert all_almost_equal([x, y, z], [x_arr, y_arr, z_arr])
# non-aliased right
z_arr = arithmetic_op(y_arr, x_arr)
z = arithmetic_op(y, x)
assert all_almost_equal([x, y, z], [x_arr, y_arr, z_arr])
# aliased operation
z_arr = arithmetic_op(y_arr, y_arr)
z = arithmetic_op(y, y)
assert all_almost_equal([x, y, z], [x_arr, y_arr, z_arr])
def test_primal_dual_l1():
"""Verify that the correct value is returned for l1 dist optimization.
Solves the optimization problem
min_x ||x - data_1||_1 + 0.5 ||x - data_2||_1
which has optimum value data_1.
"""
# Define the space
space = odl.rn(5)
# Operator
L = [odl.IdentityOperator(space)]
# Data
data_1 = odl.util.testutils.noise_element(space)
data_2 = odl.util.testutils.noise_element(space)
# Proximals
f = odl.solvers.L1Norm(space).translated(data_1)
g = [0.5 * odl.solvers.L1Norm(space).translated(data_2)]
# Solve with f term dominating
x = space.zero()
douglas_rachford_pd(x, f, g, L, tau=3.0, sigma=[1.0], niter=10)
assert all_almost_equal(x, data_1, places=2)
def test_metric():
H = odl.rn(2)
v11 = H.element([1, 2])
v12 = H.element([5, 3])
v21 = H.element([1, 2])
v22 = H.element([8, 9])
# 1-norm
HxH = odl.ProductSpace(H, H, exponent=1.0)
w1 = HxH.element([v11, v12])
w2 = HxH.element([v21, v22])
assert (HxH.dist(w1,
w2) == pytest.approx(H.dist(v11, v21) + H.dist(v12, v22)))
# 2-norm
HxH = odl.ProductSpace(H, H, exponent=2.0)
w1 = HxH.element([v11, v12])
w2 = HxH.element([v21, v22])
assert (HxH.dist(w1, w2) == pytest.approx(
(H.dist(v11, v21)**2 + H.dist(v12, v22)**2)**0.5))
# inf norm
HxH = odl.ProductSpace(H, H, exponent=float('inf'))
w1 = HxH.element([v11, v12])
w2 = HxH.element([v21, v22])
assert (HxH.dist(w1, w2) == pytest.approx(
max(H.dist(v11, v21), H.dist(v12, v22))))
def test_functional():
r3 = odl.rn(3)
x = r3.element([1, 2, 3])
op = SumFunctional(r3)
assert op(x) == 6
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_nonlinear_functional():
r3 = odl.rn(3)
x = r3.element([1, 2, 3])
op = SumSquaredFunctional(r3)
assert almost_equal(op(x), np.sum(x ** 2))
def test_nearest_interpolation_1d_variants():
intv = odl.IntervalProd(0, 1)
part = odl.uniform_partition_fromintv(intv, 5, nodes_on_bdry=False)
# Coordinate vectors are:
# [0.1, 0.3, 0.5, 0.7, 0.9]
space = odl.FunctionSpace(intv)
dspace = odl.rn(part.size)
# 'left' variant
interp_op = NearestInterpolation(space, part, dspace, variant='left')
function = interp_op([0, 1, 2, 3, 4])
# Testing two midpoints and the extreme values
pts = np.array([0.4, 0.8, 0.0, 1.0])
true_arr = [1, 3, 0, 4]
assert all_equal(function(pts), true_arr)
# 'right' variant
interp_op = NearestInterpolation(space, part, dspace, variant='right')
function = interp_op([0, 1, 2, 3, 4])
# Testing two midpoints and the extreme values
pts = np.array([0.4, 0.8, 0.0, 1.0])
true_arr = [2, 4, 0, 4]
assert all_equal(function(pts), true_arr)
def test_arithmetic():
"""Test that all standard arithmetic works."""
space = odl.rn(3)
# Create elements needed for later
functional = odl.solvers.L2Norm(space).translated([1, 2, 3])
functional2 = odl.solvers.L2NormSquared(space)
operator = odl.IdentityOperator(space) - space.element([4, 5, 6])
x = noise_element(functional.domain)
y = noise_element(functional.domain)
scalar = np.pi
# Simple tests here, more in depth comes later
assert functional(x) == functional(x)
assert functional(x) != functional2(x)
assert (scalar * functional)(x) == scalar * functional(x)
assert (scalar * (scalar * functional))(x) == scalar**2 * functional(x)
assert (functional * scalar)(x) == functional(scalar * x)
assert ((functional * scalar) * scalar)(x) == functional(scalar**2 * x)
assert (functional + functional2)(x) == functional(x) + functional2(x)
assert (functional - functional2)(x) == functional(x) - functional2(x)
assert (functional * operator)(x) == functional(operator(x))
assert all_almost_equal((y * functional)(x), y * functional(x))
assert all_almost_equal((y * (y * functional))(x), (y * y) * functional(x))
assert all_almost_equal((functional * y)(x), functional(y * x))
assert all_almost_equal(((functional * y) * y)(x), functional((y * y) * x))
def __init__(self, q):
assert all(qi.space == q[0].space for qi in q[1:])
super(KrylovSpaceEmbedding, self).__init__(domain=odl.rn(len(q)),
range=q[0].space,
linear=True)
self.q = q
def test_primal_dual_l1():
"""Verify that the correct value is returned for l1 dist optimization.
Solves the optimization problem
min_x ||x - data_1||_1 + 0.5 ||x - data_2||_1
which has optimum value data_1.
"""
# Define the space
space = odl.rn(5)
# Operator
L = [odl.IdentityOperator(space)]
# Data
data_1 = odl.util.testutils.noise_element(space)
data_2 = odl.util.testutils.noise_element(space)
# Proximals
f = odl.solvers.L1Norm(space).translated(data_1)
g = [0.5 * odl.solvers.L1Norm(space).translated(data_2)]
# Solve with f term dominating
x = space.zero()
douglas_rachford_pd(x, f, g, L, tau=3.0, sigma=[1.0], niter=10)
assert all_almost_equal(x, data_1, places=2)
def test_discretizedspace_init():
"""Test initialization and basic properties of DiscretizedSpace."""
# Real space
part = odl.uniform_partition([0, 0], [1, 1], (2, 4))
tspace = odl.rn(part.shape)
discr = DiscretizedSpace(part, tspace)
assert discr.tspace == tspace
assert discr.partition == part
assert discr.exponent == tspace.exponent
assert discr.axis_labels == ('$x$', '$y$')
assert discr.is_real
# Complex space
tspace_c = odl.cn(part.shape)
discr = DiscretizedSpace(part, tspace_c)
assert discr.is_complex
# Make sure repr shows something
assert repr(discr) != ''
# Error scenarios
part_1d = odl.uniform_partition(0, 1, 2)
with pytest.raises(ValueError):
DiscretizedSpace(part_1d, tspace) # wrong dimensionality
part_diffshp = odl.uniform_partition([0, 0], [1, 1], (3, 4))
with pytest.raises(ValueError):
DiscretizedSpace(part_diffshp, tspace) # shape mismatch
def test_weighted_proximal_L1_norm_close(space):
"""Test for the weighted proximal of the L1 norm near zero"""
# Set the space.
space = odl.rn(5)
# Define the functional on the space.
func = odl.solvers.L1Norm(space)
# Set the stepsize.
sigma = [0.1, 0.2, 0.5, 1.0, 2.0]
# Set the starting point.
x = 0.5 * space.one()
# Calculate the proximal point in-place and out-of-place
p_ip = space.element()
func.proximal(sigma)(x, out=p_ip)
p_oop = func.proximal(sigma)(x)
# Both should contain the same vector now.
assert all_almost_equal(p_ip, p_oop)
# Check if this equals the expected result.
expected_result = [0.4, 0.3, 0.0, 0.0, 0.0]
assert all_almost_equal(expected_result, p_ip)
def test_matrix_op_call(matrix):
"""Validate result from calls to matrix operators against Numpy."""
dense_matrix = matrix
sparse_matrix = scipy.sparse.coo_matrix(dense_matrix)
# Default 1d case
dmat_op = MatrixOperator(dense_matrix)
smat_op = MatrixOperator(sparse_matrix)
xarr, x = noise_elements(dmat_op.domain)
true_result = dense_matrix.dot(xarr)
assert all_almost_equal(dmat_op(x), true_result)
assert all_almost_equal(smat_op(x), true_result)
out = dmat_op.range.element()
dmat_op(x, out=out)
assert all_almost_equal(out, true_result)
smat_op(x, out=out)
assert all_almost_equal(out, true_result)
# Multi-dimensional case
domain = odl.rn((2, 2, 4))
mat_op = MatrixOperator(dense_matrix, domain, axis=2)
xarr, x = noise_elements(mat_op.domain)
true_result = np.moveaxis(np.tensordot(dense_matrix, xarr, (1, 2)), 0, 2)
assert all_almost_equal(mat_op(x), true_result)
out = mat_op.range.element()
mat_op(x, out=out)
assert all_almost_equal(out, true_result)
def test_functional_scale():
r3 = odl.rn(3)
Aop = SumFunctional(r3)
x = r3.element([1, 2, 3])
y = 1
# Test a range of scalars (scalar multiplication could implement
# optimizations for (-1, 0, 1).
scalars = [-1.432, -1, 0, 1, 3.14]
for scale in scalars:
C = OperatorRightScalarMult(Aop, scale)
assert C.is_linear
assert C.adjoint.is_linear
assert C(x) == scale * np.sum(x)
assert all_almost_equal(C.adjoint(y), scale * y * np.ones(3))
assert all_almost_equal(C.adjoint.adjoint(x), C(x))
# Using operator overloading
assert (scale * Aop)(x) == scale * np.sum(x)
assert (Aop * scale)(x) == scale * np.sum(x)
assert all_almost_equal((scale * Aop).adjoint(y),
scale * y * np.ones(3))
assert all_almost_equal((Aop * scale).adjoint(y),
scale * y * np.ones(3))
def test_collocation_interpolation_identity():
"""Check if collocation is left-inverse to interpolation."""
# Interpolation followed by collocation on the same grid should be
# the identity
rect = odl.IntervalProd([0, 0], [1, 1])
part = odl.uniform_partition_fromintv(rect, [4, 2])
space = odl.FunctionSpace(rect)
tspace = odl.rn(part.shape)
coll_op = PointCollocation(space, part, tspace)
interp_ops = [
NearestInterpolation(space, part, tspace, variant='left'),
NearestInterpolation(space, part, tspace, variant='right'),
LinearInterpolation(space, part, tspace),
PerAxisInterpolation(space,
part,
tspace,
schemes=['linear', 'nearest'])
]
values = np.arange(1, 9, dtype='float64').reshape(tspace.shape)
for interp_op in interp_ops:
ident_values = coll_op(interp_op(values))
assert all_almost_equal(ident_values, values)
def test_forward_backward_with_lin_ops():
"""Test for the forward-backward solver with linear operatros.
The test is done by minimizing ||x - b||_2^2 + ||alpha * x||_2^2. The
general problem is of the form
``min_x f(x) + sum_i g_i(L_i x) + h(x)``
and here we take f = 0, g = ||.||_2^2, L = alpha * IndentityOperator,
and h = ||. - b||_2^2.
"""
space = odl.rn(10)
alpha = 0.1
b = noise_element(space)
lin_ops = [alpha * odl.IdentityOperator(space)]
g = [odl.solvers.L2NormSquared(space)]
f = odl.solvers.ZeroFunctional(space)
# Gradient of two-norm square
h = odl.solvers.L2NormSquared(space).translated(b)
x = noise_element(space)
# Explicit solution: x_hat = (I^T * I + (alpha*I)^T * (alpha*I))^-1 * (I*b)
x_global_min = b / (1 + alpha**2)
forward_backward_pd(x, f, g, lin_ops, h, tau=0.5, sigma=[1.0], niter=20)
assert all_almost_equal(x, x_global_min, places=LOW_ACCURACY)
def test_nearest_interpolation_2d_float():
"""Test nearest neighbor interpolation in 2d."""
rect = odl.IntervalProd([0, 0], [1, 1])
part = odl.uniform_partition_fromintv(rect, [4, 2], nodes_on_bdry=False)
# Coordinate vectors are:
# [0.125, 0.375, 0.625, 0.875], [0.25, 0.75]
fspace = odl.FunctionSpace(rect)
tspace = odl.rn(part.shape)
interp_op = NearestInterpolation(fspace, part, tspace)
function = interp_op(np.reshape([0, 1, 2, 3, 4, 5, 6, 7], part.shape))
# Evaluate at single point
val = function([0.3, 0.6]) # closest to index (1, 1) -> 3
assert val == 3.0
# Input array, with and without output array
pts = np.array([[0.3, 0.6], [1.0, 1.0]])
true_arr = [3, 7]
assert all_equal(function(pts.T), true_arr)
out = np.empty(2, dtype='float64')
function(pts.T, out=out)
assert all_equal(out, true_arr)
# Input meshgrid, with and without output array
mg = sparse_meshgrid([0.3, 1.0], [0.4, 1.0])
# Indices: (1, 3) x (0, 1)
true_mg = [[2, 3], [6, 7]]
assert all_equal(function(mg), true_mg)
out = np.empty((2, 2), dtype='float64')
function(mg, out=out)
assert all_equal(out, true_mg)
assert repr(interp_op) != ''
def _test_getslice(slice):
# Validate get against python list behaviour
r6 = odl.rn(6)
y = [0, 1, 2, 3, 4, 5]
x = r6.element(y)
assert all_equal(x[slice].data, y[slice])
def test_primal_dual_with_li():
"""Test for the forward-backward solver with infimal convolution.
The test is done by minimizing the functional ``(g @ l)(x)``, where
``(g @ l)(x) = inf_y { g(y) + l(x - y) }``,
g is the indicator function on [-3, -1], and l(x) = 1/2||x||_2^2.
The optimal solution to this problem is given by x in [-3, -1].
"""
# Parameter values for the box constraint
upper_lim = -1
lower_lim = -3
space = odl.rn(1)
lin_ops = [odl.IdentityOperator(space)]
g = [odl.solvers.IndicatorBox(space, lower=lower_lim, upper=upper_lim)]
f = odl.solvers.ZeroFunctional(space)
l = [odl.solvers.L2NormSquared(space)]
# Centering around a point further away from [-3,-1].
x = space.element(10)
douglas_rachford_pd(x, f, g, lin_ops, tau=0.5, sigma=[1.0], niter=20, l=l)
assert lower_lim - 10**-LOW_ACCURACY <= float(x)
assert float(x) <= upper_lim + 10**-LOW_ACCURACY
def test_element_1d(exponent):
discr = odl.uniform_discr(0, 1, 3, impl='numpy', exponent=exponent)
weight = 1.0 if exponent == float('inf') else discr.cell_volume
dspace = odl.rn(3, exponent=exponent, weighting=weight)
elem = discr.element()
assert isinstance(elem, odl.DiscreteLpElement)
assert elem.ntuple in dspace
def test_operators(arithmetic_op):
# Test of the operators `+`, `-`, etc work as expected by numpy
space = odl.rn(3)
pspace = odl.ProductSpace(space, 2)
# Interactions with scalars
for scalar in [-31.2, -1, 0, 1, 2.13]:
# Left op
x_arr, x = noise_elements(pspace)
if scalar == 0 and arithmetic_op in [operator.truediv,
operator.itruediv]:
# Check for correct zero division behaviour
with pytest.raises(ZeroDivisionError):
y = arithmetic_op(x, scalar)
else:
y_arr = arithmetic_op(x_arr, scalar)
y = arithmetic_op(x, scalar)
assert all_almost_equal([x, y], [x_arr, y_arr])
# Right op
x_arr, x = noise_elements(pspace)
y_arr = arithmetic_op(scalar, x_arr)
y = arithmetic_op(scalar, x)
assert all_almost_equal([x, y], [x_arr, y_arr])
# Verify that the statement z=op(x, y) gives equivalent results to NumPy
x_arr, x = noise_elements(space, 1)
y_arr, y = noise_elements(pspace, 1)
# non-aliased left
if arithmetic_op in [operator.iadd,
operator.isub,
operator.itruediv,
operator.imul]:
# Check for correct error since in-place op is not possible here
with pytest.raises(TypeError):
z = arithmetic_op(x, y)
else:
z_arr = arithmetic_op(x_arr, y_arr)
z = arithmetic_op(x, y)
assert all_almost_equal([x, y, z], [x_arr, y_arr, z_arr])
# non-aliased right
z_arr = arithmetic_op(y_arr, x_arr)
z = arithmetic_op(y, x)
assert all_almost_equal([x, y, z], [x_arr, y_arr, z_arr])
# aliased operation
z_arr = arithmetic_op(y_arr, y_arr)
z = arithmetic_op(y, y)
assert all_almost_equal([x, y, z], [x_arr, y_arr, z_arr])
def test_collocation_interpolation_identity():
# Check if interpolation followed by collocation on the same grid
# is the identity
rect = odl.IntervalProd([0, 0], [1, 1])
part = odl.uniform_partition_fromintv(rect, [4, 2])
space = odl.FunctionSpace(rect)
dspace = odl.rn(part.size)
coll_op_c = PointCollocation(space, part, dspace, order='C')
coll_op_f = PointCollocation(space, part, dspace, order='F')
interp_ops_c = [
NearestInterpolation(space, part, dspace, variant='left', order='C'),
NearestInterpolation(space, part, dspace, variant='right', order='C'),
LinearInterpolation(space, part, dspace, order='C'),
PerAxisInterpolation(space, part, dspace, order='C',
schemes=['linear', 'nearest'])]
interp_ops_f = [
NearestInterpolation(space, part, dspace, variant='left', order='F'),
NearestInterpolation(space, part, dspace, variant='right', order='F'),
LinearInterpolation(space, part, dspace, order='F'),
PerAxisInterpolation(space, part, dspace, order='F',
schemes=['linear', 'nearest'])]
values = np.arange(1, 9, dtype='float64')
for interp_op_c in interp_ops_c:
ident_values = coll_op_c(interp_op_c(values))
assert all_almost_equal(ident_values, values)
for interp_op_f in interp_ops_f:
ident_values = coll_op_f(interp_op_f(values))
assert all_almost_equal(ident_values, values)
def test_nonlinear_functional():
r3 = odl.rn(3)
x = r3.element([1, 2, 3])
op = SumSquaredFunctional(r3)
assert op(x) == pytest.approx(np.sum(x ** 2))
def test_nearest_interpolation_2d_float():
rect = odl.IntervalProd([0, 0], [1, 1])
part = odl.uniform_partition_fromintv(rect, [4, 2], nodes_on_bdry=False)
# Coordinate vectors are:
# [0.125, 0.375, 0.625, 0.875], [0.25, 0.75]
space = odl.FunctionSpace(rect)
dspace = odl.rn(part.size)
interp_op = NearestInterpolation(space, part, dspace)
function = interp_op([0, 1, 2, 3, 4, 5, 6, 7])
# Evaluate at single point
val = function([0.3, 0.6]) # closest to index (1, 1) -> 3
assert val == 3.0
# Input array, with and without output array
pts = np.array([[0.3, 0.6],
[1.0, 1.0]])
true_arr = [3, 7]
assert all_equal(function(pts.T), true_arr)
out = np.empty(2, dtype='float64')
function(pts.T, out=out)
assert all_equal(out, true_arr)
# Input meshgrid, with and without output array
mg = sparse_meshgrid([0.3, 1.0], [0.4, 1.0])
# Indices: (1, 3) x (0, 1)
true_mg = [[2, 3],
[6, 7]]
assert all_equal(function(mg), true_mg)
out = np.empty((2, 2), dtype='float64')
function(mg, out=out)
assert all_equal(out, true_mg)
def test_primal_dual_no_operator():
"""Verify that the correct value is returned when there is no operator.
Solves the optimization problem
min_x ||x - data_1||_1
which has optimum value data_1.
"""
# Define the space
space = odl.rn(5)
# Operator
L = []
# Data
data_1 = odl.util.testutils.noise_element(space)
# Proximals
f = odl.solvers.L1Norm(space).translated(data_1)
g = []
# Solve with f term dominating
x = space.zero()
douglas_rachford_pd(x, f, g, L, tau=3.0, sigma=[], niter=10)
assert all_almost_equal(x, data_1, ndigits=2)