def test_resizing_op_mixed_uni_nonuni():
"""Check if resizing along uniform axes in mixed discretizations works."""
nonuni_part = odl.nonuniform_partition([0, 1, 4])
uni_part = odl.uniform_partition(-1, 1, 4)
part = uni_part.append(nonuni_part, uni_part, nonuni_part)
fspace = odl.FunctionSpace(odl.IntervalProd(part.min_pt, part.max_pt))
tspace = odl.rn(part.shape)
space = odl.DiscreteLp(fspace, part, tspace)
# Keep non-uniform axes fixed
res_op = odl.ResizingOperator(space, ran_shp=(6, 3, 6, 3))
assert res_op.axes == (0, 2)
assert res_op.offset == (1, 0, 1, 0)
# Evaluation test with a simpler case
part = uni_part.append(nonuni_part)
fspace = odl.FunctionSpace(odl.IntervalProd(part.min_pt, part.max_pt))
tspace = odl.rn(part.shape)
space = odl.DiscreteLp(fspace, part, tspace)
res_op = odl.ResizingOperator(space, ran_shp=(6, 3))
result = res_op(space.one())
true_result = [[0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1],
[0, 0, 0]]
assert np.array_equal(result, true_result)
# Test adjoint
elem = noise_element(space)
res_elem = noise_element(res_op.range)
inner1 = res_op(elem).inner(res_elem)
inner2 = elem.inner(res_op.adjoint(res_elem))
assert almost_equal(inner1, inner2)
def test_dspace_type_numpy():
# Plain function set -> Ntuples-like
fset = odl.FunctionSet(odl.Interval(0, 1), odl.Strings(2))
assert dspace_type(fset, 'numpy') == odl.Ntuples, None
assert dspace_type(fset, 'numpy', np.int) == odl.Ntuples
# Real space
rspc = odl.FunctionSpace(odl.Interval(0, 1), field=odl.RealNumbers())
assert dspace_type(rspc, 'numpy') == odl.Fn
assert dspace_type(rspc, 'numpy', np.float32) == odl.Fn
assert dspace_type(rspc, 'numpy', np.int) == odl.Fn
with pytest.raises(TypeError):
dspace_type(rspc, 'numpy', np.complex)
with pytest.raises(TypeError):
dspace_type(rspc, 'numpy', np.dtype('<U2'))
# Complex space
cspc = odl.FunctionSpace(odl.Interval(0, 1), field=odl.ComplexNumbers())
assert dspace_type(cspc, 'numpy') == odl.Fn
assert dspace_type(cspc, 'numpy', np.complex64) == odl.Fn
with pytest.raises(TypeError):
dspace_type(cspc, 'numpy', np.float)
with pytest.raises(TypeError):
assert dspace_type(cspc, 'numpy', np.int)
with pytest.raises(TypeError):
dspace_type(cspc, 'numpy', np.dtype('<U2'))
def test_dspace_type_cuda():
# Plain function set -> Ntuples-like
fset = odl.FunctionSet(odl.Interval(0, 1), odl.Strings(2))
assert dspace_type(fset, 'cuda') == odl.CudaNtuples
assert dspace_type(fset, 'cuda', np.int) == odl.CudaNtuples
# Real space
rspc = odl.FunctionSpace(odl.Interval(0, 1), field=odl.RealNumbers())
assert dspace_type(rspc, 'cuda') == odl.CudaFn
assert dspace_type(rspc, 'cuda', np.float64) == odl.CudaFn
assert dspace_type(rspc, 'cuda', np.int) == odl.CudaFn
with pytest.raises(TypeError):
dspace_type(rspc, 'cuda', np.complex)
with pytest.raises(TypeError):
dspace_type(rspc, 'cuda', np.dtype('<U2'))
# Complex space (not implemented)
cspc = odl.FunctionSpace(odl.Interval(0, 1), field=odl.ComplexNumbers())
with pytest.raises(NotImplementedError):
dspace_type(cspc, 'cuda')
with pytest.raises(NotImplementedError):
dspace_type(cspc, 'cuda', np.complex64)
with pytest.raises(TypeError):
dspace_type(cspc, 'cuda', np.float)
with pytest.raises(TypeError):
assert dspace_type(cspc, 'cuda', np.int)
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_discretelp_init():
"""Test initialization and basic properties of DiscreteLp."""
# Real space
fspace = odl.FunctionSpace(odl.IntervalProd([0, 0], [1, 1]))
part = odl.uniform_partition_fromintv(fspace.domain, (2, 4))
tspace = odl.rn(part.shape)
discr = DiscreteLp(fspace, part, tspace)
assert discr.fspace == fspace
assert discr.tspace == tspace
assert discr.partition == part
assert discr.interp == 'nearest'
assert discr.interp_byaxis == ('nearest', 'nearest')
assert discr.exponent == tspace.exponent
assert discr.axis_labels == ('$x$', '$y$')
assert discr.is_real
discr = DiscreteLp(fspace, part, tspace, interp='linear')
assert discr.interp == 'linear'
assert discr.interp_byaxis == ('linear', 'linear')
discr = DiscreteLp(fspace, part, tspace, interp=['nearest', 'linear'])
assert discr.interp == ('nearest', 'linear')
assert discr.interp_byaxis == ('nearest', 'linear')
# Complex space
fspace_c = odl.FunctionSpace(odl.IntervalProd([0, 0], [1, 1]),
out_dtype=complex)
tspace_c = odl.cn(part.shape)
discr = DiscreteLp(fspace_c, part, tspace_c)
assert discr.is_complex
# Make sure repr shows something
assert repr(discr)
# Error scenarios
with pytest.raises(ValueError):
DiscreteLp(fspace, part, tspace_c) # mixes real & complex
with pytest.raises(ValueError):
DiscreteLp(fspace_c, part, tspace) # mixes complex & real
part_1d = odl.uniform_partition(0, 1, 2)
with pytest.raises(ValueError):
DiscreteLp(fspace, part_1d, tspace) # wrong dimensionality
part_diffshp = odl.uniform_partition_fromintv(fspace.domain, (3, 4))
with pytest.raises(ValueError):
DiscreteLp(fspace, part_diffshp, tspace) # shape mismatch
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_rectangle():
# Continuous definition of problem
space = odl.FunctionSpace(odl.Rectangle([0, 0], [1, 1]))
# Complicated functions to check performance
n = 5
m = 7
# Discretization
d = odl.uniform_discr(space, (n, m), impl='cuda')
fun = d.element([[0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
diff = ForwardDiff2D(d)
derivative = diff(fun)
assert all_almost_equal(
derivative[0].asarray(),
[[0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 1, -1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0]])
assert all_almost_equal(
derivative[1].asarray(),
[[0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0]])
# Verify that the adjoint is ok
# -gradient.T(gradient(x)) is the laplacian
laplacian = -diff.adjoint(derivative)
assert all_almost_equal(
laplacian.asarray(),
[[0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0], [0, 1, -4, 1, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0]])
def test_nearest_interpolation_1d_variants():
"""Test nearest neighbor interpolation variants in 1d."""
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]
fspace = odl.FunctionSpace(intv)
tspace = odl.rn(part.shape)
# 'left' variant
interp_op = NearestInterpolation(fspace, part, tspace, variant='left')
assert repr(interp_op) != ''
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(fspace, part, tspace, variant='right')
assert repr(interp_op) != ''
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_nearest_interpolation_2d_string():
"""Test nearest neighbor interpolation in 2d with string values."""
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, out_dtype='U1')
tspace = odl.tensor_space(part.shape, dtype='U1')
interp_op = NearestInterpolation(fspace, part, tspace)
values = np.array([c for c in 'mystring']).reshape(tspace.shape)
function = interp_op(values)
# Evaluate at single point
val = function([0.3, 0.6]) # closest to index (1, 1) -> 3
assert val == 't'
# Input array, with and without output array
pts = np.array([[0.3, 0.6], [1.0, 1.0]])
true_arr = ['t', 'g']
assert all_equal(function(pts.T), true_arr)
out = np.empty(2, dtype='U1')
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 = [['s', 't'], ['n', 'g']]
assert all_equal(function(mg), true_mg)
out = np.empty((2, 2), dtype='U1')
function(mg, out=out)
assert all_equal(out, true_mg)
assert repr(interp_op) != ''
def test_bwt1d(wbasis):
# Verify that the operator works as axpected
# 1D test
n = 16
x = np.zeros(n)
x[5:10] = 1
nscales = 2
# Define a discretized domain
domain = odl.FunctionSpace(odl.Interval([-1], [1]))
nPoints = np.array([n])
disc_domain = odl.uniform_discr_fromspace(domain, nPoints)
disc_phantom = disc_domain.element(x)
# Create the discrete wavelet transform operator.
# Only the domain of the operator needs to be defined
Wop = BiorthWaveletTransform(disc_domain, nscales, wbasis)
# Compute the discrete wavelet transform of discrete imput image
coeffs = Wop(disc_phantom)
# Compute the inverse wavelet transform
reconstruction = Wop.inverse(coeffs)
# Verify that reconstructions lie in correct discretized domain
assert reconstruction in disc_domain
assert all_almost_equal(reconstruction.asarray(), x)
def test_bwt2d():
# 2D test
n = 16
x = np.zeros((n, n))
x[5:10, 5:10] = 1
wbasis = 'josbiorth5'
nscales = 3
# Define a discretized domain
domain = odl.FunctionSpace(odl.Rectangle([-1, -1], [1, 1]))
nPoints = np.array([n, n])
disc_domain = odl.uniform_discr_fromspace(domain, nPoints)
disc_phantom = disc_domain.element(x)
# Create the discrete wavelet transform operator.
# Only the domain of the operator needs to be defined
Bop = BiorthWaveletTransform(disc_domain, nscales, wbasis)
Bop2 = InverseAdjBiorthWaveletTransform(disc_domain, nscales, wbasis)
# Compute the discrete wavelet transform of discrete imput image
coeffs = Bop(disc_phantom)
coeffs2 = Bop2(disc_phantom)
reconstruction = Bop.inverse(coeffs)
reconstruction2 = Bop2.inverse(coeffs2)
assert all_almost_equal(reconstruction.asarray(), x)
assert all_almost_equal(reconstruction2.asarray(), x)
def test_nearest_interpolation_1d_variants():
intv = odl.Interval(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_nearest_interpolation_1d_complex(odl_tspace_impl):
"""Test nearest neighbor interpolation in 1d with complex values."""
impl = odl_tspace_impl # TODO: not used!
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]
fspace = odl.FunctionSpace(intv, out_dtype=complex)
tspace = odl.cn(part.shape)
interp_op = NearestInterpolation(fspace, part, tspace)
function = interp_op([0 + 1j, 1 + 2j, 2 + 3j, 3 + 4j, 4 + 5j])
# Evaluate at single point
val = function(0.35) # closest to index 1 -> 1 + 2j
assert val == 1.0 + 2.0j
# Input array, with and without output array
pts = np.array([0.4, 0.0, 0.65, 0.95])
true_arr = [1 + 2j, 0 + 1j, 3 + 4j, 4 + 5j]
assert all_equal(function(pts), true_arr)
# Should also work with a (1, N) array
pts = pts[None, :]
assert all_equal(function(pts), true_arr)
out = np.empty(4, dtype='complex128')
function(pts, out=out)
assert all_equal(out, true_arr)
# Input meshgrid, with and without output array
# Same as array for 1d
mg = sparse_meshgrid([0.4, 0.0, 0.65, 0.95])
true_mg = [1 + 2j, 0 + 1j, 3 + 4j, 4 + 5j]
assert all_equal(function(mg), true_mg)
function(mg, out=out)
assert all_equal(out, true_mg)
assert repr(interp_op) != ''
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_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_nearest_interpolation_1d_complex(fn_impl):
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, field=odl.ComplexNumbers())
dspace = odl.cn(part.size)
interp_op = NearestInterpolation(space, part, dspace)
function = interp_op([0 + 1j, 1 + 2j, 2 + 3j, 3 + 4j, 4 + 5j])
# Evaluate at single point
val = function(0.35) # closest to index 1 -> 1 + 2j
assert val == 1.0 + 2.0j
# Input array, with and without output array
pts = np.array([0.4, 0.0, 0.65, 0.95])
true_arr = [1 + 2j, 0 + 1j, 3 + 4j, 4 + 5j]
assert all_equal(function(pts), true_arr)
# Should also work with a (1, N) array
pts = pts[None, :]
assert all_equal(function(pts), true_arr)
out = np.empty(4, dtype='complex128')
function(pts, out=out)
assert all_equal(out, true_arr)
# Input meshgrid, with and without output array
# Same as array for 1d
mg = sparse_meshgrid([0.4, 0.0, 0.65, 0.95])
true_mg = [1 + 2j, 0 + 1j, 3 + 4j, 4 + 5j]
assert all_equal(function(mg), true_mg)
function(mg, out=out)
assert all_equal(out, true_mg)
def test_per_axis_interpolation():
"""Test different interpolation schemes per axis."""
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)
schemes = ['linear', 'nearest']
variants = [None, 'right']
interp_op = PerAxisInterpolation(fspace,
part,
tspace,
schemes=schemes,
nn_variants=variants)
values = np.arange(1, 9, dtype='float64').reshape(part.shape)
function = interp_op(values)
rvals = values.reshape([4, 2])
# Evaluate at single point
val = function([0.3, 0.5])
l1 = (0.3 - 0.125) / (0.375 - 0.125)
# 0.5 equally far from both neighbors -> 'right' chooses 0.75
true_val = (1 - l1) * rvals[0, 1] + l1 * rvals[1, 1]
assert val == pytest.approx(true_val)
# Input array, with and without output array
pts = np.array([[0.3, 0.6], [0.1, 0.25], [1.0, 1.0]])
l1 = (0.3 - 0.125) / (0.375 - 0.125)
true_val_1 = (1 - l1) * rvals[0, 1] + l1 * rvals[1, 1]
l1 = (0.125 - 0.1) / (0.375 - 0.125)
true_val_2 = (1 - l1) * rvals[0, 0] # only lower left contributes
l1 = (1.0 - 0.875) / (0.875 - 0.625)
true_val_3 = (1 - l1) * rvals[3, 1] # lower left only
true_arr = [true_val_1, true_val_2, true_val_3]
assert all_equal(function(pts.T), true_arr)
out = np.empty(3, 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, 0.85])
# Indices: (1, 3) x (0, 1)
lx1 = (0.3 - 0.125) / (0.375 - 0.125)
lx2 = (1.0 - 0.875) / (0.875 - 0.625)
true_val_11 = (1 - lx1) * rvals[0, 0] + lx1 * rvals[1, 0]
true_val_12 = ((1 - lx1) * rvals[0, 1] + lx1 * rvals[1, 1])
true_val_21 = (1 - lx2) * rvals[3, 0]
true_val_22 = (1 - lx2) * rvals[3, 1]
true_mg = [[true_val_11, true_val_12], [true_val_21, true_val_22]]
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_collocation_cuda():
rect = odl.Rectangle([0, 0], [1, 1])
part = odl.uniform_partition_fromintv(rect, [4, 2])
space = odl.FunctionSpace(rect)
dspace = odl.CudaRn(part.size)
coll_op = PointCollocation(space, part, dspace)
interp_op = LinearInterpolation(space, part, dspace)
values = np.arange(1, 9, dtype='float64')
ident_values = coll_op(interp_op(values))
assert all_almost_equal(ident_values, values)
def test_norm_interval(exponent):
# Test the function f(x) = x^2 on the interval (0, 1). Its
# L^p-norm is (1 + 2*p)^(-1/p) for finite p and 1 for p=inf
p = exponent
fspace = odl.FunctionSpace(odl.IntervalProd(0, 1))
lpdiscr = odl.uniform_discr_fromspace(fspace, 10, exponent=p)
testfunc = fspace.element(lambda x: x**2)
discr_testfunc = lpdiscr.element(testfunc)
if p == float('inf'):
assert discr_testfunc.norm() <= 1 # Max at boundary not hit
else:
true_norm = (1 + 2 * p)**(-1 / p)
assert discr_testfunc.norm() == pytest.approx(true_norm, rel=1e-2)
def test_norm_nonuniform():
"""Check if norms are correct in non-uniform discretizations."""
fspace = odl.FunctionSpace(odl.IntervalProd(0, 5))
part = odl.nonuniform_partition([0, 2, 3, 5], min_pt=0, max_pt=5)
weights = part.cell_sizes_vecs[0]
tspace = odl.rn(part.size, weighting=weights)
discr = odl.DiscreteLp(fspace, part, tspace)
sqrt = discr.element(lambda x: np.sqrt(x))
# Exact norm is the square root of the integral from 0 to 5 of x,
# which is sqrt(5**2 / 2)
exact_norm = np.sqrt(5**2 / 2.0)
norm = sqrt.norm()
assert norm == pytest.approx(exact_norm)
def test_inner_nonuniform():
"""Check if inner products are correct in non-uniform discretizations."""
fspace = odl.FunctionSpace(odl.IntervalProd(0, 5))
part = odl.nonuniform_partition([0, 2, 3, 5], min_pt=0, max_pt=5)
weights = part.cell_sizes_vecs[0]
tspace = odl.rn(part.size, weighting=weights)
discr = odl.DiscreteLp(fspace, part, tspace)
one = discr.one()
linear = discr.element(lambda x: x)
# Exact inner product is the integral from 0 to 5 of x, which is 5**2 / 2
exact_inner = 5**2 / 2.0
inner = one.inner(linear)
assert inner == pytest.approx(exact_inner)
def test_fwd_diff():
# Continuous definition of problem
space = odl.FunctionSpace(odl.Interval(0, 1))
# Discretization
n = 6
d = odl.uniform_discr(space, n, impl='cuda')
fun = d.element([1, 2, 5, 3, 2, 1])
# Create operator
diff = ForwardDiff(d)
assert all_almost_equal(diff(fun), [0, 3, -2, -1, -1, 0])
assert all_almost_equal(diff.adjoint(fun), [0, -1, -3, 2, 1, 0])
assert all_almost_equal(diff.adjoint(diff(fun)), [0, -3, 5, -1, 0, 0])
def ExpOp_builder(bin_param, filter_space, interp):
# Create binning scheme
if interp == 'Full':
spf_space = filter_space
Exp_op = odl.IdentityOperator(filter_space)
elif interp == 'uniform':
# Create binning scheme
dpix = np.size(filter_space)
dsize = filter_space.max_pt
filt_bin_space = odl.uniform_discr(-dsize, dsize, dpix // (bin_param))
spf_space = odl.uniform_discr(0, dsize, dpix // (2 * bin_param))
resamp = odl.Resampling(filt_bin_space, filter_space)
sym = SymOp(spf_space, filt_bin_space)
Exp_op = resamp * sym
else:
if interp == 'constant':
interp = 'nearest'
elif interp == 'linear':
pass
else:
raise ValueError('unknown `expansion operator type` ({})'
''.format(interp))
B = ExpBin(bin_param, np.size(filter_space)) * \
filter_space.weighting.const
B[-1] -= 1 / 2 * filter_space.weighting.const
# Create sparse filter space
spf_part = odl.nonuniform_partition(B, min_pt=0, max_pt=B[-1])
spf_weight = np.ravel(
np.multiply.reduce(np.meshgrid(*spf_part.cell_sizes_vecs)))
spf_fspace = odl.FunctionSpace(spf_part.set)
spf_space = odl.DiscreteLp(spf_fspace,
spf_part,
odl.rn(spf_part.size, weighting=spf_weight),
interp=interp)
filt_pos_part = odl.uniform_partition(0, B[-1],
int(np.size(filter_space) / 2))
filt_pos_space = odl.uniform_discr_frompartition(filt_pos_part,
dtype='float64')
lin_interp = odl.Resampling(spf_space, filt_pos_space)
# Create symmetry operator
sym = SymOp(filt_pos_space, filter_space)
# Create sparse filter operator
Exp_op = sym * lin_interp
return spf_space, Exp_op
def test_norm_rectangle(exponent):
# Test the function f(x) = x_0^2 * x_1^3 on (0, 1) x (-1, 1). Its
# L^p-norm is ((1 + 2*p) * (1 + 3 * p) / 2)^(-1/p) for finite p
# and 1 for p=inf
p = exponent
fspace = odl.FunctionSpace(odl.IntervalProd([0, -1], [1, 1]))
lpdiscr = odl.uniform_discr_fromspace(fspace, (20, 30), exponent=p)
testfunc = fspace.element(lambda x: x[0]**2 * x[1]**3)
discr_testfunc = lpdiscr.element(testfunc)
if p == float('inf'):
assert discr_testfunc.norm() <= 1 # Max at boundary not hit
else:
true_norm = ((1 + 2 * p) * (1 + 3 * p) / 2)**(-1 / p)
assert discr_testfunc.norm() == pytest.approx(true_norm, rel=1e-2)
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), out_dtype=complex)
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_linear_interpolation_1d():
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)
interp_op = LinearInterpolation(space, part, dspace)
function = interp_op([1, 2, 3, 4, 5])
# Evaluate at single point
val = function(0.35)
true_val = 0.75 * 2 + 0.25 * 3
assert almost_equal(val, true_val)
# Input array, with and without output array
pts = np.array([0.4, 0.0, 0.65, 0.95])
true_arr = [2.5, 0.5, 3.75, 3.75]
assert all_almost_equal(function(pts), true_arr)
def projector(index):
axis = np.roll(np.array([1, 0, 0]), index)
axis2 = np.roll(axis, 1)
geom = odl.tomo.Parallel3dGeometry(angle_intvl, dparams, angle_grid,
det_grid, axis=axis,
origin_to_det=axis2)
# Projection space
proj_space = odl.FunctionSpace(geom.params)
discr_data = odl.util.phantom.indicate_proj_axis(discr_vol_space, 0.5)
# `DiscreteLp` projection space
proj_shape = geom.grid.shape
discr_proj_space = odl.uniform_discr_fromspace(proj_space, proj_shape,
dtype='float32')
# Forward
proj_data = odl.tomo.astra_cuda_forward_projector(
discr_data, geom, discr_proj_space)
return proj_data
def performace_example():
# Create a space of functions on the interval [0, 1].
fspace = odl.FunctionSpace(odl.IntervalProd(0, 1))
# Simple function, already supports vectorization.
f_vec = fspace.element(lambda x: x ** 2)
# If 'vectorized=False' is used, odl automatically vectorizes with
# the help of numpy.vectorize. This will be very slow, though, since
# the implementation is basically a Python loop.
f_novec = fspace.element(lambda x: x ** 2, vectorized=False)
# We test both versions with 10000 evaluation points. The natively
# vectorized version should be much faster than the one using
# numpy.vectorize.
points = np.linspace(0, 1, 10000)
print('Vectorized runtime: {:5f}'
''.format(timeit.timeit(lambda: f_vec(points), number=100)))
print('Non-vectorized runtime: {:5f}'
''.format(timeit.timeit(lambda: f_novec(points), number=100)))
def test_resizing_op_raise():
# domain not a uniformely discretized Lp
with pytest.raises(TypeError):
odl.ResizingOperator(odl.rn(5), ran_shp=(10, ))
grid = odl.RectGrid([0, 2, 3])
part = odl.RectPartition(odl.IntervalProd(0, 3), grid)
fspace = odl.FunctionSpace(odl.IntervalProd(0, 3))
dspace = odl.rn(3)
space = odl.DiscreteLp(fspace, part, dspace)
with pytest.raises(ValueError):
odl.ResizingOperator(space, ran_shp=(10, ))
# different cell sides in domain and range
space = odl.uniform_discr(0, 1, 10)
res_space = odl.uniform_discr(0, 1, 15)
with pytest.raises(ValueError):
odl.ResizingOperator(space, res_space)
# non-integer multiple of cell sides used as shift (grid of the
# resized space shifted)
space = odl.uniform_discr(0, 1, 5)
res_space = odl.uniform_discr(-0.5, 1.5, 10)
with pytest.raises(ValueError):
odl.ResizingOperator(space, res_space)
# need either range or ran_shp
with pytest.raises(ValueError):
odl.ResizingOperator(space)
# offset cannot be combined with range
space = odl.uniform_discr([0, -1], [1, 1], (10, 5))
res_space = odl.uniform_discr([0, -3], [2, 3], (20, 15))
with pytest.raises(ValueError):
odl.ResizingOperator(space, res_space, offset=(0, 0))
# bad pad_mode
with pytest.raises(ValueError):
odl.ResizingOperator(space, res_space, pad_mode='something')
def test_yzproj():
# `DiscreteLp` volume space
vol_shape = (100,) * 3
discr_vol_space = odl.uniform_discr([-50] * 3, [50] * 3, vol_shape,
dtype='float32')
# Angles: 0 and pi/2
angle_intvl = odl.Interval(0, np.pi / 2)
angle_grid = odl.uniform_sampling(angle_intvl, 2, as_midp=False)
# agrid = angle_grid.points() / np.pi
# Detector
dparams = odl.Rectangle([-50] * 2, [50] * 2)
det_grid = odl.uniform_sampling(dparams, (100, 100))
axis = (0, 1, 0)
origin_to_det = (0, 0, 1)
geom = odl.tomo.Parallel3dGeometry(angle_intvl, dparams,
angle_grid,
det_grid, axis=axis,
origin_to_det=origin_to_det)
# Projection space
proj_space = odl.FunctionSpace(geom.params)
discr_data = odl.util.phantom.indicate_proj_axis(discr_vol_space, 0.5)
# `DiscreteLp` projection space
proj_shape = geom.grid.shape
discr_proj_space = odl.uniform_discr_fromspace(proj_space,
proj_shape,
dtype='float32')
# Forward
proj_data = odl.tomo.astra_cuda_forward_projector(discr_data,
geom,
discr_proj_space)
plt.switch_backend('qt4agg')
proj_data.show(indices=np.s_[0, :, :])
plt.show()