def test_linear_interpolation_2d():
"""Test linear interpolation in 2d."""
coord_vecs = [[0.125, 0.375, 0.625, 0.875], [0.25, 0.75]]
f = np.array([[1, 2], [3, 4], [5, 6], [7, 8]], dtype='float64')
interpolator = linear_interpolator(f, coord_vecs)
# Evaluate at single point
val = interpolator([0.3, 0.6])
l1 = (0.3 - 0.125) / (0.375 - 0.125)
l2 = (0.6 - 0.25) / (0.75 - 0.25)
true_val = ((1 - l1) * (1 - l2) * f[0, 0] + (1 - l1) * l2 * f[0, 1] + l1 *
(1 - l2) * f[1, 0] + l1 * l2 * f[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)
l2 = (0.6 - 0.25) / (0.75 - 0.25)
true_val_1 = ((1 - l1) * (1 - l2) * f[0, 0] + (1 - l1) * l2 * f[0, 1] +
l1 * (1 - l2) * f[1, 0] + l1 * l2 * f[1, 1])
l1 = (0.125 - 0.1) / (0.375 - 0.125)
# l2 = 0
true_val_2 = (1 - l1) * f[0, 0] # only lower left contributes
l1 = (1.0 - 0.875) / (0.875 - 0.625)
l2 = (1.0 - 0.75) / (0.75 - 0.25)
true_val_3 = (1 - l1) * (1 - l2) * f[3, 1] # lower left only
true_arr = [true_val_1, true_val_2, true_val_3]
assert all_equal(interpolator(pts.T), true_arr)
out = np.empty(3, dtype='float64')
interpolator(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, 0.75])
# Indices: (1, 3) x (0, 1)
lx1 = (0.3 - 0.125) / (0.375 - 0.125)
lx2 = (1.0 - 0.875) / (0.875 - 0.625)
ly1 = (0.4 - 0.25) / (0.75 - 0.25)
# ly2 = 0
true_val_11 = ((1 - lx1) * (1 - ly1) * f[0, 0] +
(1 - lx1) * ly1 * f[0, 1] + lx1 * (1 - ly1) * f[1, 0] +
lx1 * ly1 * f[1, 1])
true_val_12 = ((1 - lx1) * f[0, 1] + lx1 * f[1, 1] # ly2 = 0
)
true_val_21 = ((1 - lx2) * (1 - ly1) * f[3, 0] +
(1 - lx2) * ly1 * f[3, 1] # high node 1.0, no upper
)
true_val_22 = (1 - lx2) * f[3, 1] # ly2 = 0, no upper for 1.0
true_mg = [[true_val_11, true_val_12], [true_val_21, true_val_22]]
assert all_equal(interpolator(mg), true_mg)
out = np.empty((2, 2), dtype='float64')
interpolator(mg, out=out)
assert all_equal(out, true_mg)
def test_linear_interpolation_1d():
"""Test linear interpolation in 1d."""
coord_vecs = [[0.1, 0.3, 0.5, 0.7, 0.9]]
f = np.array([1, 2, 3, 4, 5], dtype="float64")
interpolator = linear_interpolator(f, coord_vecs)
# Evaluate at single point
val = interpolator(0.35)
true_val = 0.75 * 2 + 0.25 * 3
assert val == pytest.approx(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(interpolator(pts), true_arr)
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
coord_vecs = [[0.125, 0.375, 0.625, 0.875], [0.25, 0.75]]
f = np.array([[1, 2], [3, 4], [5, 6], [7, 8]], dtype='float64')
interpolators = [
nearest_interpolator(f, coord_vecs),
linear_interpolator(f, coord_vecs),
per_axis_interpolator(f, coord_vecs, interp=['linear', 'nearest']),
]
for interpolator in interpolators:
mg = sparse_meshgrid(*coord_vecs)
ident_f = point_collocation(interpolator, mg)
assert all_almost_equal(ident_f, f)
def interpolator(x, out=None):
x = (x[0], np.clip(x[1], min_x, max_x))
interpolator = linear_interpolator(
sinogram, skimage_range.grid.coord_vectors
)
return interpolator(x, out=out)
def load_projections(folder, indices=None):
"""Load geometry and data stored in Mayo format from folder.
Parameters
----------
folder : str
Path to the folder where the Mayo DICOM files are stored.
indices : optional
Indices of the projections to load.
Accepts advanced indexing such as slice or list of indices.
Returns
-------
geometry : ConeBeamGeometry
Geometry corresponding to the Mayo projector.
proj_data : `numpy.ndarray`
Projection data, given as the line integral of the linear attenuation
coefficient (g/cm^3). Its unit is thus g/cm^2.
"""
datasets, data_array = _read_projections(folder, indices)
# Get the angles
angles = [d.DetectorFocalCenterAngularPosition for d in datasets]
angles = -np.unwrap(angles) - np.pi # different definition of angles
# Set minimum and maximum corners
shape = np.array([
datasets[0].NumberofDetectorColumns, datasets[0].NumberofDetectorRows
])
pixel_size = np.array([
datasets[0].DetectorElementTransverseSpacing,
datasets[0].DetectorElementAxialSpacing
])
# Correct from center of pixel to corner of pixel
minp = -(np.array(datasets[0].DetectorCentralElement) - 0.5) * pixel_size
maxp = minp + shape * pixel_size
# Select geometry parameters
src_radius = datasets[0].DetectorFocalCenterRadialDistance
det_radius = (datasets[0].ConstantRadialDistance -
datasets[0].DetectorFocalCenterRadialDistance)
# For unknown reasons, mayo does not include the tag
# "TableFeedPerRotation", which is what we want.
# Instead we manually compute the pitch
pitch = ((datasets[-1].DetectorFocalCenterAxialPosition -
datasets[0].DetectorFocalCenterAxialPosition) /
((np.max(angles) - np.min(angles)) / (2 * np.pi)))
# Get flying focal spot data
offset_axial = np.array([d.SourceAxialPositionShift for d in datasets])
offset_angular = np.array([d.SourceAngularPositionShift for d in datasets])
offset_radial = np.array([d.SourceRadialDistanceShift for d in datasets])
# TODO(adler-j): Implement proper handling of flying focal spot.
# Currently we do not fully account for it, merely making some "first
# order corrections" to the detector position and radial offset.
# Update angles with flying focal spot (in plane direction).
# This increases the resolution of the reconstructions.
angles = angles - offset_angular
# We correct for the mean offset due to the rotated angles, we need to
# shift the detector.
offset_detector_by_angles = det_radius * np.mean(offset_angular)
minp[0] -= offset_detector_by_angles
maxp[0] -= offset_detector_by_angles
# We currently apply only the mean of the offsets
src_radius = src_radius + np.mean(offset_radial)
# Partially compensate for a movement of the source by moving the object
# instead. We need to rescale by the magnification to get the correct
# change in the detector. This approximation is only exactly valid on the
# axis of rotation.
mean_offset_along_axis_for_ffz = np.mean(offset_axial) * (
src_radius / (src_radius + det_radius))
# Create partition for detector
detector_partition = odl.uniform_partition(minp, maxp, shape)
# Convert offset to odl definitions
offset_along_axis = (mean_offset_along_axis_for_ffz +
datasets[0].DetectorFocalCenterAxialPosition -
angles[0] / (2 * np.pi) * pitch)
# Assemble geometry
angle_partition = odl.nonuniform_partition(angles)
geometry = odl.tomo.ConeBeamGeometry(angle_partition,
detector_partition,
src_radius=src_radius,
det_radius=det_radius,
pitch=pitch,
offset_along_axis=offset_along_axis)
# Create a *temporary* ray transform (we need its range)
spc = odl.uniform_discr([-1] * 3, [1] * 3, [32] * 3)
ray_trafo = odl.tomo.RayTransform(spc, geometry, interp='linear')
# convert coordinates
theta, up, vp = ray_trafo.range.grid.meshgrid
d = src_radius + det_radius
u = d * np.arctan(up / d)
v = d / np.sqrt(d**2 + up**2) * vp
# Calculate projection data in rectangular coordinates since we have no
# backend that supports cylindrical
interpolator = linear_interpolator(data_array,
ray_trafo.range.coord_vectors)
proj_data = interpolator((theta, u, v))
return geometry, proj_data.asarray()
def project_data(data, old_detector, new_detector):
"""Transforms data one detector to another using linear interpolation.
Parameters
----------
data : `numpy.ndarray`
Data sampled on a partition of the detector.
old_detector : Detector
The detector on which the data was sampled.
new_detector : Detector
The detector to which the data is projected.
Returns
-------
resampled_data : `numpy.ndarray`
Resampled data, which corresponds to a new detector
Examples
--------
Transforming a flat detector to a circular. In this example
a flat detector has a range [-1, 1], with uniform discretization
[-1, -0.5, 0, 0.5, 1]. The circular detector has a range [-pi/4, pi/4]
with uniform discretization [-pi/4, -pi/8, 0, pi/8, pi/4], which
corresponds to [-1, -0.41, 0, 0.41, 1] on the flat detector,
since tg(pi/8) approx. 0.41. Thus, values at points -pi/8 and pi/8 obtained
through interpolation are -0.83 and 0.83 (approx. 0.41/0.5).
>>> part = odl.uniform_partition(-1, 1, 5, nodes_on_bdry=True)
>>> det = odl.tomo.Flat1dDetector(part, axis=[1, 0])
>>> det.partition.meshgrid[0]
array([-1. , -0.5, 0. , 0.5, 1. ])
>>> data = np.arange(-2, 3)
>>> new_det = flat_to_curved(det, radius=1)
>>> new_data = project_data(data, det, new_det)
>>> np.round(new_data, 2)
array([-2. , -0.83, 0. , 0.83, 2. ])
Transforming a circular detector to a flat. In this example
a circular detector has a range [-pi/4, pi/4] with uniform discretization
[-0.79, -0.39, 0, 0.39, 0.79]. The corresponding flat detector has
uniform discretization [-1, -0.5, 0, 0.5, 1], which corresponds to points
[-0.79, -0.46, 0, 0.46, 0.79] on the circular detector,
since arctg(0.5) = 0.46. Thus, values at points -0.5 and 0.5 obtained
through interpolation are -1.18 and 1.18
(approx. (0.79-0.46)/0.39*1 + (0.46-0.39)/0.39*2).
>>> part = odl.uniform_partition(-np.pi / 4, np.pi / 4, 5,
... nodes_on_bdry=True)
>>> det = odl.tomo.CircularDetector(part, axis=[1, 0], radius=1)
>>> data = np.arange(-2, 3)
>>> new_det = curved_to_flat(det)
>>> new_data = project_data(data, det, new_det)
>>> np.round(new_data, 2)
array([-2. , -1.18, 0. , 1.18, 2. ])
Previous example extended to 2D cylindrical detector with height 2 and
uniform partition along height axis [-1, -0.5, 0, 0.5, 1].
The height partition of corresponding 2D flat detector is
[-1.41, -0.71, 0., 0.71, 1.41].
We can see that points that are closer to side edges of the cylinder are
are projected higher on the flat detector.
>>> part = odl.uniform_partition([-np.pi / 4, -1], [np.pi / 4, 1], (5, 5),
... nodes_on_bdry=True)
>>> det = odl.tomo.CylindricalDetector(part,
... axes=[[1, 0, 0], [0, 0, 1]],
... radius=1)
>>> data = np.zeros((5,5))
>>> data[:, 1:4] = 1
>>> new_det = curved_to_flat(det)
>>> np.round(new_det.partition.meshgrid[1], 2)
array([[-1.41, -0.71, 0. , 0.71, 1.41]])
>>> new_data = project_data(data, det, new_det)
>>> np.round(new_data.T, 2)
array([[ 0. , 0. , 0. , 0. , 0. ],
[ 1. , 0.74, 0.59, 0.74, 1. ],
[ 1. , 1. , 1. , 1. , 1. ],
[ 1. , 0.74, 0.59, 0.74, 1. ],
[ 0. , 0. , 0. , 0. , 0. ]])
Now projecting this back to curved detector.
>>> new_data = project_data(new_data, new_det, det)
>>> np.round(new_data.T, 2)
array([[ 0. , 0.33, 0.34, 0.33, 0. ],
[ 1. , 0.78, 0.71, 0.78, 1. ],
[ 1. , 1. , 1. , 1. , 1. ],
[ 1. , 0.78, 0.71, 0.78, 1. ],
[ 0. , 0.33, 0.34, 0.33, 0. ]])
The method is vectorized, i.e., it can be called for multiple observations
of values on the detector (most often corresponding to different angles):
>>> part = odl.uniform_partition(-1, 1, 3, nodes_on_bdry=True)
>>> det = odl.tomo.Flat1dDetector(part, axis=[1, 0])
>>> data_row = np.arange(-1, 2)
>>> data = np.stack([data_row] * 2, axis=0)
>>> new_det = flat_to_curved(det, 1)
>>> new_data = project_data(data, det, new_det)
>>> np.round(new_data, 2)
array([[-1., 0., 1.],
[-1., 0., 1.]])
"""
assert isinstance(old_detector, Detector)
assert isinstance(new_detector, Detector)
part = old_detector.partition
d = len(part.shape)
if d == 1 and any(old_detector.axis != new_detector.axis):
NotImplementedError('Detectors are axis not the same, {} and {}'
''.format(old_detector.axis, new_detector.axis))
elif d > 1 and (any(old_detector.axes[0] != new_detector.axes[0])
or any(old_detector.axes[1] != new_detector.axes[1])):
NotImplementedError('Detectors are axis not the same, {} and {}'
''.format(old_detector.axes, new_detector.axes))
data = np.asarray(data, dtype=float)
if data.shape[-d:] != part.shape:
raise ValueError('Last dimensions of `data.shape` must '
'correspond to the detector partitioning, '
'got {} and {}'.format(data.shape[-d:], part.shape))
# find out if there are multiple data points
if d < len(data.shape):
n = data.shape[0]
else:
n = 1
if n > 1:
# extend detectors partition for multiple samples
data_part = uniform_partition(np.append([0], part.min_pt),
np.append([n], part.max_pt),
np.append([n], part.shape),
nodes_on_bdry=part.nodes_on_bdry)
else:
data_part = part
if isinstance(old_detector, Flat1dDetector):
assert isinstance(new_detector, CircularDetector)
phi = new_detector.partition.meshgrid[0]
u = new_detector.radius * np.tan(phi)
interpolator = linear_interpolator(data, data_part.coord_vectors)
if n > 1:
i = data_part.meshgrid[0]
u = u.reshape(1, -1)
resampled_data = interpolator((i, u))
else:
resampled_data = interpolator(u)
elif isinstance(old_detector, CircularDetector):
assert isinstance(new_detector, Flat1dDetector)
u = new_detector.partition.meshgrid
phi = np.arctan2(u, old_detector.radius)
interpolator = linear_interpolator(data, data_part.coord_vectors)
if n > 1:
i = data_part.meshgrid[0]
phi = phi.reshape(-1, 1)
resampled_data = interpolator((i, phi))
else:
resampled_data = interpolator(phi)
elif isinstance(old_detector, Flat2dDetector):
assert isinstance(new_detector,
(CylindricalDetector, SphericalDetector))
r = new_detector.radius
if isinstance(new_detector, CylindricalDetector):
phi, h = new_detector.partition.meshgrid
u = r * np.tan(phi)
v = h / r * np.sqrt(r * r + u * u)
else:
phi, theta = new_detector.partition.meshgrid
u = r * np.tan(phi)
v = np.tan(theta) * np.sqrt(r * r + u * u)
interpolator = linear_interpolator(data, data_part.coord_vectors)
if n > 1:
i = data_part.meshgrid[0]
u = np.expand_dims(u, axis=0)
v = np.expand_dims(v, axis=0)
resampled_data = interpolator((i, u, v))
else:
u = np.repeat(u, v.shape[1], axis=-1)
coord = np.stack([u, v], axis=-1).reshape((-1, 2))
resampled_data = interpolator(coord.T).reshape(
new_detector.partition.shape)
elif isinstance(old_detector, CylindricalDetector):
assert isinstance(new_detector, (Flat2dDetector, SphericalDetector))
r = old_detector.radius
if isinstance(new_detector, Flat2dDetector):
u, v = new_detector.partition.meshgrid
phi = np.arctan2(u, r)
h = v * r / np.sqrt(r * r + u * u)
else:
phi, theta = new_detector.partition.meshgrid
h = r * np.tan(theta)
interpolator = linear_interpolator(data, data_part.coord_vectors)
if n > 1:
i = data_part.meshgrid[0]
phi = np.expand_dims(phi, axis=0)
h = np.expand_dims(h, axis=0)
resampled_data = interpolator((i, phi, h))
else:
phi = np.repeat(phi, h.shape[1], axis=-1)
if h.shape[0] != phi.shape[0]:
h = np.repeat(h, phi.shape[0], axis=0)
coord = np.stack([phi, h], axis=-1).reshape((-1, 2))
resampled_data = interpolator(coord.T).reshape(
new_detector.partition.shape)
elif isinstance(old_detector, SphericalDetector):
assert isinstance(new_detector, (Flat2dDetector, CylindricalDetector))
r = old_detector.radius
if isinstance(new_detector, Flat2dDetector):
u, v = new_detector.partition.meshgrid
phi = np.arctan2(u, r)
theta = np.arctan2(v, np.sqrt(r * r + u * u))
else:
phi, h = new_detector.partition.meshgrid
theta = np.arctan2(h, r)
interpolator = linear_interpolator(data, data_part.coord_vectors)
if n > 1:
i = data_part.meshgrid[0]
phi = np.expand_dims(phi, axis=0)
theta = np.expand_dims(theta, axis=0)
resampled_data = interpolator((i, phi, theta))
else:
phi = np.repeat(phi, theta.shape[1], axis=-1)
if theta.shape[0] != phi.shape[0]:
theta = np.repeat(theta, phi.shape[0], axis=0)
coord = np.stack([phi, theta], axis=-1).reshape((-1, 2))
resampled_data = interpolator(coord.T).reshape(
new_detector.partition.shape)
else:
NotImplementedError('Data transformation between detectors {} and {}'
'is not implemented'.format(
old_detector, new_detector))
return resampled_data