"""Three-dimensional radial projection sampling."""
__all__ = ["radial_proj"]
import math
import numpy as np
# this is for stupid Sphinx
try:
from ... import _design
except Exception:
pass
from ..._types import Header
[docs]def radial_proj(shape, nviews=None, order="ga", **kwargs):
r"""
Design a 3D radial projectiontrajectory.
The trajectory consists of a 2D radial trajectory, whose plane
is rotated to cover the 3D k-space. In-plane rotations
are sequential. Plane rotation types are specified
via the ``order`` argument.
Parameters
----------
shape : Iterable[int]
Matrix shape ``(in-plane, contrasts=1, echoes=1)``.
nviews : int, optional
Number of spokes (in-plane, radial).
The default is ``$\pi$ * (shape[0], shape[1])`` if ``shape[2] == 1``,
otherwise it is ``($\pi$ * shape[0], 1)``.
order : str, optional
Radial plane rotation type.
These can be:
* ``ga``: Pseudo golden angle variation of periodicity ``377``.
* ``ga::multiaxis``: Pseudo golden angle, i.e., same as ``ga`` but views are repeated 3 times on orthogonal axes.
* ``ga-sh``: Shuffled pseudo golden angle.
* ``ga-sh::multiaxis``: Multiaxis shuffled pseudo golden angle, i.e., same as ``ga-sh`` but views are repeated 3 times on orthogonal axes.
The default is ``ga``.
Keyword Arguments
-----------------
variant : str
Type of radial trajectory. Allowed values are:
* ``fullspoke``: starts at the edge of k-space and ends on the opposite side (default).
* ``center-out``: starts at the center of k-space and ends at the edge.
Returns
-------
head : Header
Acquisition header corresponding to the generated sampling pattern.
Example
-------
>>> import deepmr
We can create a Nyquist-sampled 3D radial trajectory for a matrix of ``(128, 128, 128)`` voxels by:
>>> head = deepmr.radial_proj(128)
An undersampled trajectory can be generated by specifying the ``nviews`` argument:
>>> head = deepmr.radial_proj(128, nviews=64)
Multiple contrasts with different sampling (e.g., for MR Fingerprinting) can be achieved by providing
a tuple of ints as the ``shape`` argument:
>>> head = deepmr.radial_proj((128, 420))
>>> head.traj.shape
torch.Size([420, 402, 128, 2])
corresponding to 420 different contrasts, each sampled with a different fully sampled plane.
Similarly, multiple echoes (with fixed sampling) can be specified as:
>>> head = deepmr.radial_proj((128, 1, 8))
>>> head.traj.shape
torch.Size([8, 161604, 128, 2])
corresponding to a 8-echoes fully sampled k-spaces, e.g., for QSM and T2* mapping.
Notes
-----
The returned ``head`` (:func:`deepmr.Header`) is a structure with the following fields:
* shape (torch.Tensor):
This is the expected image size of shape ``(nz, ny, nx)``.
* t (torch.Tensor):
This is the readout sampling time ``(0, t_read)`` in ``ms``.
with shape ``(nsamples,)``.
* traj (torch.Tensor):
This is the k-space trajectory normalized as ``(-0.5 * shape, 0.5 * shape)``
with shape ``(ncontrasts, nviews, nsamples, 2)``.
* dcf (torch.Tensor):
This is the k-space sampling density compensation factor
with shape ``(ncontrasts, nviews, nsamples)``.
* TE (torch.Tensor):
This is the Echo Times array. Assumes a k-space raster time of ``1 us``
and minimal echo spacing.
"""
# expand shape if needed
if np.isscalar(shape):
shape = [shape, 1]
else:
shape = list(shape)
while len(shape) < 3:
shape = shape + [1]
# default views
if nviews is None:
if shape[1] == 1:
nviews = int(math.pi * shape[0])
else:
nviews = 1
# expand nviews if needed
if np.isscalar(nviews):
nviews = [int(math.pi * shape[0]), nviews]
else:
nviews = list(nviews)
# assume 1mm iso
fov = shape[0]
# design single interleaf spiral
tmp, _ = _design.radial(fov, shape[0], 1, 1, **kwargs)
# generate angles
ncontrasts = shape[1]
dphi = 360.0 / nviews[0]
dtheta = (1 - 233 / 377) * 360.0
# build rotation angles
j = np.arange(ncontrasts * nviews[1])
i = np.arange(nviews[0])
j = np.tile(j, nviews[0])
i = np.repeat(i, ncontrasts * nviews[1])
# radial angle
if order[:5] == "ga-sh":
theta = (i + j) * dtheta
else:
theta = j * dtheta
# in-plane angle
phi = i * dphi
# convert to radians
theta = np.deg2rad(theta) # angles in radians
phi = np.deg2rad(phi) # angles in radians
# perform rotation
axis = np.zeros_like(theta, dtype=int) # rotation axis
Rx = _design.angleaxis2rotmat(theta, [1, 0, 0]) # whole-plane rotation about x
Rz = _design.angleaxis2rotmat(phi, [0, 0, 1]) # in-plane rotation about z
# put together full rotation matrix
rot = np.einsum("...ij,...jk->...ik", Rx, Rz)
# get trajectory
traj = tmp["kr"] * tmp["mtx"]
traj = np.concatenate((traj, 0 * traj[..., [0]]), axis=-1)
traj = _design.projection(traj[0].T, rot)
traj = traj.swapaxes(-2, -1).T
traj = traj.reshape(nviews[0], nviews[1], ncontrasts, *traj.shape[-2:])
traj = traj.transpose(2, 1, 0, *np.arange(3, len(traj.shape)))
traj = traj.reshape(ncontrasts, -1, *traj.shape[3:])
# get dcf
dcf = tmp["dcf"]
dcf = _design.angular_compensation(dcf, traj.reshape(-1, *traj.shape[-2:]), axis)
dcf = dcf.reshape(*traj.shape[:-1])
# apply multiaxis
if order[-9:] == "multiaxis":
# expand trajectory
traj1 = np.stack((traj[..., 2], traj[..., 0], traj[..., 1]), axis=-1)
traj2 = np.stack((traj[..., 1], traj[..., 2], traj[..., 0]), axis=-1)
traj = np.concatenate((traj, traj1, traj2), axis=-3)
# expand dcf
dcf = np.concatenate((dcf, dcf, dcf), axis=-2)
# renormalize dcf
tabs = (traj[0, 0] ** 2).sum(axis=-1) ** 0.5
k0_idx = np.argmin(tabs)
nshots = nviews[0] * nviews[1] * ncontrasts
# impose that center of k-space weight is 1 / nshots
scale = 1.0 / (dcf[[k0_idx]] + 0.000001) / nshots
dcf = scale * dcf
# expand echoes
nechoes = shape[-1]
traj = np.repeat(traj, nechoes, axis=0)
dcf = np.repeat(dcf, nechoes, axis=0)
# get shape
shape = [shape[0]] + list(tmp["mtx"])
# get time
t = tmp["t"]
# calculate TE
min_te = float(tmp["te"][0])
TE = np.arange(nechoes, dtype=np.float32) * t[-1] + min_te
# get indexes
head = Header(shape, t=t, traj=traj, dcf=dcf, TE=TE)
head.torch()
return head
