"""Three-dimensional spiral projection sampling."""
__all__ = ["spiral_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 spiral_proj(shape, accel=None, nintl=1, order="ga", **kwargs):
r"""
Design a constant- or multi-density spiral projection.
The trajectory consists of a 2D spiral, 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)``.
accel : Iterable[int], optional
Acceleration factors ``(in-plane, radial)``.
Range from ``1`` (fully sampled) to ``nintl`` / ``$\pi$ * shape[0]``.
The default is ``(1, 1)`` if ``ncontrast == 1`` and ``(1, ``$\pi$ * shape[0])``
if ``ncontrast > 1``.
nintl : int, optional
Number of interleaves to fully sample a plane.
The default is ``1``.
order : str, optional
Spiral 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
-----------------
moco_shape : int
Matrix size for inner-most (motion navigation) spiral.
The default is ``None``.
acs_shape : int
Matrix size for intermediate inner (coil sensitivity estimation) spiral.
The default is ``None``.
acs_nintl : int
Number of interleaves to fully sample intermediate inner spiral.
The default is ``1``.
variant : str
Type of spiral. Allowed values are:
* ``center-out``: starts at the center of k-space and ends at the edge (default).
* ``reverse``: starts at the edge of k-space and ends at the center.
* ``in-out``: starts at the edge of k-space and ends on the opposite side (two 180° rotated arms back-to-back).
Returns
-------
head : Header
Acquisition header corresponding to the generated spiral.
Example
-------
>>> import deepmr
We can create a single-shot spiral projection for a matrix of ``(128, 128, 128)`` voxels by:
>>> head = deepmr.spiral_proj(128)
An in-plane multi-shot trajectory can be generated by specifying the ``nintl`` argument:
>>> head = deepmr.spiral_proj(128, nintl=48)
Both spirals have constant density. If we want a dual density we can use ``acs_shape`` and ``acs_nintl`` arguments.
For example, if we want an inner ``(32, 32)`` k-space region sampled with a 4 interleaves spiral, this can be obtained as:
>>> head = deepmr.spiral_proj(128, nintl=48, acs_shape=32, acs_nintl=4)
This inner region can be used e.g., for Parallel Imaging calibration. Similarly, a triple density spiral can
be obtained by using the ``moco_shape`` argument:
>>> head = deepmr.spiral_proj(128, nintl=48, acs_shape=32, acs_nintl=4, moco_shape=8)
The generated spiral will have an innermost ``(8, 8)`` single-shot k-space region (e.g., for PROPELLER-like motion correction),
an intermediate ``(32, 32)`` k-space region fully covered by 4 spiral shots and an outer ``(128, 128)`` region fully covered by 48 interleaves.
In-plane acceleration can be specified using the ``accel`` argument. For example, the following
>>> head = deepmr.spiral_proj(128, nintl=48, accel=4)
will generate the following trajectory:
>>> head.traj.shape
torch.Size([1, 4824, 538, 2])
i.e., a 48-interleaves trajectory with an in-plane acceleration factor of 4 (i.e., 12 interleaves),
repeated for 402 planes covering the 3D k-space sphere.
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.spiral_proj((128, 420), nintl=48)
>>> head.traj.shape
torch.Size([420, 48, 538, 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.spiral_proj((128, 1, 8), nintl=48)
>>> head.traj.shape
torch.Size([8, 19296, 538, 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, 3)``.
* 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 accel
if accel is None:
if shape[1] == 1:
accel = 1
else:
accel = int(math.pi * shape[0])
# expand accel if needed
if np.isscalar(accel):
accel = [1, accel]
else:
accel = list(accel)
# assume 1mm iso
fov = shape[0]
# design single interleaf spiral
tmp, _ = _design.spiral(fov, shape[0], 1, nintl, **kwargs)
# generate angles
ncontrasts = shape[1]
nplanes = max(int((math.pi * shape[0]) // accel[1]), 1)
nviews = max(int(nintl // accel[0]), 1)
dphi = 360.0 / nintl
dtheta = (1 - 233 / 377) * 360.0
# build rotation angles
j = np.arange(ncontrasts * nplanes)
i = np.arange(nviews)
j = np.tile(j, nviews)
i = np.repeat(i, ncontrasts * nplanes)
# 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]) # radial 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, nplanes, 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 * ncontrasts * nplanes
# 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
# extra args
user = {}
user["moco_shape"] = tmp["moco"]["mtx"]
user["acs_shape"] = tmp["acs"]["mtx"]
# get indexes
head = Header(shape, t=t, traj=traj, dcf=dcf, TE=TE, user=user)
head.torch()
return head
