About

GRAPPA is a popular parallel imaging reconstruction algorithm. Unfortunately there aren't a lot of easy to use Python implementations of it or its many variants available, so I decided to release this simple package.

There are also a couple reference SENSE-like implementations that have made their way into the package. This is to be expected -- a lot of later parallel imaging algorithms have hints of both GRAPPA- and SENSE-like inspirations.

Included in the pygrappa module are the following:

• GRAPPA [1]: grappa()
• VC-GRAPPA [2]: vcgrappa()
• iGRAPPA [3]: igrappa()
• hp-GRAPPA [4]: hpgrappa()
• Segmented-GRAPPA [5]: seggrappa()
• TGRAPPA [6]: tgrappa()
• Slice-GRAPPA [7]: slicegrappa()
• Split-Slice-GRAPPA [8]: splitslicegrappa()
• GRAPPA operator [9], [13]: grappaop(), radialgrappaop()
• Through-time GRAPPA [11]: ttgrappa()
• PARS [12]: pars()
• GROG [14]: grog()
• NL-GRAPPA [15]: nlgrappa()
• g-factor map: gfactor()
• SENSE [16]: sense1d()
• CG-SENSE [17]: cgsense()

Installation

This package is developed in Ubuntu 18.04 using Python 3.6.8. That's not to say it won't work on other things. You should submit an issue when it doesn't work like it says it should. The whole idea was to have an easy to use, pip-install-able GRAPPA module, so let's try to do that.

In general, it's a good idea to work inside virtual environments. I create and activate mine like this:

python3 -m venv /venvs/pygrappa
source /venvs/pygrappa/bin/activate

More information can be found in the venv documentation: https://docs.python.org/3/library/venv.html

Installation under a Unix-based platform should then be as easy as:

pip install pygrappa

You will need a C/C++ compiler that supports the C++14 standard. See INSTALLATION.rst for more info on installing under Windows.

Examples

See the examples module. It has several scripts showing basic usage. Docstrings are also a great resource -- check them out for all possible arguments and usage info.

You can run examples from the command line by calling them like this:

python -m pygrappa.examples.[example-name]

# For example, if I wanted to try out TGRAPPA:
python -m pygrappa.examples.basic_tgrappa

Usage

pygrappa.grappa() implements GRAPPA ([1]) for arbitrarily sampled Cartesian datasets. It is called with undersampled k-space data and calibration data (usually a fully sampled portion of the center of k-space). The unsampled points in k-space should be exactly 0:

from pygrappa import grappa

# These next two lines are to show you the sizes of kspace and
# calib -- you need to bring your own data.  It doesn't matter
# where the coil dimension is, you just need to let 'grappa' know
# when you call it by providing the 'coil_axis' argument
sx, sy, ncoils = kspace.shape[:]
cx, cy, ncoils = calib.shape[:]

# Here's the actual reconstruction
res = grappa(kspace, calib, kernel_size=(5, 5), coil_axis=-1)

# Here's the resulting shape of the reconstruction.  The coil
# axis will end up in the same place you provided it in
sx, sy, ncoils = res.shape[:]

If calibration data is in the k-space data, simply extract it (make sure to call the ndarray.copy() method, may break if using reference to the original k-space data):

from pygrappa import grappa

sx, sy, ncoils = kspace.shape[:] # center 20 lines are ACS
ctr, pd = int(sy/2), 10
calib = kspace[:, ctr-pd:ctr+pad, :].copy() # call copy()!

# coil_axis=-1 is default, so if coil dimension is last we don't
# need to explicity provide it
res = grappa(kspace, calib, kernel_size=(5, 5))
sx, sy, ncoils = res.shape[:]

A very similar GRAPPA implementation with the same interface can be called like so:

from pygrappa import cgrappa
res = cgrappa(kspace, calib, kernel_size=(5, 5), coil_axis=-1)

This function uses much of the same code as the Python grappa() implementation, but has certain parts written in C++ and all compiled using Cython. It runs about twice as fast. It will probably become the default GRAPPA implementation in future releases.

vcgrappa() is a VC-GRAPPA ([2]) implementation that simply constructs conjugate virtual coils, appends them to the coil dimension, and passes everything through to cgrappa(). The function signature is identical to pygrappa.grappa().

igrappa() is an Iterative-GRAPPA ([3]) implementation that can be called as follows:

from pygrappa import igrappa
res = igrappa(kspace, calib, kernel_size=(5, 5))

# You can also provide the reference kspace to get the MSE at
# each iteration, showing you the performance.  Regularization
# parameter k (as described in paper) can also be provided:
res, mse = igrappa(kspace, calib, k=0.6, ref=ref_kspace)

igrappa() makes calls to cgrappa() on the back end.

hpgrappa() implements the High-Pass GRAPPA (hp-GRAPPA) algorithm ([4]). It requires FOV to construct an appropriate high pass filter. It can be called as:

from pygrappa import hpgrappa
res = hpgrappa(kspace, calib, fov=(FOV_x, FOV_y))

seggrappa() is a generalized Segmented GRAPPA implementation ([5]). It is supplied a list of calibration regions, cgrappa is run for each, and all the reconstructions are averaged together to yield the final image. It can be called with all the normal cgrappa arguments:

from pygrappa import seggrappa

cx1, cy1, ncoil = calib1.shape[:]
cx2, cy2, ncoil = calib2.shape[:]
res = seggrappa(kspace, [calib1, calib2])

TGRAPPA is a Temporal GRAPPA implementation ([6]) and does not require calibration data. It can be called as:

from pygrappa import tgrappa

sx, sy, ncoils, nt = kspace.shape[:]
res = tgrappa(
    kspace, calib_size=(20, 20), kernel_size=(5, 5),
    coil_axis=-2, time_axis=-1)

Calibration region size and kernel size must be provided. The calibration regions will be constructed in a greedy manner: once enough time frames have been consumed to create an entire ACS, GRAPPA will be run. TGRAPPA uses the cgrappa implementation for its speed.

slicegrappa() is a Slice-GRAPPA ([7]) implementation that can be called like:

from pygrappa import slicegrappa

sx, sy, ncoils, nt = kspace.shape[:]
sx, sy, ncoils, sl = calib.shape[:]
res = slicegrappa(kspace, calib, kernel_size=(5, 5), prior='sim')

kspace is assumed to SMS-like with multiple collapsed slices and multiple time frames that each need to be separated. calib are the individual slices' kspace data at the same size/resolution. prior tells the Slice-GRAPPA algorithm how to construct the sources, that is, how to solve T = S W, where T are the targets (calibration data), S are the sources, and W are GRAPPA weights. prior='sim' creates S by simulating the SMS acquisition, i.e., S = sum(calib, slice_axis). prior='kspace' uses the first time frame from the kspace data, i.e., S = kspace[1st time frame]. The result is an array containing all target slices for all time frames in kspace.

Similarly, Split-Slice-GRAPPA ([8]) can be called like so:

from pygrappa import splitslicegrappa as ssgrappa

sx, sy, ncoils, nt = kspace.shape[:]
sx, sy, ncoils, sl = calib.shape[:]
res = ssgrappa(kspace, calib, kernel_size=(5, 5))

# Note that pygrappa.splitslicegrappa is an alias for
# pygrappa.slicegrappa(split=True), so it can also be called
# like this:
from pygrappa import slicegrappa
res = slicegrappa(kspace, calib, kernel_size=(5, 5), split=True)

grappaop returns two unit GRAPPA operators ([9], [10]) found from a 2D Cartesian calibration dataset:

from pygrappa import grappaop

sx, sy, ncoils = calib.shape[:]
Gx, Gy = grappaop(calib, coil_axis=-1)

See the examples to see how to use the GRAPPA operators to reconstruct datasets.

Similarly, radialgrappaop() returns two unit GRAPPA operators [13] found from a radial calibration dataset:

from pygrappa import radialgrappaop
sx, nr = kx.shape[:] # sx: number of samples along each spoke
sx, nr = ky.shape[:] # nr: number of rays/spokes
sx, nr, nc = k.shape[:] # nc is number of coils

Gx, Gy = radialgrappaop(kx, ky, k)

For large number of coils, warnings will appear about matrix logarithms and exponents, but I think it should be fine.

ttgrappa implements the through-time GRAPPA algorithm ([11]). It accepts arbitrary k-space sampling locations and measurements along with corresponding fully sampled calibration data. The kernel is specified by the number of points desired, not a tuple as is usually the case:

from pygrappa import ttgrappa

# kx, ky are both 1D arrays describing the points (kx, ky)
# sampled in kspace.  kspace is a matrix with two dimensions:
# (meas., coil) corresponding to the measurements takes at each
# (kx, ky) from each coil.  (cx, cy) and calib are similarly
# supplied.  kernel_size is the number of nearest neighbors used
# for the least squares fit.  25 corresponds to a kernel size of
# (5, 5) for Cartesian GRAPPA:

res = ttgrappa(kx, ky, kspace, cx, cy, calib, kernel_size=25)

PARS [12] is an older parallel imaging algorithm, but it checks out. It can be called like so:

from pygrappa import pars

# Notice we provide the image domain coil sensitivity maps: sens
res = pars(kx, ky, kspace, sens, kernel_radius=.8, coil_axis=-1)

# You can use kernel_size instead of kernel_radius, but it seems
# that kernel_radius gives better reconstructions.

In general, PARS is slower in this Python implementation because the size of the kernels change from target point to target point, so we have to loop over every single one. Notice that pars returns the image domain reconstruction on the Cartesian grid, not interpolated k-space as most methods in this package do.

GROG [14] is called with trajectory information and unit GRAPPA operators Gx and Gy:

from pygrappa import grog

# (N, M) is the resolution of the desired Cartesian grid
res = grog(kx, ky, k, N, M, Gx, Gy)

# Precomputations of fractional matrix powers can be accelerated
# using a prime factorization technique submitted to ISMRM 2020:
res = grog(kx, ky, k, N, M, Gx, Gy, use_primefac=True)

See examples.basic_radialgrappaop.py for usage example.

Esoterically, forward and inverse gridding are supported out of the box with this implementation of GROG, i.e., non-Cartesian -> Cartesian can be reversed. It's not perfect and I've never heard of anyone doing this via GROG, but check out examples.inverse_grog for more info.

NL-GRAPPA uses machine learning feature augmentation to reduce model- based reconstruction error [15]. It can be called like so:

from pygrappa import nlgrappa
res = nlgrappa(
    kspace, calib, ml_kernel_args={'cross_term_neighbors': 2})

You might need to play around with the arguments to get good images.

g-factor maps show geometry factor and a general sense of how well parallel imaging techniques like GRAPPA will work. Coil sensitivities must be known for to use this function as well as integer acceleration factors in x and y:

from pygrappa import gfactor
g = gfactor(sens, Rx, Ry)

SENSE implements the algorithm described in [16] for unwrapping aliased images along a single axis. Coil sensitivity maps must be provided. Coil images may be provided in image domain or k-space with the approprite flag:

from pygrappa import sense1d
res = sense1d(im, sens, Rx=2, coil_axis=-1)

# Or, kspace data for coil images may be provided:
res = sense1d(kspace, sens, Rx=2, coil_axis=-1, imspace=False)

CG-SENSE implements a Cartesian version of the algorithm described in [17]. It works for arbitrary undersampling of Cartesian datasets. Undersampled k-space and coil sensitivity maps are provided:

from pygrappa import cgsense
res = cgsense(kspace, sens, coil_axis=-1)

Although SENSE is more commonly known as an image domain parallel imaging reconstruction technique, it is useful to include in this package for comparison to kernel based and hybrid reconstructions.

References

[1]	(1, 2) Griswold, Mark A., et al. "Generalized autocalibrating partially parallel acquisitions (GRAPPA)." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 47.6 (2002): 1202-1210.

[2]	(1, 2) Blaimer, Martin, et al. "Virtual coil concept for improved parallel MRI employing conjugate symmetric signals." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 61.1 (2009): 93-102.

[3]	(1, 2) Zhao, Tiejun, and Xiaoping Hu. "Iterative GRAPPA (iGRAPPA) for improved parallel imaging reconstruction." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 59.4 (2008): 903-907.

[4]	(1, 2) Huang, Feng, et al. "High‐pass GRAPPA: An image support reduction technique for improved partially parallel imaging." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 59.3 (2008): 642-649.

[5]	(1, 2) Park, Jaeseok, et al. "Artifact and noise suppression in GRAPPA imaging using improved k‐space coil calibration and variable density sampling." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 53.1 (2005): 186-193.

[6]	(1, 2) Breuer, Felix A., et al. "Dynamic autocalibrated parallel imaging using temporal GRAPPA (TGRAPPA)." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 53.4 (2005): 981-985.

[7]	(1, 2) Setsompop, Kawin, et al. "Blipped‐controlled aliasing in parallel imaging for simultaneous multislice echo planar imaging with reduced g‐factor penalty." Magnetic resonance in medicine 67.5 (2012): 1210-1224.

[8]	(1, 2) Cauley, Stephen F., et al. "Interslice leakage artifact reduction technique for simultaneous multislice acquisitions." Magnetic resonance in medicine 72.1 (2014): 93-102.

[9]	(1, 2) Griswold, Mark A., et al. "Parallel magnetic resonance imaging using the GRAPPA operator formalism." Magnetic resonance in medicine 54.6 (2005): 1553-1556.

[10]	Blaimer, Martin, et al. "2D‐GRAPPA‐operator for faster 3D parallel MRI." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 56.6 (2006): 1359-1364.

[11]	(1, 2) Seiberlich, Nicole, et al. "Improved radial GRAPPA calibration for real‐time free‐breathing cardiac imaging." Magnetic resonance in medicine 65.2 (2011): 492-505.

[12]

(1, 2) Yeh, Ernest N., et al. "3Parallel magnetic resonance imaging with adaptive radius in k‐space (PARS): Constrained image reconstruction using k‐space locality in radiofrequency coil encoded data." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 53.6 (2005): 1383-1392.

[13]	(1, 2) Seiberlich, Nicole, et al. "Self‐calibrating GRAPPA operator gridding for radial and spiral trajectories." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 59.4 (2008): 930-935.

[14]	(1, 2) Seiberlich, Nicole, et al. "Self‐calibrating GRAPPA operator gridding for radial and spiral trajectories." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 59.4 (2008): 930-935.

[15]	(1, 2) Chang, Yuchou, Dong Liang, and Leslie Ying. "Nonlinear GRAPPA: A kernel approach to parallel MRI reconstruction." Magnetic resonance in medicine 68.3 (2012): 730-740.

[16]	(1, 2) Pruessmann, Klaas P., et al. "SENSE: sensitivity encoding for fast MRI." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 42.5 (1999): 952-962.

[17]	(1, 2) Pruessmann, Klaas P., et al. "Advances in sensitivity encoding with arbitrary k‐space trajectories." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 46.4 (2001): 638-651.