Madison Kretzler^{1}, Jesse Hamilton^{2}, Mark Griswold^{1,2,3}, and Nicole Seiberlich^{1,2,3}

a-f SPARSE is a new technique which enables the k-t SPARSE approach to be used for radial trajectories in the Radon domain. Its use for cardiac imaging retrospectively accelerated by a factor of 4 is presented.

By
using the transform concept presented in a-f BLAST, as seen in Figure 1, the
Radon transform of the randomly collected radial data can be generated by
performing a Fourier transform only along the read-out direction. A subsequent Fourier transform performed along
the projection direction leads to an “aliased space”. In this a-t domain the transformed radial
data contains aliasing artifacts similar to those that appear in randomly
undersampled Cartesian data. As in k-t
SPARSE, a Fourier transform can then be applied along the temporal direction,
resulting in a representation of the data in a-f space. Due to the random radial sampling pattern
this a-f space is sparse, with the type of incoherent aliasing artifacts that
can be removed using a compressed sensing reconstruction. As in k-t SPARSE, a-f SPARSE uses iterative
soft-thresholding to solve $$$\widehat{\bf{d}}=argmin_d\left\{\parallel{\bf{F}_{2D} \bf{d}-\bf{m}}\parallel_2^2+\lambda\parallel{\bf{F}_t\bf{d}}\parallel_1\right\}$$$
where $$$\bf{m}$$$ is the collected
dynamic data in the radial k-space, $$$\bf{d}$$$ is the a-t series to
be reconstructed, $$$\bf{F}_{2D}$$$
is the 2D spatial
Fourier transform, $$$\bf{F}_{t}$$$ is the temporal
Fourier transform, $$$\lambda$$$
is the weighting
parameter, and $$$\widehat{\bf{d}}$$$ is the reconstructed
a-t series. This
formulation is similar to that employed in k-t SPARSE-SENSE^{4} (although without the use of coil sensitivity
information) where the *l _{1}*-norm
enforces the sparsity in the temporal Fourier domain and the

This technique was applied to a cardiac phantom with a heartrate of 60 bpm, matrix size 144x144, and 100 time frames. The data were sampled along both a Cartesian trajectory and a radial trajectory and undersampled to acceleration factors of R=2, 3, 4, 6, and 8. Both k-t and a-f SPARSE reconstructions were applied to the accelerated data (Cartesian and radial, respectively) in order to compare the RMSE values and computation time. No coil sensitivity information was used for the reconstructions. Additionally, a-f SPARSE was tested on retrospectively sampled in-vivo radial cardiac datasets. a-f SPARSE was applied to in-vivo cardiac breathheld cine scans which were downsampled to mimic different acceleration factors. RMSE values were calculated using the fully-sampled cine images. These data were collected along a radial trajectory on a Siemens Skyra 3T whole-body scanner with a bSSFP sequence using TR = 29 ms, TE = 1.5ms, BW = 1 kHz, FoV = 300mm, spatial resolution = 2.3x2.3x8.0 mm^{3}, flip-angle = 57 degrees.

1. Lustig, M., Santos, J. M., Donoho, D. & Pauly, J. M. k-t SPARSE: high frame rate dynamic MRI exploiting spatio-temporal sparsity. Proc. ISMRM, Seattle 50, 2420 (2006).

2. Feng, L. et al. Golden-angle radial sparse parallel MRI: Combination of compressed sensing, parallel imaging, and golden-angle radial sampling for fast and flexible dynamic volumetric MRI. Magn. Reson. Med. 717, 707–717 (2013).

3. Kretzler, M., Hamilton, J., Griswold, M. & Seiberlich, N. a-f BLAST: A Non-Iterative Radial k-t BLAST Reconstruction in Radon Space. in ISMRM (2016).

4. Otazo, R., Kim, D., Axel, L. & Sodickson, D. K. Combination of Compressed Sensing and Parallel Imaging for Highly Accelerated First-Pass Cardiac Perfusion MRI. Magn. Reson. Med. 29, 997–1003 (2010).

5. Fessler, J. A. & Sutton, B. P. Nonuniform fast fourier transforms using min-max interpolation. IEEE Trans. Signal Process. 51, 560–574 (2003).

Randomly
undersampled radial data (far-left) can be transformed into a space with
aliasing similar to that of randomly undersampled Cartesian data in the x-f
space (far-right) by performing Fourier transforms along the readout,
projection, and temporal directions. The
aliasing in this a-f space is sparse and incoherent allowing it to be resolved
using the compressed sensing reconstruction, k-t SPARSE.

Flow chart of the
a-f SPARSE reconstruction. The dynamic
image (bottom-left) is randomly radially sampled and sorted into the radial
k-space (top-left). A 2D Fourier
transform is performed to transform the data to the aliased space (middle-top)
then the through time Fourier transform is used to transform the data into a
sparse domain, a-f space (top-right).
The a-f SPARSE reconstruction is performed and the reconstructed image
is produced (bottom-middle). The inverse
Fourier transform was then used to transform the reconstructed data to the
radial k-space data which were then gridded using the Fessler Toolbox^{5} to generate the reconstructed
images.

(Top-left) Graph of RMSE versus varying
acceleration factors for k-t SPARSE and a-f SPARSE. The graph shows a-f SPARSE outperforming k-t
SPARSE for all accelerations. (Top-right) Bar graph of the SPARSE reconstruction times versus acceleration factor
for k-t and a-f SPARSE. The graph shows
significantly faster reconstruction times for a-f SPARSE. (Bottom) Example
images of the phantom reconstruction for an acceleration factor of R=4 for both
k-t and a-f SPARSE.

An
example of a-f SPARSE, showing (top) the fully-sampled in-vivo cardiac cine, R=4 zero-filled, R=4 reconstructed images,
and x-t images of the through-time heartbeat for each (bottom).

Gif
of a-f SPARSE R=4 reconstruction from Figure 4 showing the cine, zero-filled,
and a-f SPARSE reconstruction (from left to right).