Loubna EL GUEDDARI^{1}, Carole LAZARUS^{1}, Hanaé CARRIE^{1}, Alexandre VIGNAUD^{2}, and Philippe CIUCIU^{1}

Compressed Sensing has allowed a significant reduction of acquisition times in MRI. However, to maintain high signal-to-noise ratio during acquisition, CS is usually combined with parallel imaging (PI). Here, we propose a new self-calibrating MRI reconstruction framework that handles non-Cartesian CS and PI. Sensitivity maps are estimated from the data in the center of k-space while MR images are iteratively reconstructed by minimizing a nonsmooth criterion using the proximal optimized gradient method, which converges faster than FISTA. Comparison with L1-ESPIRiT suggests that our approach performs better both visually and numerically on 8-fold accelerated Human brain data collected at 7 Tesla.

**
Setup.
**Four healthy volunteers were scanned with a 7T system (Siemens
Healthineers, Erlangen, Germany) and a 1Tx/32Rx head coil (Nova
Medical, Wilmington, USA). All subjects signed a written informed
consent form and were enrolled in the study under the approval of our
institutional review board.

**Sequence
and k-space trajectories.** A
modified 2D T2*-weighted interleaved GRE sequence was acquired for an
in-plane resolution of 390 ㎛
with
the following parameters: TR=550ms, TE=30ms and FA=25° for one axial
slice of 3 mm-thickness, matrix size N=512x512. The Sparkling
trajectory (**Fig.
1**)
was composed of 64 shots, each comprising 512 samples during a
readout of 30.72 ms. Although Sparkling was implemented prospectively, here we used this sampling scheme retrospectively from fully sampled Cartesian data. Hence, the number of measurements was M=32,768 and
the subsampling factor R=N/M=8.

**Sensitivity
maps extraction.** Sensitivity
maps information lies in the low-frequency domain, hence
variable-density trajectories like radial or sparkling intrinsically
handle this information and allow self-calibration without fully
sampling the k-space center. Our sensitivity map estimation method
thus extracts the 10% central surface of the measured k-space. Then,
low frequency NxN coil images were reconstructed applying the NFFT
operator to the data completed by zero-filling. Third, the square
root of the Sum of Squares (SSOS) was computed. Fourth, the
sensitivity maps were estimated by the pixelwise ratio between image
coils and the SSOS.

**Reconstruction.**
The CS-PI reconstruction problem consists of minimizing
a penalized least square criterion involving the data collected over
the 32 channels and a L1-norm penalty term promoting sparsity in the
wavelet domain. The balance between the two is controlled by paramter
λ>0 whose optimal setting was performed using a grid search
procedure over [10^{-7}, 10^{-4}]. Forward-Backward, FISTA
and POGM optimization algorithms are summarized in **Fig. 2**.

**
Sensitivity
maps. Fig. 3**
illustrates
three sensitivity maps extracted using either our method or
L1-ESPIRIT. Because of the SVD decomposition involved in L1-ESPIRIT,
the sensitivity profiles are smoother as compared to ours, which
clearly delineate the FOV part illuminated by each receiver coil.
Moreover, our approach is faster since it costs 1min as compared to
10min for L1-ESPIRIT on the same architecture and Matlab-R2017
software.

**MR
image reconstruction.** MR
images were reconstructed from Sparkling data either using our
approach or L1-ESPIRiT. Although full FOV images look very similar
(**Fig.**
**4(a)-(c)**),
the respective zooms (**Fig.
4(d)-(f)**)
show that the dark stripes in the white matter are lost in the
L1-ESPIRiT image whereas they are well preserved using our
self-calibrating solution. In addition, our POGM algorithm converged
in 2.5 min whereas L1-ESPIRiT took about 5 min.

**Convergence
speed.** The
same experimental setup was used to compare the three algorithms. As
reported in **Fig.
5**,
FISTA and POGM decrease faster than FB even though they show some
“Nesterov ripples”. Interestingly, POGM decreases a little bit
faster than FISTA during the first tens of iterations.

**Fig. 3: Estimation of sensitivity maps.** Top: 3 out of 32 sensitivity maps extracted using our method. Bottom: Three consistent sensitivity maps yielded by the ESPIRiT algorithm based on eigenvalue decomposition.