Self-calibrating nonlinear MR image reconstruction algorithms for variable density sampling and parallel imaging
Loubna EL GUEDDARI1, Carole LAZARUS1, Hanaé CARRIE1, Alexandre VIGNAUD2, and Philippe CIUCIU1

1CEA/NeuroSpin & INRIA Parietal, Gif-sur-Yvette, France, 2CEA/NeuroSpin, Gif-sur-Yvette, France


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.


Reducing scan times in Magnetic Resonance Imaging (MRI) is essential to explore higher spatial resolution. Common methods to speed up MR acquisition rely either on deterministic or pseudo-random subsampling of k-space. This can be achieved using either parallel imaging (PI) [1] or compressed sensing (CS) [2], or both [3-4] to benefit from high SNR using a multiple receiver coil. Variable density sampling (VDS) is required to achieve high acceleration factors in MRI scans [1,5-6]. In prospective CS, VDS is implemented along non-Cartesian k-space trajectories (radial, spiral, …). Although recent CS-PI reconstruction algorithms [4,7] are able to deal with multichannel non-Cartesian data, they usually proceed with a gridding step [8] to get a Cartesian k-space before performing MR image reconstruction. Here, we propose a new self-calibrating approach to MR image reconstruction in the CS-PI context. An automated and fast procedure for extracting the sensitivity maps is proposed using the original non-Cartesian data and the Nonequispaced fast Fourier transform (NFFT) [9]. Second, we implement the Proximal Optimized Gradient Method (POGM) to solve the CS-PI reconstruction problem. To illustrate its advantages over ESPIRiT [10], the proposed method is tested to reconstruct high resolution 2D T2* images from 8-fold undersampled variable-density Sparkling [11] data at 7 Tesla using a 32-channel receiver coil.

Materials and Methods

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.

Discussion and Conclusion

Compared to the state-of-the-art, the proposed self-calibrating method to the CS-PI reconstruction problem is more efficient and the sensitivity profiles are easier to interpret. Based on these estimates, we have shown that POGM converges faster than FB and FISTA. On in vivo Human brain T2* data collected at 7 Tesla, we have also demonstrated that our approach is both more accurate and efficient than L1-ESPIRiT. Future work will be devoted to further speed up reconstruction by coupling POGM with B1-based surrogates as proposed in [12] for FISTA in Cartesian acquisition scenarios.


We would like to thank Prof. Jeffrey Fessler who provided significant insight during his stay at NeuroSpin in June 2017. This research program was supported by DRF Impulsion grant in 2016 (COSMIC, P.I.: P.C.).


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Fig. 1: Non-Cartesian multi-shot 8-fold subsampled Sparkling trajectories. One segment out of 64 and composed of 512 samples is outlined in red.

Fig. 2: Pseudo-code of the Proximal Optimized Gradient Method (POGM) for minimizing the penalized least squares objective function. The main difference compared to FISTA lies in the momentum rule which is given between lines 8 and 11.

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.

Fig. 4: Reconstructed MR images. (a) Cartesian reference, (b) Self-calibrating POGM-based and (c) L1 -ESPIRiT reconstructions from 8-fold accelerated prospective CS based on Sparkling trajectories. (d)-(f) respective zooms in the red square. The same optimal (in terms of SSIM metric) regularization parameter value λ=10−5 was used both in POGM and L1-ESPIRiT algorithms for minimizing Eq. (1).

Fig. 5: Numerical convergence of reconstruction algorithms. Comparison of convergence speeds for different optimization algorithms: Forward-Backward (FB), Fast Iterative Soft Thresholding (FISTA) and POGM. For the sake of fairness, the same estimated sensitivity maps were injected in all reconstruction algorithms. To track numerical convergence, we computed at each iteration k a normalized cost function ε(k) = 10 log_10 ||F(x_k) - F(x^*)||^2/||F(x^*)||^2 where F is the global cost function to minimize composed of the data consistency term and the L1-norm penalization in the wavelet domain.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)