Analysis vs Synthesis-based Regularization for Combined Compressed Sensing and Parallel MRI Reconstruction at 7 Tesla

Compressed Sensing (CS) has allowed a significant reduction of acquisition times in MRI, especially in the high spatial resolution (e.g., 400 $\mu{\mathrm{m}}$) context. Nonlinear CS reconstruction usually relies on analysis (e.g., Total Variation) or synthesis (e.g., wavelet) based priors and $\ell_{1}$ regularization to promote sparsity in the transform domain. Here, we compare the performance of several orthogonal wavelet transforms with those of tight frames for MR image reconstruction in the CS setting combined with parallel imaging (multiple receiver coil). We show that overcomplete dictionaries such as the fast curvelet transform provide improved image quality as compared to orthogonal transforms. For doing so, we rely on an analysis-based formulation where the underlying $\ell_{1}$ regularized criterion is minimized using a primal dual splitting method (e.g., Condat-V $\tilde{u}$ algorithm). Validation is performed on ex-vivo baboon brain $T^{*}_{2}$ MRI data collected at 7 Tesla and restrospectively under-sampled using non-Cartesian schemes (radial and Sparkling). We show that multiscale analysis priors based on tight frames instead of orthogonal transforms achieve better image quality (pSNR, SSIM) in particular at low signal-to-noise ratio.