Wave-CAIPI utilizes sinusoidal gradients during each readout period to traverse a corkscrew trajectory and spreads aliasing in all spatial directions. This allows 10-fold acceleration in 3D (1,2) and Simultaneous MultiSlice (SMS) imaging (3) with low image artifacts. CS-Wave was recently proposed (4), which extended Wave-encoding through Poisson sampling and wavelet regularization to achieve further accelerations. In this work, we propose an optimized CS-Wave with hybrid sampling that synergistically combines controlled aliasing at low spatial frequencies with incoherent aliasing at high frequencies. This is inspired by the performance gain of hybrid undersampling in (5). Additionally, efficient reconstruction via variable splitting of consistency and regularization terms is developed. At 15-fold acceleration, proposed CS-Wave provides 20% RMSE improvement over Wave-CAIPI, which nearly doubles the improvement from previous CS-Wave approach. This permits whole-brain QSM at 1×1×2mm³ resolution in 25s. CS-Wave is also combined with SMS Echo-Shift (6,7) to achieve further increase in acceleration to 30-fold by minimizing sequence dead-time, thus enabling multi-orientation QSM at 1.5mm isotropic resolution in 72s.

Despite following a non-Cartesian trajectory, Wave encoding can be expressed in Cartesian k-space through point spread function (psf) formalism: k = MF_yzPF_xSm, where k is the k-space data, M represents hybrid undersampling, F_yz and F_x denote Fourier transforms along k_y-k_z and k_x. P is the psf in hybrid (k_x,y,z) space, S are the coil sensitivities, and m is the unknown image.

Regularization is incorporated via

1/2||MF_yzPF_xSm – k||₂² + λ||Rm||₁

where R is wavelet or gradient operator. By introducing auxiliary variables c = F_yzPF_xSm, and r = Rm, and applying ADMM (8,9), closed-form updates are obtained:

c = (M+αI)^–1·[Mk + α(F_yzPF_xSm+d_c)]

r = max(|Rm+d_r|–λ/β, 0)·sign(Rm+d_r)

where d_c and d_r are dual variables, and α and β are Lagrangian parameters. Image update is found by solving

(αS^HS+βR^HR)m = αS^HF_x^HP^HF_yz^H(c–d_c) + βR^H(r–d_r)

This system can be inverted in closed-form for wavelet transform where R^HR = I, and solved for gradient transform (TV) using conjugate gradient with the diagonal preconditioner (S^HS+6βI)^–1.

(i) 3D-GRE Wave data were acquired at 3T and 7T using 32-channel receiver with parameters: matrix=224×222×120, resolution=1×1×2mm³. For 3T: TE/TR=13.3/26ms, and for 7T: TE/TR=10.9/27ms.

(ii) SMS Echo-Shift Wave data were acquired at 3T, where 2-fold improvement in encoding efficiency is achieved by echo-shifting even and odd slices data to minimize dead-time from long-TE. Resolution was 1.5mm isotropic with matrix=160×160×96, TE/TR=35/47ms. Three volumes were collected at different head orientations.

For fast computation, SVD coil compression was applied (10). ESPIRiT calibration (11) from the center 16×16×16 k-space was employed for coil sensitivity estimation.

(i) 3D-GRE Wave data: were retrospectively undersampled using

i) Wave-CAIPI under-sampling at R=5×3,

ii) Poisson CS-Wave sampling at R=15 (4), and

iii) Proposed hybrid CS-Wave sampling with controlled aliasing at R=3×3 in central 25% k-space and variable-density Poisson mask (12) at outer k-space for R=15-fold total acceleration (Fig.1).

Generalized SENSE was used for Wave-CAIPI reconstruction (1,13) and proposed efficient algorithm was employed for CS-Wave reconstruction.

(ii) Echo-Shift Wave data: were undersampled to provide 30-fold total acceleration (R_cs-wave×R_echo-shift=15×2) and reconstructed using proposed algorithm.

Phase data were processed with BET brain-masking (14), Laplacian unwrapping (15), and harmonic background removal (16,17) to obtain the tissue phase.

(i) QSM for 3D-GRE: employed single-step reconstruction that directly relates the unwrapped phase to the underlying susceptibility via TV regularization.

(ii) QSM for Echo-Shift: combined complementary information in phase images from 3-directions using COSMOS (18), thus obviating the need for regularization.

Fig.1 compares proposed CS-Wave and Wave-CAIPI at 7T. RMSEs were 8.6% for CS-Wave with TV-penalty and hybrid sampling (9.1% with previously proposed wavelet and Poisson sampling, not shown) and 10.2% for Wave-CAIPI.

Fig.2 shows 3T results, where the RMSEs were 7.4% for proposed CS-Wave with TV-penalty and hybrid sampling (7.9% with wavelet and Poisson), and 8.9% for Wave-CAIPI. Processing times in Matlab were 8.4min for TV-regularized CS-Wave (3.8min using wavelet) and 1.8min for Wave-CAIPI.

Figs.3&4 demonstrate tissue phase and QSM images, derived from these accelerated 3D-GRE reconstructions.

Fig.5 presents Echo-Shift CS-Wave reconstructions at 3-orientations, and high quality QSM and phase images.

The proposed CS-Wave enables improved mitigation of artifacts and noise amplification. This was made possible by efficient ADMM reconstruction, and hybrid sampling that simultaneously accommodates sensitivity encoding and sparsity prior. Compared to previous CS-Wave that achieves 12% improvement over Wave-CAIPI, proposed strategy attains 20% RMSE reduction. This leads to improved phase and QSM from single-orientation at 1×1×2mm³ resolution in 25s. Combining CS-Wave with Echo-Shift permits 30-fold acceleration for multi-orientation QSM, enabling a 72s protocol with 3-rotations. Future work will explore complementary sampling across orientations with joint reconstruction for further gain.