While compressed-sensing regularizers enable interpretable reconstruction of 4D Flow CMR through transparent variational objectives, their hand-crafted nature is too restrictive under high acceleration. State-of-the-art learning-based approaches mitigate this, but typically encode regularization implicitly through unrolled network modules, which limits their interpretability. To address this limitation, we propose CLEAR, designed to combine the interpretability of compressed sensing with the flexibility of learned models. To the best of our knowledge, it is the first learned regularizer for a 4D reconstruction task. In the ultra-accelerated 10×--50× regime of the CMRx4DFlow2026 challenge, CLEAR outperforms compressed sensing locally low-rank (LLR) and the popular variational network FlowVN, while using less than 10k parameters and preserving an interpretable regularization structure.
Figures & tables
Method
Direct R1
Generalizability S1
Generalizability S2
RelErr↓
AngErr↓
nRMSE↓
SSIM↑
RelErr↓
AngErr↓
nRMSE↓
SSIM↑
RelErr↓
AngErr↓
nRMSE↓
SSIM↑
LLR
0.583
39.815
0.127
0.799
0.522
34.764
0.128
0.857
0.465
31.951
0.102
0.863
FlowVN
0.526
41.562
0.059
0.927
0.626
40.599
0.110
0.909
0.803
44.044
0.120
0.854
CLEAR (ours)
0.396
30.214
0.042
0.959
0.354
23.211
0.046
0.983
0.293
24.800
0.033
0.989
Table 1: Quantitative performance, computational complexity, and acceleration-conditioning ablation. (a) Leaderboard results on Direct R1 and the generalizability sets S1 and S2. (b) Model complexity and reconstruction time for a four-point-encoded, 30× -accelerated, 10-coil k -space case on a 28×84×108×18 spatio-temporal grid, measured on an NVIDIA RTX PRO 6000 Blackwell GPU. (c) Comparison of acceleration-specific CLEAR models (at R=10,20,30,40,50 ) and the acceleration-conditioned CLEAR model on Direct R1. The best value in each comparison is shown in bold.
Figure 1: Convergence and stopping-tolerance sensitivity of CLEAR. (a) RelErr and nRMSE during batched reconstruction of the four velocity-encoded images in a 4D Flow CMR case, with markers indicating different nmAPG stopping tolerances. (b) Decrease of the objective ( 3 ) during reconstruction of a single velocity-encoded image.
Figure 2: Qualitative results on a 30× undersampled aortic case. Corresponding slice locations are illustrated with red dashed lines, indicating cross-section of the aorta and systolic peak. Scatter plot of velocity magnitude over manually segmented aorta (contour shown in magenta) is given together with correlation analysis ( y=ax+b ).
Figure 3: Visualization of the learned components in CLEAR. Only filter coefficients with magnitudes larger than 15% of the maximum magnitude are displayed for clarity.
Figure S4: Illustration of the proposed regularizer architecture.
Figure S5: CLEAR’s learned components (complex filters and base potential) in the t⊥ 3D branch. Only filter coefficients with magnitudes larger than 15% of the maximum magnitude filter coefficient are displayed for clarity.
Figure S6: CLEAR’s learned components (complex filters and base potential) in the z⊥ 3D branch. Only filter coefficients with magnitudes larger than 15% of the maximum magnitude filter coefficient are displayed for clarity.
Figure S7: CLEAR’s learned components (complex filters and base potential) in the y⊥ 3D branch. Only filter coefficients with magnitudes larger than 15% of the maximum magnitude filter coefficient are displayed for clarity.
Figure S8: CLEAR’s learned components (complex filters and base potential) in the x⊥ 3D branch. Only filter coefficients with magnitudes larger than 15% of the maximum magnitude filter coefficient are displayed for clarity.
4D flow MRI measures time-resolved, three-directional blood velocity but requires long acquisition times, and its diagnostic signal is carried by the phase difference between velocity encodings, not by image magnitude. Recent work has developed a per-encoding variational network to address image reconstruction in this field. In this work, we incorporate a joint supervised and self-supervised training regime and utilize both magnitude and velocity data during supervision. At the same time, we add multiple acquisition-robust and conditioning strategies based on the acceleration factors. On the CMRx4DFlow~2026 aortic dataset (1.5 and 3T), our model lowers RelErr by 38--50% and AngErr by 7.0--8.7∘ against a training-matched baseline across R=10--50, improving on every held-out subject at every acceleration. Our model also shows strong generalization ability to transfer on out-of-distribution data by employing the joint training scheme, with an increase of 7.2% in SSIM and decrease of 38% and 31% in AngErr and RelErr respectively.
Mengyuan Xue, Bochun Mei
Department of Radiology and Biomedical Imaging, University of California, San Francisco, San Francisco, CA, USA
We present FlowMoDL, an unrolled neural network for highly accelerated 4D flow MRI reconstruction that directly optimizes for both anatomical magnitude and phase-derived velocity accuracy. Building on the MoDL framework, FlowMoDL alternates a learned (3+1)D spatiotemporal denoiser with conjugate-gradient data-consistency updates based on the SENSE forward model. A novel dual-pathway conditioning scheme adapts the denoiser features and data-consistency weighting, enabling a single model to handle varying acceleration factors (10× to 50×). To ensure physiological accuracy, the network is trained using a deep-supervision composite loss that explicitly penalizes velocity magnitude and angular errors, stabilized by a curriculum schedule. We evaluate FlowMoDL on the multi-center CMRx4DFlow dataset against classical and deep-learning baselines (CG-SENSE, MoDL, FlowVN, and FlowMRI-Net). A key advantage of FlowMoDL is its superior gradient step efficiency. When evaluated under an equivalent, limited budget of gradient steps, competing flow-specific networks degrade significantly. In contrast, FlowMoDL robustly converges and strictly outperforms all competitors across all acceleration factors in magnitude SSIM, nRMSE, relative velocity error, and angular error, successfully recovering sharp structural details and temporally coherent velocity fields.
Institute of Information Processing (tnt), Leibniz University Hannover, Germany · L3S, Leibniz University Hannover, Hannover, Germany · Department of Data Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany +2
Generative priors reconstruct accelerated 3D MRI well but pay heavily at deployment: tens of network evaluations per volume, and protocol-specific hyperparameter tuning. A third hidden cost is the scanner's fixed sampling pattern. We treat the whole operating point as an input. A 3D MeanFlow patch network (a one-step flow model) is fine-tuned end-to-end through a warm-started, five-iteration differentiable conjugate-gradient projection. A small hypernetwork maps the operating point (data-consistency weight, acceleration, and the Cartesian sampling pattern itself) to the network's per-channel modulation. Three findings follow. (i) Learning the acquisition is worth more than any other operating point: on clinical knee data, the learned mask gains up to +2.34 dB over the protocol's variable-density mask. This gain requires the solver: with a feed-forward reconstructor the same learned mask hurts at 4x (-1.7 dB), but with the data-consistency projection it adds +4.6 dB. (ii) One evaluation is highly effective: it beats a 20-step patch-diffusion prior by up to +3.1 dB on brain and +2.9 dB on knee. Three to five evaluations extend the front to +6 dB while using a quarter of the prior's network calls. (iii) Fully sampled targets are optional: trained self-supervised on a split of acquired samples, the reconstructor matches its supervised twin at 4x on real data. Finally, we report what failed and why: subject-adaptive acquisition from measured energy, combining self-supervision with learned acquisition, and amortising the data-consistency weight.
Ruibo Wang
Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2600 AA Delft, Netherlands