CLEAR: Complex Learned Explicit Analytical Regularization for Ultra-Accelerated 4D Flow CMR Reconstruction

German Shâma Wache; Sebastian Neumayer

cs.CVSep 19, 2026

CLEAR: Complex Learned Explicit Analytical Regularization for Ultra-Accelerated 4D Flow CMR Reconstruction

Authors: German Shâma Wache, Sebastian Neumayer

Organizations: Chemnitz University of Technology, Germany

Abstract

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 $10\times$ -- $50\times$ 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
Method	$\mathrm{RelErr}\!\downarrow$	$\mathrm{AngErr}\!\downarrow$	$\mathrm{nRMSE}\!\downarrow$	$\mathrm{SSIM}\!\uparrow$	$\mathrm{RelErr}\!\downarrow$	$\mathrm{AngErr}\!\downarrow$	$\mathrm{nRMSE}\!\downarrow$	$\mathrm{SSIM}\!\uparrow$	$\mathrm{RelErr}\!\downarrow$	$\mathrm{AngErr}\!\downarrow$	$\mathrm{nRMSE}\!\downarrow$	$\mathrm{SSIM}\!\uparrow$
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\times

-accelerated, 10-coil

k

-space case on a

28\times 84\times 108\times 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.

Panel of Figure 1 — 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 $30\times$ 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 $t^{\perp}$ 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 $z^{\perp}$ 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 $y^{\perp}$ 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 $x^{\perp}$ 3D branch. Only filter coefficients with magnitudes larger than $15\%$ of the maximum magnitude filter coefficient are displayed for clarity.

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Sep 29, 2026J/K move · Enter open · S save

Joint Supervised and Self-Supervised Training with Acquisition-Robust Techniques for Accelerated 4D Flow MRI Reconstruction

Mengyuan Xue, Bochun Mei

Department of Radiology and Biomedical Imaging, University of California, San Francisco, San Francisco, CA, USA

4D Reconstruction Accelerated MRI Reconstruction Magnetic Resonance Imaging Cardiac MRI

Open paper AskPDF arXiv

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^\circ

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.

Abstract

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