Weakly Convex Ridge Regularization for 3D Non-Cartesian MRI Reconstruction

German Shâma Wache; Chaithya G R; Asma Tanabene; Sebastian Neumayer

cs.CVMar 28, 2026

Weakly Convex Ridge Regularization for 3D Non-Cartesian MRI Reconstruction

Authors: German Shâma Wache, Chaithya G R, Asma Tanabene, Sebastian Neumayer

Organizations: Faculty of Mathematics, Chemnitz University of Technology, Germany · NeuroSpin, Frédéric Joliot Institute for Life Sciences, CEA Paris-Saclay, France · MIND, Inria Saclay Centre, France · Siemens Healthineers, France

Abstract

While highly accelerated non-Cartesian acquisition protocols significantly reduce scan time, they often entail long reconstruction delays. Deep learning based reconstruction methods can alleviate this, but often lack stability and robustness to distribution shifts. As an alternative, we train a rotation invariant weakly convex ridge regularizer (WCRR). The resulting variational reconstruction approach is benchmarked against state of the art methods on retrospectively simulated data and (out of distribution) on prospective GoLF SPARKLING and CAIPIRINHA acquisitions. Our approach consistently outperforms widely used baselines and achieves performance comparable to Plug and Play reconstruction with a state of the art 3D DRUNet denoiser, while offering substantially improved computational efficiency and robustness to acquisition changes. In summary, WCRR unifies the strengths of principled variational methods and modern deep learning based approaches.

Figures & tables

Figure 1: Illustration of WCRR: The (rotated) inputs are processed by a filter bank $\{\mathbf{W}_{1},\ldots,\mathbf{W}_{J}\}$ . The extracted features are penalized based on the potentials $\{\psi_{1},\ldots,\psi_{J}\}$ . Usually, the filters extract high-frequency information.

Figure 2: Overview of the training and reconstruction pipeline. The WCRR is trained on a Gaussian denoising task, and then used for MRI reconstruction. This involves only tuning of the scalar hyperparameters $\sigma$ and $\lambda$ of the reconstruction model on a (small) validation set. The coil sensitivity maps are estimated using the ESPIRiT algorithm [ 62 ] on the central $24\times 24$ k-space data (simulating ACS acquisition).

Figure 3: (A) 3D GoLF-SPARKLING trajectory with GRAPPA acceleration. The green portion highlights the central Cartesian readouts, and the blue one depicts the non-Cartesian SPARKLING parts. Slices with $k_{x}=0$ , $k_{z}=0$ , and $k_{y}=0$ are given in (B), (C), and (D), respectively.

Method	12-coil			32-coil			# Param.	Train. Time
	Masked	Masked	Recon. Time	Masked	Masked	Recon. Time
	PSNR	SSIM	Recon. Time	PSNR	SSIM	Recon. Time
GRAPPA + DCp	$28.72\pm 1.91$	$0.8901\pm 0.00194$	5 s	$35.47\pm 0.91$	$0.9601\pm 0.0077$	11 s	–	–
$\ell_{1}$ -wavelet	$30.30\pm 1.54$	$0.9233\pm 0.0123$	37 s	$36.21\pm 0.82$	$0.9732\pm 0.0050$	1 min 21 s	–	–
TV	$31.56\pm 1.59$	$0.9402\pm 0.0093$	1 min 10 s	$37.14\pm 0.82$	$0.9786\pm 0.0039$	2 min 23 s	–	–
DRUNet3D	$31.02\pm 1.30$	$0.9394\pm 0.0059$	2 s	$34.61\pm 0.72$	$0.9699\pm 0.0051$	4 s	96,543,168	41 h

Table 1: Mean and standard deviation of masked PSNR (dB) and masked SSIM over the 12-coil and 32-coil (unseen generalization task) test data for each method. Runtime for a single reconstruction, parameter count and training time are also provided when applicable. In each column, the best value is highlighted in bold, and the second-best is underlined.

Figure 4: WCRR convergence curves for a representative volume. (Left) Relative error between consecutive iterates (tolerance). (Middle) Energy functional. (Right) Masked PSNR. The dashed vertical line highlights where Algorithm 1 terminates with tolerance

5\text{\times}{10}^{-3}

Figure 5: Cross-section (5th slice before the mid-slice) of a 12-coil representative volume along the sagittal , coronal , and transversal axes. The highlighted region is shown magnified in the corresponding inset. The shorthands m-psnr and m-ssim refer to the masked metrics. Additionally, m-nmse refers to the masked normalized mean squared error.

Version[# learnable filters ]	Masked	Masked	Recon. Time	# Param.
Version[# learnable filters ]	PSNR	SSIM	Recon. Time	# Param.
non-rotation-invariant[ $16$ filters ]	$31.82\pm 1.56$	$0.9446\pm 0.0080$	28 s	4,729
rotation-invariant[ $16$ filters ] (ours)	$32.04\pm 1.54$	$\underline{0.9469}\pm 0.0077$	34 s	4,729
non-rotation-invariant[ $64$ filters ]	$\underline{32.06}\pm 1.54$	$\underline{0.9469}\pm 0.0077$	32 s	15,673
rotation-invariant[ $64$ filters ]	$\textbf{32.11}\pm 1.53$	$\textbf{0.9475}\pm 0.0075$	44 s	15,673

Table 2: Quantitative comparison between the rotation-invariant and the non-rotation-invariant versions of WCRR. Mean and standard deviation for masked PSNR (dB) and masked SSIM are given over the 12-coil testing set. Runtime for a single reconstruction and parameter count are also provided. In each column, the best value is highlighted in bold, and the second-best is underlined.

Figure 6: Mid-plane reconstructions for prospective GS-SPARKLING accelerated acquisitions with $2\times 2$ GRAPPA (1 min) with (B) SENSE, (C) $\ell_{1}$ -wavelets, (D) TV, (E) NC-PDNet, (F) DPIR, (G) Score-DiffMBIR and (H) WCRR. A reference volume (A) from a complete scan is also shown. We showcase sagittal , coronal , and transversal views along with zoomed insets. The arrows highlight representative local differences between reconstructions.

Figure 7: Mid-plane reconstructions for prospective CAIPIRINHA $3\times 2$ accelerated acquisitions (1 min) with (B) SENSE, (C) $\ell_{1}$ -wavelets, (D) TV, (E) NC-PDNet, (F) DPIR, (G) Score-DiffMBIR and (H) WCRR. A reference volume (A) from a complete scan is also shown. We showcase sagittal , coronal , and transversal views along with zoomed insets. The arrows highlight representative local differences between reconstructions.

Figure 8: Visualization of the learned rotation-invariant WCRR prior. The top rows show the learned 3D filters, where only weights with magnitude larger than $15\%$ of the maximum absolute filter weight are displayed for clarity. The bottom plot shows the shared potential $\phi_{\beta}$ , with individual filter potentials obtained by rescaling this shared potential with the learned coefficients $\alpha_{i}$ while preserving the same shape. The localized positive and negative filter weights and the symmetric potential support the interpretation of WCRR as a learned sparsity prior.

Figure S.1.9: Sensitivity of the parameter $\sigma$ on a representative WCRR reconstruction (with fixed $\lambda=0.01$ ). Only a transversal slice is displayed.

Figure S.1.10: Sensitivity of the parameter $\lambda$ on a representative WCRR reconstruction (with fixed $\sigma=0.03$ ). Only a transversal slice is displayed.

Figure S.2.11: Quantitative metrics (averaged across the retrospective 12-coil testing set) vs training set size, for the end-to-end methods (DRUNet3D and NC-PDNet) and WCRR.

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Learning spatially adaptive sparsity level maps for arbitrary convolutional dictionaries

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State-of-the-art learned reconstruction methods often rely on black-box modules that, despite their strong performance, raise questions about their interpretability and robustness. Here, we build on a recently proposed image reconstruction method, which is based on embedding data-driven information into a model-based convolutional dictionary regularization via neural network-inferred spatially adaptive sparsity level maps. By means of improved network design and dedicated training strategies, we extend the method to achieve filter-permutation invariance as well as the possibility to change the convolutional dictionary at inference time. We apply our method to low-field MRI and compare it to several other recent deep learning-based methods, also on in vivo data, where the benefit of using a different dictionary is demonstrated. We further assess the method's robustness when tested on in- and out-of-distribution data. When tested on the latter, the proposed method suffers less from the data distribution shift compared to the other learned methods, which we attribute to its reduced reliance on training data due to its underlying model-based reconstruction component.

Abstract

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