Moner: Motion Correction in Undersampled Radial MRI with Unsupervised Neural Representation

Thank you for the valuable comments. We are encouraged by your positive feedback. Below, we provide detailed point-by-point responses to address your concerns.

W1: It is not clear how the motion trajectories, e.g., in Figure 7 are simulated – more information should be provided. Are there fixed intervals after which motion changes direction? In between those events does every spoke has its own motion state meaning that 300 spokes result intro estimating 300*3=900 unknowns? How does this model change when the number of spokes changes due to a different acceleration factor?

A: Our motion simulation is based on the concept of motion stages [1][2]. Specifically, all spokes are divided into $N$ motion stages, where spokes in the same motion stage share identical motion trajectories. Our Moner model assigns 3 motion parameters per stage and thus totally estimates $N\times$3 unknowns. In our experiment, we use 360 and 180 spokes for acceleration factors (AFs) of 2$\times$ and 4$\times$, respectively. These spokes are divided into $N=18$ motion stages, resulting in the estimation of $18×3=54$ unknowns. We have added this information in the revised submission (Sec. 4.1).

Additionally, we evaluate our method across different numbers of motion stages ($N=\{18, 72\}$), corresponding to $18\times 3=54, 72\times 3=216$ unknowns, and simulate different types of motion trajectories (including random, periodic, and abrupt). Note that all hyperparameters for our model remained consistent with those reported in the original submission.

As shown in Table R1, our Moner achieves excellent and stable reconstruction performance, indicating minor variations in PSNR ($\pm1$ dB) and SSIM ($\pm$0.2), and statistical tests show no significant differences. Qualitative results are provided in Figure R3 of the Supplementary Material, where reconstructed MR images across all conditions appear visually consistent, further confirming the robustness of our method.

Table R1: Quantitative results of our Moner on the fastMRI dataset for AF=2x and MR=±5 when simulating different types of motion trajectories and different numbers of motion stages. Results of t-test based statistical tests comoaring Row #1 (Random & N=18) to other settings are presented by **(p<0.01), *(p<0.05), and ▼ (no significant, p≥0.05).

[1] Spieker, Veronika, et al. "Deep learning for retrospective motion correction in MRI: a comprehensive review." IEEE Transactions on Medical Imaging (2023).

[2] Levac, Brett, et al. "Accelerated motion correction with deep generative diffusion models." Magnetic Resonance in Medicine 92.2 (2024): 853-868.

W2: Section 4.1 does not contain enough information about the dataset used in the experiments. More information should be provided either in the main body or the appendix. 25 test slices is not much depending on how the data is selected. Are the 25 slices the entire slices from e.g. two subjects or are they 25 mid-slices from 25 subjects? The fastMRI dataset is very diverse. Are the subjects for training and testing taken from the same contrasts, e.g., T2 weighted or FLAIR? What is the main difference of the MoDL dataset to the data used from the fastMRI dataset?

A: In our experiment, the training, validation, and test sets of the fastMRI dataset are randomly selected from different subjects along the axial direction to ensure no data leakage between sets. These sets include a diverse combination of three contrasts: T1w, T2w, and FLAIR images. For the MoDL dataset, we used 20 T2w brain MR slices from 20 distinct subjects along the sagittal direction. This difference in orientation (axial vs. sagittal) and the exclusive use of T2w images introduce a distribution shift. We thus believe that, despite the limited number, these slices sufficiently represent the data diversity. We have added the dataset information in the revised submission (Sec. 4.1) to improve clarity. Additionally, we have a plan to evaluate the model on additional slices to further demonstrate the model generalization.

Q2: It would be helpful to add scores to the qualitative comparisons in e.g. Figure 4 to provide a better feeling how differences in the scores translate to perceivable differences.

A: Thanks for your constructive suggestions. We have included the quantitative metrics (PSNR/SSIM) in Figures 4, 5, 6, and 8 of the revised submission.

Q3: Why is the diffusion model baseline categorized as self-supervised e.g. in line 309? As the model requires fully sampled data for training I would consider it as a supervised baseline.

A: Thank you for pointing this out. Unlike the traditional end-to-end learning paradigm, which relies on extensive paired MRI data, diffusion models can effectively capture data distributions using only numerous high-quality MR images. We thus categorize the diffusion-based Score-MoCo as a self-supervised model. However, we also agree with the reviewer that the diffusion model should be classified as a supervised baseline due to its heavy reliance on fully sampled MRI data. In the revised submission, we modify the categorization of Score-MoCo to supervised.

Q4: How many spokes are acquired for the acceleration factors of 2 and 4 respectively?

A: In our experimental setup, each spoke has a length of 511, corresponding to an imaging FOV of 511$\times$511. The fully sampled radial k-space data consists of approximately 720 views. Therefore, for acceleration factors (AFs) of 2 and 4, the number of spokes is 360 and 180, respectively. This information has been included in the revised submission (Sec. 4.1) to improve clarity and reproducibility.

W3: The comparison of the computational speed would benefit from a more detailed discussion. Where do the differences in terms of speed come from? How can the TV method with 200 epochs be slower than the INR with 4000 epochs? What is responsible for the speed difference compared to the diffusion model – the size of the networks or the number of steps? (What is the size of the MLP that is used?)

A: The computational efficiency of our Moner model mainly benefits from two factors:

• Hash Encoding for Compact and Efficient Representations: Hash encoding [3][4] explicitly learns features for each position in the imaging space, significantly enhancing the ability of the subsequent tiny MLP network (a two-layer structure) to fit high-frequency image details. This design accelerates convergence and reduces the computational overhead typically associated with training larger networks for high-resolution image reconstruction.

• Ray Tracing-Based Optimization: Our method employs a ray tracing-based optimization pipeline. Specifically, for each projection $\boldsymbol{g}(\theta_i,\rho)$, multiple coordinates along the corresponding ray $\boldsymbol{r}(\theta_i,\rho)$ are sampled and input into the INR network. In each optimization epoch, our Moner only computes forward and backward passes for these sampled positions, rather than over the entire imaging space (i.e., FOV) as in TV and Score-MoCo. Consequently, TV with 200 epochs is slower than our Moner with 4000 epochs. Additionally, Score-MoCo relies on a deep U-Net or Transformer to iteratively denoise the noisy image, and backward optimization for both motion parameters and MR images is performed in each iteration. These factors significantly slow down its optimization.

[3] Müller, Thomas, et al. "Instant neural graphics primitives with a multiresolution hash encoding." ACM transactions on graphics (TOG) 41.4 (2022): 1-15.

[4] Barron, Jonathan T., et al. "Zip-nerf: Anti-aliased grid-based neural radiance fields." Proceedings of the IEEE/CVF International Conference on Computer Vision. 2023.

Q1: Is formulating the loss in the projection space specific to the radial sampling trajectory or can it also be used for e.g. Cartesian sampling as well?

A: Radial sampling benefits from each spoke passing through the center of Fourier frequency space, where the 1D inverse Fourier transform of a spoke corresponds to an integral projection. In contrast, Cartesian sampling acquires k-space data on a regular grid, with each spoke parallel to the readout axis (i.e., k_x) and adjacent spokes separated by a fixed interval along the phase encoding axis (i.e., k_y). According to Fourier transform theory, translating a signal in k-space induces a phase shift in the spatial domain.

Technically, to extend our Moner model to Cartesian sampling, we propose applying a phase shift to the network-predicted MR image before integrating it into the projection forward model. However, Cartesian sampling lacks the redundant low-frequency information inherent in radial sampling (i.e., central k-space coverage by spokes), which could decrease optimization stability. Therefore, Cartesian sampling with specifically designed trajectories to simultaneously scan high-frequency and low-frequency spokes [5][6] may be required. Moreover, adopting an alternating optimization strategy could help enhance model convergence and performance under Cartesian sampling patterns.

[5] Levac, Brett, Ajil Jalal, and Jonathan I. Tamir. "Accelerated motion correction for MRI using score-based generative models." 2023 IEEE 20th International Symposium on Biomedical Imaging (ISBI). IEEE, 2023.

[6] Levac, Brett, et al. "Accelerated motion correction with deep generative diffusion models." Magnetic Resonance in Medicine 92.2 (2024): 853-868.

Thank you for the insigthful rebuttal. The additional experiments and information provided look good. I raised my score accordingly.

Thank you for taking the time to review our work. We greatly appreciate your positive feedback. Below, we provide detailed point-by-point responses to address your concerns.

W1: Experiments were only conducted on brain MRI datasets with similar contrast/MR sequence. Readers would be interested in how good the proposed method is on MRI images of different contrasts or taken at different body parts. Specifically, will the optimization still converge stably with the same optimization strategy (e.g. hash encoding) and hyperparameter choice? In addition, how does noise level affect the optimization process?

A: Thank you for the insightful comment. In our experiments, the fastMRI dataset includes diverse contrasts, such as T1w, T2w, and FLAIR images, as shown in Figures 4 and 13. To improve clarity, this information has been explicitly added to the revised manuscript (Sec. 4.1).

We agree that experiments using different body parts are essential for assessing model generalization. To address this, we additionally test our method on 10 knee T2w MR images from the fastMRI dataset. Gaussian noises (SNR=40 dB) are also simulated in the k-space data. Importantly, all hyperparameters and optimization strategies, including Coarse2fine hash encoding, were consistent with those used in our original experiments.

As shown in Table R1, our Moner achieves consistent reconstruction quality on this new dataset, confirming its robustness. Qualitative results are provided in Figure R1 of the Supplementary Material, further supporting the stability and effectiveness of our method.

Table R1: Quantitative results (PSNR/SSIM) of NuIFFT and our Moner on 10 knee samples of the fastMRI dataset for MR=±10 and AF=2x. Results of t-test based statistical tests comparing our Model to NuIFFT are presented by ** (p<0.01), *(p<0.05), and ▼ (no significant, p≥0.05).

W2: How will trans-plane motion affect the stability of optimization?

A: Thank you for the question. Our current Moner model is designed for 2D MRI motion correction (MoCo) and does not account for trans-plane motion in 3D space. To address this limitation, we extended Moner to a 3D version for 3D MRI MoCo. This extension explicitly models motion across all three spatial dimensions. Experimental results, provided in the revised submission (Sec. A.1), demonstrate that 3D Moner achieves consistent and accurate motion-corrected reconstructions, highlighting its robustness in handling trans-plane motion.

W3: In Fig. 8, I notice that the reconstructed image looks blurry and lacks some thin structures like vessels. In other figures, images are too small to observe such details. I suggest authors show zoomed-in images from both baselines and the proposed method for better comparison.

A: Thanks for your constructive suggestions. We have included zoomed-in images for all comparison methods in Figure 4 to provide a more detailed comparison. In Figure 8, our focus is to show the influence of the Coarse2fine hash encoding strategy on the performance of our model. There are two main reasons for the ‘looks blurry’ mentioned by the reviewers:

• Our experimental setup involves undersampling (AF = 2$\times$ in Figure 8), so the proposed unsupervised method, Moner, faces challenges not only from motion artifacts but also from the undersampling issue.

• Since the reconstructed images and GTs may not be in the same space, we perform the rigid registration to align the reconstructions to GTs before calculating quantitative metrics. While the registration involves interpolation operators (e.g., linear interpolation) and thus may slightly cause blurring effects.

W4: Authors may want to report FOV and pixel spacing to let readers get a sense of the magnitude of motion ranges used in this paper.

A: In our experiment, the imaging FOV is 511$\times$511 mm$^2$, and the pixel spacing is 1$\times$1 mm$^2$. This information has been added to the revised submission (Sec. 4.1) to improve clarity and provide context for the motion ranges.

Q1: Could the authors elaborate on potential strategies to adapt Moner for 3D MRI data or non-radial sampling patterns?

A: Please refer to our response to W1 for additional details on this topic.

Q2: How might Moner’s performance change if the motion were non-rigid or occurred within acquisition frames?

A: Thank you for the insightful question. The proposed Moner currently employs a quasi-static motion model, which operates under two key assumptions: 1) the motion is rigid, and 2) motion occurs only between spokes, while remaining static within each spoke. Consequently, the current version of Moner cannot directly address non-rigid motion or motion occurring within acquisition frames.

However, the quasi-static model remains both reasonable and practically significant for many clinical applications. Radial MRI scans acquire a single spoke in just a few milliseconds, during which motion is generally negligible, and rigid motion is common in clinical MRI scans of the brain and legs.

To handle non-rigid motion, Moner could be extended by incorporating non-rigid motion models. For instance, in 4D dynamic CT imaging, previous work has introduced deformation fields to address non-rigid motion [3]. Similarly, Moner could be enhanced by integrating deformation field modeling into the optimization process of the MLP network. This approach could allow Moner to estimate and compensate for more complex motion patterns during reconstruction.

[3] Reed, Albert W., et al. "Dynamic ct reconstruction from limited views with implicit neural representations and parametric motion fields." Proceedings of the IEEE/CVF International Conference on Computer Vision. 2021.

Q3: What specific benefits does the coarse-to-fine hash encoding offer over traditional encoding techniques for motion correction?

A: Hash encoding assigns learnable features to each position in the imaging space, unlike traditional encoding techniques such as Fourier or position encoding, which rely on pre-defined mathematical transformations. This adaptiveness allows hash encoding to use a compact MLP (typically 2–4 layers) to approximate high-frequency signals effectively, enabling faster convergence and improved reconstruction quality.

However, traditional hash encoding’s strong ability to fit high-frequency signals can sometimes lead to overfitting, negatively affecting motion correction (MoCo) performance. To address this, we introduce a coarse-to-fine strategy that balances detail preservation with MoCo robustness. The strategy starts with a low-frequency approximation in the early stages and progressively refines the reconstruction, allowing better alignment between motion modeling and image reconstruction.

As shown in Table R2, the coarse2fine hash encoding achieves the best performances in both image reconstruction quality and optimization speed. Figure R2 of the Supplementary Material shows the quantitative results. These results demonstrate that advanced encoding schemes, particularly coarse-to-fine hash encoding, can significantly enhance both image reconstruction and motion correction in MRI.

Table R2: Qualitative results of our Moner with two different encoding modules on the fastMRI dataset for AF=2x and MR=±5.

We sincerely appreciate your time and effort in reviewing our work. Your constructive feedback motivates us to further improve our study. Below, we address your comments in detail.

W1: Extension to 3D and Non-Radial Sampling Patterns.

A1 (Extention to 3D Radial MRI): We have extended the proposed Moner to the 3D stack-of-radial sampling pattern. In 3D radial MRI, the 2D inverse Fourier transform of a spoke array from a specific view corresponds to its respective 2D projections, enabling the extension of Moner to address the problem of parallel back-projection in 3D space.

To validate this extension, we conducted simulation experiments on a 3D brain MRI volume with dimensions of 240$\times$240$\times$240. As shown in Table R1, our Moner achieves reconstruction quality consistent with the reference images, demonstrating its feasibility and effectiveness for 3D radial MRI. For a more detailed discussion on the extension to 3D radial MRI, please refer to the revised submission (Sec. A.1).

Table R1: Quantitative results of 3D MR image by NuIFFT and our Moner the 3D brain data. Results of t-test statistical tests comparing our Moner to NuIFFT are denoted by $^{**}$ ($p$-value $<$ 0.01), $^{*}$ ($p$-value $<$ 0.05), and $^\blacktriangledown$ (not significant, $p$-value $\ge$ 0.05).

A2 (Non-radial Sampling Patterns, e.g., Cartesian sampling): Our Moner is originally designed for radial MRI, as each spoke in radial sampling passes through the center of k-space. The 1D inverse Fourier transform of a spoke corresponds to a projection integral, allowing the reconstruction of radial MRI to be reformulated as a back-projection problem. In contrast, Cartesian sampling acquires k-space data on a regular grid, where each spoke is parallel to the readout axis (i.e., k_x), and adjacent spokes are separated by fixed intervals along the phase-encoding axis (i.e., k_y). According to Fourier transform theory, a translation in the frequency domain results in a phase shift in the spatial domain. This makes it challenging to apply the current Moner directly to Cartesian sampling.

Technically, one approach to extending Moner to Cartesian sampling is to apply a phase shift to the MR image predicted by the MLP network before integrating it into the projection-based forward model. However, Cartesian sampling lacks the inherent redundancy of low-frequency information present in radial sampling, which could reduce the efficiency and stability of the model's optimization. Therefore, Cartesian sampling with specifically designed trajectories to simultaneously scan high-frequency and low-frequency spokes [1][2] may be required. Moreover, adopting an alternating optimization strategy could help enhance model convergence and performance under Cartesian sampling patterns.

[1] Levac, Brett, Ajil Jalal, and Jonathan I. Tamir. "Accelerated motion correction for MRI using score-based generative models." 2023 IEEE 20th International Symposium on Biomedical Imaging (ISBI). IEEE, 2023.

[2] Levac, Brett, et al. "Accelerated motion correction with deep generative diffusion models." Magnetic Resonance in Medicine 92.2 (2024): 853-868.

W2: Reliance on Motion Assumptions: The quasi-static motion model assumes rigid motion between acquisition frames. Further clarification on its limitations in more complex motion settings could strengthen the study.

A: The quasi-static motion model assumes rigid motion between acquisition frames and is reasonable and practical for most of the clinical applications. However, this model does have some limitations, particularly in its inability to simulate some complex motion situations, including motion within acquisition frames, non-rigid motion, or complicated spin-history effects. Fortunately, there are some potential solutions to these limitations. For motion within acquisition frames, since radial MRI scans take only a few milliseconds to acquire a single spoke, intra-frame motion is usually negligible. For non-rigid motion, we can use deformation field modelling and incorporate it into the optimization process of the Moner network.

We have added this discussion in the revised submission (Sec. 5) as below:

Secondly, the quasi-static motion model assumes rigid motion between acquisition frames, but it falls short in simulating certain complex motion scenarios. These include motion occurring within acquisition frames, non-rigid motion, and intricate spin-history effects. ... Also, for non-rigid motion, Moner can be extended by incorporating deformation field modeling (Reed et al., 2021).

Thank you for the valuable addition of a in-vivo motion corrupted case. You mention how subjects were instructed to move. Could you comment on the estimated motion parameters for the motion-corrected reconstruction and if this aligns with your original motion corruption setup?

W3: Within the results, "significant improvement" (l., 365/l.431) is often claimed, the standard deviation (e.g. in table 2) and statistical tests should be reported to support this statement. Similarly for the improvement claim regarding Table 4: Statistical testing would be helpful, since the standard deviation reported opens up questions the actual improvement

A: Thank you for your constructive suggestions. In our original submission, only mean PSNR and SSIM values were reported due to space limitations. In the revised submission, we now provide comprehensive results, including Mean$\pm$STD and statistical t-tests, in Tables 2 and 7 of the revised submission. The key observations are as follows:

• Compared to NuIFFT, TV, and DRN-DCMB, our model shows statistically significant improvements across all cases ($p<0.01$).

• On the fastMRI dataset, the current SOTA method, Score-MoCo (pre-trained on fastMRI), slightly outperforms Moner (by approximately 1 dB in PSNR), but the improvement is not statistically significant ($p≥0.05$).

• On the out-of-domain (OOD) MoDL dataset, our unsupervised Moner achieves steady and statistically significant improvements ($p<0.05$) over Score-MoCo in most cases.

For our ablation studies, the statistical t-tests in Tables 3, 4, and 5 of the revised submission reveal the following:

• The motion model is a critical component, yielding significant improvements ($p<0.01$).

• The projection-based optimization significantly enhances MoCo accuracy ($p<0.01$) and moderately improves MRI reconstruction ($p≥0.05$).

• The coarse-to-fine strategy plays an essential role in preventing MoCo degradation from overfitting to high-frequency details.

These statistical analyses and t-tests further validate the superiority of our Moner and underline the importance of its key components. We appreciate your feedback, which has helped us enhance the clarity and rigor of our results.

W4: For full reproducibility, it would be nice to mention all post-processing steps conducted before quantitative evaluation (nect to registration mentioned in A1, e.g. if normalization was conducted, etc.?)

A: Thank you for the suggestion. In our workflow, post-processing is limited to the registration step mentioned in Sec. A1. While during data pre-processing, the GT images are normalized to the range [0, 1] before the data simulation process. These additional details have been included in the revised manuscript (Sec. 4.1) to enhance reproducibility and clarity.

W5: There are several typos which can be corrected (line326, 504,...)

A: Thank you for your thorough proofreading. We have thoroughly reviewed the manuscript and corrected all identified typos, including those on lines 326 and 504, in the revised submission. We appreciate your attention to detail, which helps improve the overall quality of our work.

Thank you for the insightful comments. We are encouraged by your recognition of our work. Below, we provide detailed point-by-point responses to address your concerns.

W1: The evaluation is exclusively conducted on simulated motion-corrupted data, which raises the question of its actual applciability in real settings. Particularly considering that motion artefacts not only arise from the physical movement, but also from resulting field inhomogenieties, etc., the simulation procedure might be too simplified and the model is likely to simply adapt to these parameters. If the work is aimed at MR motion-correction methods within a clinical workflow, testing the method on in-vivo data is mandatory.

A1 (Evaluation on Real-world Data): To evaluate the effectiveness of our Moner on real-world MRI data, we collect two T1w brain 3D scans from a single subject using a 3.0 T United Imaging Healthcare (UIH) uMR 790 scanner. To obtain motion-free data and motion-corrupted data, the subject is instructed to remain still or make abrupt movements three times during acquisition. The data acquisition is approved by the institutional review board. As shown in Figure 12 in the revised submission, our Moner reconstruction closely resembles the reference in global structures and successfully recovers tissue details. This study preliminarily validates the effectiveness of our method. We have included this study on real-world data in the revised submission (Sec. A.1).

A2 (Modeling for Non-movement Factors): MRI acquisition is a complicated process. We agree with the reviewer that many non-movement factors, such as field inhomogeneities, could affect the quality of reconstructed images. However, this work aims to develop a new method to address the MRI MoCo problem by correcting the subject's movements. We leave these modeling of non-movement factors in our further work.

W2: In this work, it is claimed multiple times, that the Fourier slice theorem is a crucial part of their work and this works novelty. Yet, the Fourier slice theorem was already introduced a while ago (Catalan 2023 et al, https://doi.org/10.48550/arXiv.2307.14363) and used in several works after. While it is definitely aiding the method, it should be cited and not included as novelty.

A: Thank you for the insightful comment. We acknowledge that the Fourier slice theorem has been introduced in prior works, such as NF-cMRI (Catalan et al., 2023), which reconstructs high-quality cardiac MR images by optimizing a neural field from undersampled radial k-space data. In their work, the Fourier slice theorem is used to transform predicted MR images from the spatial domain into the k-space domain, where the model is optimized.

In contrast, our Moner utilizes the Fourier slice theorem to directly optimize in the projection domain, specifically to address the high-dynamic range problem inherent to k-space data. This approach differs fundamentally in motivation and application. To our knowledge, leveraging the Fourier slice theorem in this manner to overcome the high-dynamic range issue has not been explored in existing literature. As demonstrated in our ablation study, the projection-based optimization significantly enhances MoCo accuracy ($p<0.01$) and moderately improves MRI reconstruction ($p≥0.05$), further validating its effectiveness.

We have cited Catalan et al. (2023) as related work on neural fields for MRI reconstruction in the revised manuscript (line 147) to ensure proper attribution and clarity.