Self-Supervised Diffusion MRI Denoising via Iterative and Stable Refinement

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Thank you for your constructive feedback.

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Q1: The baselines chosen do not include the most widely used denoising methods. A clear omission is the random-matrix theory approaches proposed by Veraart et al in a series of very highly cited papers starting with Neuroimage 2016.

A1: We will include MPPCA (Neuroimage2016) [1], Noise2Score (N2S, NeurIPS 2021) [3], Recorrupted2Recorrupted (R2R, CVPR 2021) [4], and Patch2Self2 (P2S2, CVPR 2024) [2] for further comparisons on tractography, microstructure model fitting, diffusion signal estimates, CNR and SNR metrics. Due to page limitations, we will include the quantitative results and its visualizations in the appendix. Here is an abstract of the quantitative results:

Table 1. ↑$R^2$ of microstructure model fitting on CSD & DTI. Bold and italic fonts denote the best and the second-best performance, respectively. As measured by $R^2$, Di-Fusion achieves the best results across all four different settings.

Table 2. Comparison of SNR and CNR. Bold and italic fonts denote the best and the second-best performance, respectively.

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Q2: The only quantitative results use simulations, which seem likely to be skewed towards to capabilities of the proposed algorithm.

A2: Thank you for your valuable feedback. In Patch2Self [2], Microstructure Model Fitting is evaluated using quantitative results (i.e., the $R^2$ metric in Figure 4 of our submission). In DDM2 [3], CNR and SNR metrics are considered quantitative results (in Section 4.3 "Quantitative results on SNR/CNR metrics" of our submission), while Fractional Anisotropy (in Section 4.2 "Effect on diffusion signal estimates") is regarded as both quantitative and clinically relevant. We will further emphasize the quantitative evaluation of these tasks in the main text to enhance clarity or directly include the tables as shown in A1.

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Q3: The qualitative results on actual human data are questionable as to whether they show improvement over baselines. Even if they do, these are single cherry-picked examples and it is not clear whether these are advantages that manifest over large collections of images/scenarios.

A3: We include additional qualitative results in the Appendix and directly refer to the correspoding Figure in the main paper. We believe that this would provide a more complete picture of the method's performance. Regarding performance over large collections of images or scenarios, the effectiveness of our method can be demonstrated using the quantitative results of microstructure model fitting, as well as CNR and SNR metrics.

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References:

[1] Veraart J, Novikov D S, Christiaens D, et al. Denoising of diffusion MRI using random matrix theory[J]. Neuroimage, 2016, 142: 394-406.

[2] Fadnavis S, Batson J, Garyfallidis E. Patch2Self: Denoising Diffusion MRI with Self-Supervised Learning​[J]. Advances in Neural Information Processing Systems, 2020, 33: 16293-16303.

[3] Xiang T, Yurt M, Syed A B, et al. DDM $^2$: Self-Supervised Diffusion MRI Denoising with Generative Diffusion Models[C]//The Eleventh International Conference on Learning Representations.

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Thank you for your valuable feedback.

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Q1: Refrain from large claims.

A1: We will check the claims carefully in the revised version (Specifically, "outperforms other methods" to "achieves SOTA performance in microstructure modeling, tractography tracking, and various other downstream tasks", etc.).

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Q2: Report time and memory usage, setup (GPU types, numbers and VRAM).

A2: GPU types: RTX GeForce 3090; numbers of GPUs: 1; VRAM: 5578MB. The training duration for one ${\mathcal{F}_\theta }$ is approximately five hours on a single RTX GeForce 3090 GPU. These details will be reported in the "Training details" section of "Experiment and Reproducibility Details" in Appendix D.

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Q3: Check statements.

A3: We will carefully review the statements in the updated version to ensure clarity.

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Q4: More comparisons.

A4: We will include MPPCA (Neuroimage2016) [1], Noise2Score (N2S, NeurIPS 2021) [3], Recorrupted2Recorrupted (R2R, CVPR 2021) [4], and Patch2Self2 (P2S2, CVPR 2024) [2] for further comparisons on tractography, microstructure model fitting, diffusion signal estimates, CNR and SNR metrics. Due to page limitations, we will include the quantitative results and its visualizations in the appendix. Here is an abstract of the quantitative results:

Table 1. ↑$R^2$ of microstructure model fitting on CSD & DTI. Bold and italic fonts denote the best and the second-best performance, respectively. As measured by $R^2$, Di-Fusion achieves the best results across all four different settings.

Table 2. Comparison of SNR and CNR. Bold and italic fonts denote the best and the second-best performance, respectively.

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Q5: What is the actual minimum number of B0 and DWI volumes required ? Can this work with data that have a single B0? Would that be denoised?

A5: What is the actual minimum number of B0 and DWI volumes required ? For B0 volumes, our method does not require them to perform denoising. This is evident in our implementation (config/hardi_150.json), where "valid_mask": [10,160] filters out B0 volumes. Regarding the minimum number of DWI volumes required, this can be considered from two perspectives:

1. Total volumes: We conducted an experiment using only 30 DWI volumes (Stanford HARDI dataset), and Di-Fusion was still able to perform denoising effectively under this condition. Qualitative results are presented in the Appendix of our revised version.
2. Input volumes: This is fixed to 1 in our implementation.

Can this work with data that have a single B0? Would that be denoised? Our method does not require B0 volumes for denoising. Without B0 volumes, Di-Fusion is still capable of performing denoising effectively.

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References:

[1] Veraart J, Novikov D S, Christiaens D, et al. Denoising of diffusion MRI using random matrix theory[J]. Neuroimage, 2016, 142: 394-406.

[2] Fadnavis S, Chowdhury A, Batson J, et al. Patch2Self2: Self-supervised Denoising on Coresets via Matrix Sketching[C]//Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024: 27641-27651.

[3] Kim K, Ye J C. Noise2score: tweedie’s approach to self-supervised image denoising without clean images[J]. Advances in Neural Information Processing Systems, 2021, 34: 864-874.

[4] Pang T, Zheng H, Quan Y, et al. Recorrupted-to-recorrupted: Unsupervised deep learning for image denoising[C]//Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2021: 2043-2052.

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Q2: Better signpost the extensive results in the supplementary materials.

A2: In the main paper, we directly refer to the corresponding images within the main text and provide detailed descriptions in the figure captions. Readers can easily access the specified location by clicking on the hyperlinks. We will further check to ensure that these tables and figures are properly referenced via hyperlinks to assist readers in accessing them quickly.

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Q3: Some parts of the paper read a bit odd and should be checked for oddities e.g. from the introduction 'The MRI, including ...', 'Consequently, the denoising technique plays a crucial role..'

A3: Thank you for your valuable feedback. We have reviewed the phrasing in the introduction and refined the sentences to improve readability. Specifically, for these two parts 'The MRI, including ...' is changed to 'Magnetic Resonance Imaging (MRI), including ...'; 'Consequently, the denoising technique plays a crucial role..' is changed to 'Consequently, the denoising technique is a vital processing step'. We will review the paper again to improve clarity.

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References:

[1] Pan S, Chang C W, Peng J, et al. Cycle-guided denoising diffusion probability model for 3d cross-modality mri synthesis[J]. arXiv preprint arXiv:2305.00042, 2023.

[2] Alkhouri I, Liang S, Wang R, et al. Robust Physics-based Deep MRI Reconstruction Via Diffusion Purification[J]. arXiv e-prints, 2023: arXiv: 2309.05794.

[3] Zbontar J, Knoll F, Sriram A, et al. fastMRI: An open dataset and benchmarks for accelerated MRI[J]. arXiv preprint arXiv:1811.08839, 2018.

[4] Kim K, Ye J C. Noise2score: tweedie’s approach to self-supervised image denoising without clean images[J]. Advances in Neural Information Processing Systems, 2021, 34: 864-874.

[5] Pang T, Zheng H, Quan Y, et al. Recorrupted-to-recorrupted: Unsupervised deep learning for image denoising[C]//Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2021: 2043-2052.

[6] Veraart J, Novikov D S, Christiaens D, et al. Denoising of diffusion MRI using random matrix theory[J]. Neuroimage, 2016, 142: 394-406.

[7] Fadnavis S, Chowdhury A, Batson J, et al. Patch2Self2: Self-supervised Denoising on Coresets via Matrix Sketching[C]//Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024: 27641-27651.

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Thank you for your valuable feedback.

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Q1: Did the authors consider using: https://arxiv.org/pdf/2305.00042 and https://arxiv.org/pdf/2309.05794 as baselines?

A1: https://arxiv.org/pdf/2305.00042(Cycle-guided Denoising Diffusion Probability Model for 3D Cross-modality MRI Synthesis) was not considered as baselines because: 1. They didn't release the code in their official repository[EmoryDeepBiomedicalImagingLaboratory/Cycle-guided-diffusion-model]. Our attempt to replicate their approach might lead to discrepancies in experimental details. 2. Their approach does not inherently support dMRI denoising [1], and adapting it to this context might lead to suboptimal performance or require significant modifications, which could misrepresent the true effectiveness of their method. We will add these discussion with related work in updated version.

https://arxiv.org/pdf/2309.05794(Robust Physics-based Deep MRI Reconstruction Via Diffusion Purification) proposes an interesting pipeline for MRI reconstruction (During the Diffusion Stage, noise is added to the deconstructed image to simulate the corruption process. In the subsequent Purification Stage, sampling is guided by data consistency constraints to iteratively refine the image. Finally, the refined output is passed through $MoDL_{\theta_{\rm FT}}$, which produces the final reconstruction result.). However, $MoDL_{\theta_{\rm FT}}$ was fine-tuned using 3000 purified images ("We note that the DM model was trained on the knee training dataset. We conduct our experiments on the fast MRI [3] dataset, using 3000 purified images for fine-tuning the pre-trained MoDL network." in 'Experimental Setup' section of [2]), "purified" images here are ground truth images in the fast MRI dataset. In dMRI, we don't have the "purified" images that could used to train $MoDL_{\theta_{\rm FT}}$. Directly adapting it to dMRI might lead to suboptimal performance or require significant modifications, which could misrepresent the true effectiveness of their method. We will add these discussion with related work in updated version.

We will include MPPCA (Neuroimage2016) [6], Noise2Score (NeurIPS 2021) [4], Recorrupted2Recorrupted (CVPR 2021) [5], and Patch2Self2 (CVPR 2024) [7] for further comparisons on tractography, microstructure model fitting, diffusion signal estimates, CNR and SNR metrics. Due to page limitations, we will include the quantitative results and its visualizations in the appendix. Here is an abstract of the quantitative results:

Table 1. ↑$R^2$ of microstructure model fitting on CSD & DTI. Bold and italic fonts denote the best and the second-best performance, respectively. As measured by $R^2$, Di-Fusion achieves the best results across all four different settings.

Table 2. Comparison of SNR and CNR. Bold and italic fonts denote the best and the second-best performance, respectively.

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Thank you for your valuable feedback.

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Q1: To my understanding, the primary goal of dMRI denoising is to reduce the number of gradients required during acquisition, thus accelerating DWI scanning. In downstream tasks based on DTI, the authors compare DTI metrics computed from noisy images with those from denoised images. Why did the authors not use more DWI data to compute a clean DTI metric as a reference for comparison?

A1: We did not use more DWI data to compute a clean DTI metric as a reference for comparison in our paper because

1. This was not used in prior studies [1-3]. To ensure a fair comparison of the performance across methods, we directly followed these metrics without modification;
2. In DTI, the characteristics of noise are relatively complex, as the noise in images exhibits significantly different properties. Therefore, directly averaging these images may introduce additional errors. This is likely one of the reasons why previous studies did not adopt this metric.

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References:

[1] Veraart J, Novikov D S, Christiaens D, et al. Denoising of diffusion MRI using random matrix theory[J]. Neuroimage, 2016, 142: 394-406.

[2] Fadnavis S, Chowdhury A, Batson J, et al. Patch2Self2: Self-supervised Denoising on Coresets via Matrix Sketching[C]//Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024: 27641-27651.

[3] Fadnavis S, Batson J, Garyfallidis E. Patch2Self: Denoising Diffusion MRI with Self-Supervised Learning​[J]. Advances in Neural Information Processing Systems, 2020, 33: 16293-16303.

In fact, using more DWIs data to fit DTI as a reference is a reasonable and widely adopted approach [1-3]. This is intuitive—if no denoising algorithm is available, acquiring a large number of scans can improve the accuracy of tensor signal fitting. I will withhold my score for now.

Reference

[1] Li, Zihan, et al. "DIMOND: DIffusion Model OptimizatioN with Deep Learning." Advanced Science (2024): 2307965.

[2] Li, Hongyu, et al. "SuperDTI: Ultrafast DTI and fiber tractography with deep learning." Magnetic resonance in medicine 86.6 (2021): 3334-3347.

[3] McNab, Jennifer A., et al. "Surface based analysis of diffusion orientation for identifying architectonic domains in the in vivo human cortex." Neuroimage 69 (2013): 87-100.

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Thank you for the review and feedback.

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Answering the questions:

Q1: In Eq. (5) on Line 155, the authors highlighted a specific term as the ”major difference” between xt−1 and x^bar_t−1. Could the authors clarify why this particular term is considered the primary source of difference? Furthermore, can the authors elaborate on the underlying reason(s) for the “drift” in the model and how it emerges during the reverse diffusion process?

A1: Intuitively, through the Fusion process, we gradually feed the target denoising slice to the model, guiding it to optimize towards a fixed direction during the training phase, thereby mitigating the drift.

In equation form, The training objective without the Fusion process is:

${L _ {{\rm{simple}}}}(\theta ): = {{\mathbb{E}} _ {t,x',{\xi _ {x - x'}}}}\left[ {{{\left\| {x - {\cal{F} _ \theta }(\sqrt {{{\bar \alpha } _ t}} x' + \sqrt {1 - {{\bar \alpha } _ t}} {\xi _ {x - x'}},t)} \right\|}^2}} \right]$

The sampling start from $x _ {T _ 1}=\sqrt {{{\bar \alpha } _ {T _ 1}}} x' + \sqrt {1 - {{\bar \alpha } _ {T _ 1}}} {\xi _ {x - x'}}$ (the reverse process serves as the mapping from $x'$ to $x$), the reverse process is:

$x _ {{T _ 1}-1}=\frac{{\sqrt {{{\bar \alpha } _ {{T _ 1} - 1}}}{\beta _ {T _ 1}}}}{{1 - {{\bar \alpha } _ {T _ 1}}}}{{{\cal F} _ \theta }\left( {x _ {T _ 1},{T _ 1}} \right)} + \frac{{\sqrt {{\alpha _ {T _ 1}}} \left( {1 - {{\bar \alpha } _ {{T _ 1} - 1}}} \right)}}{{1 - {{\bar \alpha } _ {T _ 1}}}}{x _ {T _ 1}} + {\sigma _ {T _ 1}}z$

This contradicts the training objective. During training, the ${T _ 1}-1$ step is trained with

${\left\| {x - {\cal{F} _ \theta }(\sqrt {{{\bar \alpha } _ {{T _ 1}-1}}} x' + \sqrt {1 - {{\bar \alpha } _ {{T _ 1}-1}}} {\xi _ {x - x'}},{{T _ 1}-1})} \right\|}^2$

If we still feed ${{\cal F} _ \theta }$ with $x _ {{T _ 1}-1}$ and ${T _ 1}-1$, which is different from the training objective, i.e.,

${x _ {{T _ 1} - 1}} = \underbrace {\frac{{\sqrt {{{\bar \alpha } _ {{T _ 1} - 1}}} {\beta _ t}}}{{1 - {{\bar \alpha } _ {T _ 1}}}}{{{\cal F} _ \theta }\left( {{x _ {T _ 1}},{T _ 1}} \right)}} _ {{\rm{major}}{\kern 1pt} {\rm{difference}}} + \frac{{\sqrt {{\alpha _ {T _ 1}}} \left( {1 - {{\bar \alpha } _ {{T _ 1} - 1}}} \right)}}{{1 - {{\bar \alpha } _ {T _ 1}}}}{x _ {T _ 1}} + {\sigma _ {T _ 1}}z \ne {{\bar x} _ {{T _ 1} - 1}} = \underbrace{\sqrt {{{\bar \alpha } _ {{T _ 1} - 1}}} x' + \sqrt {1 - {{\bar \alpha } _ {{T _ 1} - 1}}} z} _ {\rm training \ objective \ w/o \ Fusion}$

We assume that ${x _ {T _ 1}}$ is a noisy version of $x'$ (close to $x'$), and the output of $\cal F _ \theta$ approximates $x$. The reverse diffusion process gradually maps $x'$ to $x$, so the training objective should involve a combination of the two endpoints and noise, similar to DDBM [8].

In summary, The "drift" arises because, if we directly feed $x _ {{T _ 1}-1}$ and ${T _ 1}-1$ into ${{\cal F} _ \theta }$, the output would deviate slightly further from $x$. This occurs because during training, ${{\cal F} _ \theta }$ is optimized only with the objective: ${\left\| {x - {\cal{F} _ \theta }(\sqrt {{{\bar \alpha }_{{T _ 1}-1}}} x' + \sqrt {1 - {{\bar \alpha } _ {{T _ 1}-1}}} {\xi _ {x - x'}},{{T _ 1}-1})} \right\|}^2$. Importantly, $x _ {{T _ 1}-1}$ is one step closer to $x$. ($x _ {{T _ 1}-1}= \frac{{\sqrt {{{\bar \alpha } _ {{T _ 1} - 1}}}{\beta _ {T _ 1}}}}{{1 - {{\bar \alpha } _ {T _ 1}}}}{{{\cal F} _ \theta }\left( {x _ {T _ 1},{T _ 1}} \right)}+ \frac{{\sqrt {{\alpha _ {T _ 1}}} \left( {1 - {{\bar \alpha } _ {{T _ 1} - 1}}} \right)}}{{1 - {{\bar \alpha } _ {T _ 1}}}}{x _ {T _ 1}} + {\sigma _ {T _ 1}}z$), rather than simply being a noisy version of $x'$. This drift accumulates over the sampling chain, ultimately leading the result to drift toward another slice.

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Q2: According to the definition of the Fusion process in Eq. (6) and the “forward process” in Eq. (7), it appears that the starting point for the forward process changes based on t, as x_t* is dependent on t. This dependence implies that the Fusion process dynamically adjusts the starting point of the forward process at each step, which is unconventional compared to typical diffusion models. Could the authors clarify the rationale behind this design?

A2: This idea is motivated by Noise2Void [11], Noisier2Noise [10], and Noise-as-Clean [9], which use data augmentation to construct training data. Specifically, Noise2Void achieves this through a blind-spot strategy, while Noise2Noise and Noise-as-Clean add additional noise to the original noisy image. In our approach, we leverage such a dynamic combination (the Fusion process) and continuously varying noise (the "Di-" process) to provide the model with more augmented training data, thereby enhancing its robustness. We will clarify this in revision.

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