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Abstract
This study introduces a novel inversion formula for the multi-coil MRI forward operator applicable to arbitrary sampling trajectories. Traditional MRI reconstruction leverages fast Fourier transforms (FFTs) for Cartesian sampling and nonuniform FFTs for non-Cartesian patterns. However, subsampled k-space reconstruction typically relies on iterative least-squares (LS) solutions, which are computationally intensive due to the complex structure introduced by multiple coil sensitivities. We hypothesize that the MRI multi-coil forward operator exhibits the low displacement rank (LDR) property, enabling an efficient inversion using triangular Toeplitz operators with a computational complexity of $\mathcal{O}(\alpha N \log^2 N)$, with $\alpha$ being a small integer. The hypothesis is supported through numerical simulations. For demonstration of the feasibility of such inversion formula, we propose a learning-based approach to determine the necessary LDR parameters, demonstrating successful forward and inverse operator representations across various sampling patterns, including Cartesian and radial trajectories. The proposed inversion formula offers a significant acceleration in MR reconstruction, reducing computational complexity by a factor of approximately 26 compared to conventional conjugate gradient methods. The proposed inversion formula will greatly enhance reconstruction speed and simplify reconstruction pipelines, including iterative reconstructions and deep learning solutions incorporating data-consistency layers. Future work will focus on deriving the LDR parameters analytically to further streamline the inversion process. The code is available at \url{https://github.com/mikecjz/structured-nets}.
Links to Paper and Supplementary Materials
Main Paper (Open Access Version): https://papers.miccai.org/miccai-2025/paper/4944_paper.pdf
SharedIt Link: Not yet available
SpringerLink (DOI): Not yet available
Supplementary Material: Not Submitted
Link to the Code Repository
https://github.com/mikecjz/structured-nets
Link to the Dataset(s)
N/A
BibTex
@InProceedings{CheJun_Direct_MICCAI2025,
author = { Chen, Junzhou and Christodoulou, Anthony G. and Fan, Zhaoyang},
title = { { Direct Inversion Formula of the Multi-coil MR Operator under Arbitrary Trajectories } },
booktitle = {proceedings of Medical Image Computing and Computer Assisted Intervention -- MICCAI 2025},
year = {2025},
publisher = {Springer Nature Switzerland},
volume = {LNCS 15963},
month = {September},
}
Reviews
Review #1
- Please describe the contribution of the paper
The authors attempt to approximate the direct inversion of forward models in under-sampled k-space MRI reconstruction, covering both Cartesian and non-Cartesian sampling schemes. Preliminary results demonstrate the potential of this approach to enable direct inversion, which could significantly reduce the computational cost associated with conjugate gradient (CG) methods when direct inversion is otherwise unavailable.
- Please list the major strengths of the paper: you should highlight a novel formulation, an original way to use data, demonstration of clinical feasibility, a novel application, a particularly strong evaluation, or anything else that is a strong aspect of this work. Please provide details, for instance, if a method is novel, explain what aspect is novel and why this is interesting.
- Solving the direct inversion of the k-space reconstruction forward model is an important yet under-explored question in the field. As an alternative, conjugate gradient (CG) solvers have been widely used to address the inverse problem without explicitly inverting the forward operator.
- The preliminary results are promising in approximating the target using the proposed inverse operator.
- Please list the major weaknesses of the paper. Please provide details: for instance, if you state that a formulation, way of using data, demonstration of clinical feasibility, or application is not novel, then you must provide specific references to prior work.
- It is unclear whether G and H are learned per 2D slice or from an entire 2D dataset. If they are learned per slice, the motivation behind this work is fundamentally flawed, as the proposed low-rank approximated direct inversion would be impractical for real-world applications where learning per slice is infeasible.
- Following up on the previous point, it is also unclear how long it takes to learn G and H. The authors mention “4k epochs for Cartesian undersampling to converge and 24k epochs for the non-Cartesian case,” which appears to be computationally expensive.
- Overall, the work appears quite preliminary, with validation limited to a few numerical simulations on 2D Cartesian and non-Cartesian undersampling. A major motivation of the paper—namely, that the approach “simplifies the integration of data consistency layers in end-to-end deep learning-based MR reconstruction”—is not demonstrated.
- What is the computation time for the 20 CG iterations shown in Figure 3? And how long does LDR take? A direct comparison of computational cost is missing.
- Please rate the clarity and organization of this paper
Good
- Please comment on the reproducibility of the paper. Please be aware that providing code and data is a plus, but not a requirement for acceptance.
The submission does not provide sufficient information for reproducibility.
- Optional: If you have any additional comments to share with the authors, please provide them here. Please also refer to our Reviewer’s guide on what makes a good review and pay specific attention to the different assessment criteria for the different paper categories: https://conferences.miccai.org/2025/en/REVIEWER-GUIDELINES.html
N/A
- Rate the paper on a scale of 1-6, 6 being the strongest (6-4: accept; 3-1: reject). Please use the entire range of the distribution. Spreading the score helps create a distribution for decision-making.
(3) Weak Reject — could be rejected, dependent on rebuttal
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
The paper addresses an important and under-explored problem—approximating direct inversion for under-sampled k-space MRI reconstruction. While the preliminary results are promising and suggest potential computational benefits, the current work lacks clarity on key implementation details, such as how the inversion operators G and H are learned and their practical scalability. The experiments are limited to 2D simulations with no demonstration of integration into end-to-end reconstruction pipelines, which weakens the motivation and broader impact. With further development and validation, the approach may offer valuable contributions.
- Reviewer confidence
Very confident (4)
- [Post rebuttal] After reading the authors’ rebuttal, please state your final opinion of the paper.
N/A
- [Post rebuttal] Please justify your final decision from above.
N/A
Review #2
- Please describe the contribution of the paper
In this paper, an approach for faster direct inversion of L2-based objectives with a multi-coil forward operator is proposed, by exploiting a specific type of matrix structure involving “low displacement rank”. This allows for much faster direct evaluation of the inverse operator. This is evaluated in some simple simulations, and is comparable to 20 CG iterations.
- Please list the major strengths of the paper: you should highlight a novel formulation, an original way to use data, demonstration of clinical feasibility, a novel application, a particularly strong evaluation, or anything else that is a strong aspect of this work. Please provide details, for instance, if a method is novel, explain what aspect is novel and why this is interesting.
A strength of the paper is in its novelty, and identification of a potentially useful matrix structure involved in a lot of L2-based MRI data-consistency objective functions. The potential speedups of more than an order of magnitude are impressive.
- Please list the major weaknesses of the paper. Please provide details: for instance, if you state that a formulation, way of using data, demonstration of clinical feasibility, or application is not novel, then you must provide specific references to prior work.
A major weakness of the paper is that it does not concretely demonstrate a time speedup, by accounting for the training time for learning the inversion parameters, which takes several minutes in non-Cartesian cases. Whether that time is better spent simply performing the 20 CG iterations is unclear, and the paper does not compare to pre-conditioned CG. Furthermore, it is not obvious how well this learning scales with problem dimensionality (e.g. 3D reconstructions).
- Please rate the clarity and organization of this paper
Good
- Please comment on the reproducibility of the paper. Please be aware that providing code and data is a plus, but not a requirement for acceptance.
The submission does not mention open access to source code or data but provides a clear and detailed description of the algorithm to ensure reproducibility.
- Optional: If you have any additional comments to share with the authors, please provide them here. Please also refer to our Reviewer’s guide on what makes a good review and pay specific attention to the different assessment criteria for the different paper categories: https://conferences.miccai.org/2025/en/REVIEWER-GUIDELINES.html
N/A
- Rate the paper on a scale of 1-6, 6 being the strongest (6-4: accept; 3-1: reject). Please use the entire range of the distribution. Spreading the score helps create a distribution for decision-making.
(4) Weak Accept — could be accepted, dependent on rebuttal
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
The paper is interesting and novel, but it is unclear what the actual net practical speedup is, and how it scales.
- Reviewer confidence
Very confident (4)
- [Post rebuttal] After reading the authors’ rebuttal, please state your final opinion of the paper.
N/A
- [Post rebuttal] Please justify your final decision from above.
N/A
Review #3
- Please describe the contribution of the paper
This manuscript presents a novel inversion formula for the multi-coil MRI forward operator, leveraging low displacement rank (LDR) properties to achieve computational efficiency. The work is theoretically grounded and supported by empirical evidence, demonstrating significant potential to accelerate MR reconstruction. The study is well-written and easy to follow, and addresses an important gap in the field.
- Please list the major strengths of the paper: you should highlight a novel formulation, an original way to use data, demonstration of clinical feasibility, a novel application, a particularly strong evaluation, or anything else that is a strong aspect of this work. Please provide details, for instance, if a method is novel, explain what aspect is novel and why this is interesting.
- The hypothesis that the multi-coil forward operator exhibits LDR is intriguing and supported by simulations.
- The proposed inversion formula offers a computational complexity of O(Nlog^2N), a notable improvement over iterative methods (e.g., CG).
- The learning-based approach to determine LDR parameters is innovative and practical.
- Please list the major weaknesses of the paper. Please provide details: for instance, if you state that a formulation, way of using data, demonstration of clinical feasibility, or application is not novel, then you must provide specific references to prior work.
- The claim of “arbitrary trajectories” should be tempered, as the experiments focus on Cartesian and radial patterns. Broader validation (e.g., spiral, PROPELLER) would strengthen generality.
- Additionally, the novelty of LDR in structured matrices is well-established; the manuscript should better highlight how its application to multi-coil MRI is distinct from prior uses in other domains.
- Code availability is not stated (even if “to be released”).
- Please rate the clarity and organization of this paper
Satisfactory
- Please comment on the reproducibility of the paper. Please be aware that providing code and data is a plus, but not a requirement for acceptance.
The submission does not mention open access to source code or data but provides a clear and detailed description of the algorithm to ensure reproducibility.
- Optional: If you have any additional comments to share with the authors, please provide them here. Please also refer to our Reviewer’s guide on what makes a good review and pay specific attention to the different assessment criteria for the different paper categories: https://conferences.miccai.org/2025/en/REVIEWER-GUIDELINES.html
N/A
- Rate the paper on a scale of 1-6, 6 being the strongest (6-4: accept; 3-1: reject). Please use the entire range of the distribution. Spreading the score helps create a distribution for decision-making.
(4) Weak Accept — could be accepted, dependent on rebuttal
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
The learning-based approach is promising but may face skepticism due to lack of theoretical guarantees. It’s better for the authors to address this in revision.
- Reviewer confidence
Somewhat confident (2)
- [Post rebuttal] After reading the authors’ rebuttal, please state your final opinion of the paper.
N/A
- [Post rebuttal] Please justify your final decision from above.
N/A
Author Feedback
We thank the reviewers for their thoughtful and constructive feedback. We are encouraged to see consensus on the novelty and potential impact of our work, particularly in introducing a direct inversion formula for multi-coil MR reconstruction based on the established theory of low displacement rank (LDR) in Toeplitz-based operators. We view this contribution as a “zero-to-one” innovation—bringing a powerful, yet underutilized, class of linear algebra tools into the MRI community.
Many of the critiques stem from slight misinterpretations of our paper’s intent. For example, concerns about the long training time of the inversion operators assume that our goal is to deploy this learning-based inversion as-is in clinical practice [R1, R3]. In contrast, our training on simulated data is meant solely to demonstrate the existence and feasibility of such an inversion operator. Reducing training time is an important but separate challenge, which we explicitly mention as future work in the discussion section. We hope that by making this inversion formula known to the public, it will encourage and inspire further research into its practical integration into MR reconstruction pipelines.
Similarly, reviewers noted the lack of empirical timing comparisons against conjugate gradient (CG) methods[R1, R3]. This was a deliberate choice: runtime is highly dependent on hardware, whereas our theoretical analysis offers a more generalizable perspective. In theory, our LDR inversion achieves over 26× speedup relative to non-preconditioned CG. Even for the preconditioned CG, our inversion formula will cost less because its cost is less than evaluation of one forward operator. There are also some concerns regarding lack of evidence of integration into end-to-end networks [R1]. The benefits of our method for deep learning integration are theoretically evident: gradient backpropagation with CG requires multiple operator evaluations and are subject to choice of CG starting points, while LDR inversion requires just a single, lightweight pass through a learned operator, significantly reducing training complexity and instability.
Regarding generality and scalability[R1, R3, R5]: while our experiments focus on 2D Cartesian and radial trajectories, the underlying theory of LDR inversion applies broadly to Toeplitz-like operators, which are present in forward models across ALL sampling trajectories, including spiral and PROPELLER (see Ref. [4]). Extending our work to 3D and additional trajectories is indeed a valuable direction and would strengthen the paper further—but we believe our current results already establish a strong proof-of-concept.
Some reviewers also commented on the lack of statement on making our implementation available [R5]. We would like to take this opportunity and pledge that our code will be made fully public upon acceptance of this paper.
In summary, this work introduces a novel and theoretically grounded direction for MR reconstruction with promising practical implications. We hope our clarifications will assure the reviewers and area chair that this theoretical contribution merits publication and will stimulate further exploration in this exciting direction.
Meta-Review
Meta-review #1
- Your recommendation
Invite for Rebuttal
- If your recommendation is “Provisional Reject”, then summarize the factors that went into this decision. In case you deviate from the reviewers’ recommendations, explain in detail the reasons why. You do not need to provide a justification for a recommendation of “Provisional Accept” or “Invite for Rebuttal”.
N/A
- After you have reviewed the rebuttal and updated reviews, please provide your recommendation based on all reviews and the authors’ rebuttal.
Accept
- Please justify your recommendation. You may optionally write justifications for ‘accepts’, but are expected to write a justification for ‘rejects’
N/A
Meta-review #2
- After you have reviewed the rebuttal and updated reviews, please provide your recommendation based on all reviews and the authors’ rebuttal.
Accept
- Please justify your recommendation. You may optionally write justifications for ‘accepts’, but are expected to write a justification for ‘rejects’
This is quite a borderline paper. As the reviewers, I find the idea very intriguing in the context of MRI reconstruction, and think it makes an excellent topic for a poster. However, I also agree with the reviewers that the computational advantage of the proposed method over CG is not clear, and find that the author brush this concern a bit too quickly in their rebuttal. The proposed inversion is only efficient if the basis of Toeplitz matrices is known, and as they are not shown to be obtainable analytically, their estimation cost is entirely part of the reconstruction. I also find strange that 2 out of 3 trajectories tested are cartesian, with a relatively low acceleration factor that makes the non-regularized inversion problem well posed. In this context, it’s quite likely than initializing CG with the maximum-likelihood (SENSE) solution would save a few more iterations. Nonetheless, it is an interesting idea and likely to trigger discussions, and I recommend acceptance.
Meta-review #3
- After you have reviewed the rebuttal and updated reviews, please provide your recommendation based on all reviews and the authors’ rebuttal.
Accept
- Please justify your recommendation. You may optionally write justifications for ‘accepts’, but are expected to write a justification for ‘rejects’
N/A