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Abstract

Deformable medical image registration is traditionally formulated as an optimization problem. While classical methods solve this problem iteratively, recent learning-based approaches use weight-tied neural networks to mimic this process by unrolling the prediction of deformation fields in a fixed number of steps. However, classical methods typically converge after sufficient iterations, but learning-based unrolling methods lack a theoretical convergence guarantee and show instability empirically. In addition, unrolling methods have a practical bottleneck at training time: GPU memory usage grows linearly with the unrolling steps due to backpropagation through time (BPTT). To address both theoretical and practical challenges, we propose DEQReg, a novel registration framework based on Deep Equilibrium Models (DEQ), which formulates registration as an equilibrium-seeking problem, establishing a natural connection between classical optimization and learning-based unrolling methods. DEQReg maintains constant memory usage, enabling theoretically unlimited iteration steps. Through extensive evaluation on the public brain MRI and lung CT datasets, we show that DEQReg can achieve competitive registration performance, while substantially reducing memory consumption compared to state-of-the-art unrolling methods. We also reveal an intriguing phenomenon: the performance of existing unrolling methods first increases slightly then degrades irreversibly when the inference steps go beyond the training configuration. In contrast, DEQReg achieves stable convergence with its inbuilt equilibrium-seeking mechanism, bridging the gap between classical optimization-based and modern learning-based registration methods.

Links to Paper and Supplementary Materials

Main Paper (Open Access Version): https://papers.miccai.org/miccai-2025/paper/2523_paper.pdf

SharedIt Link: Not yet available

SpringerLink (DOI): Not yet available

Supplementary Material: Not Submitted

Link to the Code Repository

https://gitlab.tudelft.nl/ai4medicalimaging/deqreg

Link to the Dataset(s)

N/A

BibTex

@InProceedings{ZhaYi_Bridging_MICCAI2025,
        author = { Zhang, Yi AND Zhao, Yidong AND Tao, Qian},
        title = { { Bridging Classical and Learning-based Iterative Registration through Deep Equilibrium Models } },
        booktitle = {proceedings of Medical Image Computing and Computer Assisted Intervention -- MICCAI 2025},
        year = {2025},
        publisher = {Springer Nature Switzerland},
        volume = {LNCS 15962},
        month = {September},
        page = {88 -- 98}
}

Reviews

Review #1

  • Please describe the contribution of the paper

    The paper introduces a deformable image registration network designed to combine the convergence stability of classical optimization methods with the efficiency and flexibility of learning-based unrolling approaches. By leveraging deep equilibrium models (DEMs), the authors iteratively predict the final transformation through residual updates that satisfy fixed-point conditions. Experimental results show that the proposed DEQReg network achieves comparable accuracy to several existing unrolling-based methods with reduced computational cost.

  • 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 idea of applying a deep equilibrium model to unsupervised medical image registration and formulating the task as a fixed-point system is interesting.

    • The proposed method demonstrates improved inference stability compared to current unrolling-based methods.

  • 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.

    • While the combination of existing DEMs to registration networks is interesting, the proposed network architecture is quite similar to [22], which somewhat diminishes the novelty of the contribution.

    • Although DEQReg demonstrates low memory usage, it incurs higher computational costs during inference compared to all baseline methods (Tab. 1). This could potentially be a major limitation for real-world clinical applications, though the paper does not discuss this trade-off in detail.

    • The experimental results also do not clearly establish a significant advantage of the proposed method over existing unrolling approaches such as RIIR or GraDIRN. As shown in Table 1, the Dice score improvements are marginal (less than 1%), despite the higher inference cost associated with DEQReg.

    • Figure 3 highlights performance drops for GraDIRN and RIIR in extended inference steps, but the paper would benefit from a clearer, more intuitive explanation of why this degradation occurs. Similarly, it would be helpful to elaborate on how the DEM with fixed-point conditions improves the regularity of the deformation field.

    • The organization of the paper could be improved. Section 2.1, which primarily covers background material, should be separated from the main method section to improve clarity.

  • 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 authors claimed to release the source code and/or dataset upon acceptance of the submission.

  • 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 proposed method has several weaknesses, including limited novelty in network design (as it closely resembles existing approaches) and experimental results that do not convincingly demonstrate clear advantages over established baselines.

  • 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

    The paper proposes to learn optimisation steps for image registration with an iterative/unrolled deep network and leverage equilibrium states to enable flexible number of steps at inference time. It is evaluated on two datasets with reasonable performance.

  • 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 method is interesting and provides a fresh perspective on learning based image registration.
    • evaluation on two different tasks as well as with accuracy and smoothness metrics are welcomed
    • the concept to find the point of zero gradient rather than a descent is interesting and helps avoid the potential problems of running inference too few/too many times in practice
    • the empirical results demonstrate huge memory/performance gains over vanilla unrolling
    • the authors promise to release code and models after acceptance
  • 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.
    • there are quite some similarities to the MedIA paper “Deep Implicit Optimization enables Robust Learnable Features for Deformable Image Registration” (preprint from 2024 available at https://arxiv.org/abs/2406.07361), which also employs the implicit function theorem and uses the same two registration datasets for benchmarking (albeit achieving much higher performance of 86.2% Dice and 1.02mm TRE) - maybe the authors can comment on the differences and if they were aware of this paper before submission
    • I found the wording recurrent neural network (RNN) slightly confusing, there is no hidden state that is carried over from one NN to the next apart from the 3D deformations - hence it is rather a normal CNN that is iteratively applied
    • the visual results are poor - both resolution and size of the overlays prevent any meaningful analysis
    • state-of-the-art comparisons, e.g. against LapIRN and corrField (for NLST) are missing - 2.23mm TRE (appears to be keypoint TRE and not landmark TRE) is not very good. LapIRN reached 0.807mm (kp) and 1.673mm (lm), corrField (optimisation based) 1.830mm (lm) based on the challenge leaderboard.
    • the key idea of the approximate inverse Jacobian computation through damping is a bit unclear (see detailed comments)
  • 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 authors claimed to release the source code and/or dataset upon acceptance of the submission.

  • 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

    To better understand the concept of DEQ for registration the authors define a fixed point of a function h as x=g(x) hence g_theta=phi+f_theta is equal to f_theta(phi)=0. Hence the DNN learns a set of parameters for which this equation (given some optimal displacement phi* is (close to) zero. Training the DEQ starts from a converged fixed point phi^0=phi* how is this obtained? It then linearly interpolates intermediate steps phi^p where again the correct transformation seems to be known (in training). For the loss function in Eq. 9 I assume phi* represents the current network prediction integrated over a fixed (predetermined) number of steps whereas phi_t are intermediate ones but this does not clearly establish the link to the theoretical explanation, it more or less seems to be a weighted average of the prediction along the unrolled networks.

  • 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?

    There are a number of relevant flaws / inaccuracies that would prevent me from accepting the paper as is. Nevertheless there are some interesting aspects that could deserve discussion during the rebuttal and if the concerns are remedied at the MICCAI conference.

  • Reviewer confidence

    Very confident (4)

  • [Post rebuttal] After reading the authors’ rebuttal, please state your final opinion of the paper.

    Accept

  • [Post rebuttal] Please justify your final decision from above.

    The authors have clarified some aspects of the paper and I recommend acceptance. I would, however, encourage the authors to provide better visual examples of the registration outcome in the final paper.



Review #3

  • Please describe the contribution of the paper

    The authors propose a registration framework based on Deep Equilibrium Models (DEQ), which are weight-tight neural network models that exploit implicit differentiation to directly find stationary points of optimization problems. The motivation of the proposed learning framework (DEQReg) is to reformulate the deformable image registration problem as an equilibrium seeking problem, thus providing convergence guarantees and bridging the gap between classical and learning-based iterative registration methods. The authors evaluate the proposed framework using two publicly available datasets, and obtain highly competitive results against state-of-the-art classical and learning-based iterative frameworks.

  • 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 authors effectively leverage (DEQ) in combination with efficient implicit differentiation to formulate the deformable registration problem as an unsupervised fixed-point learning problem. Theoretically, the proposed DEQReg formulation models the behavior of infinite-step unrolling methods, while maintaining a finite (relatively low) memory footprint and providing convergence guarantees. The authors experimentally demonstrate the improved convergence behavior of DEQReg with respect to unrolling methods, all without significantly sacrificing accuracy, regularity and performance.

  • 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.

    I believe this paper has no major weaknesses. However, there is an aspect of the paper that can be improved, which is the discussion about the phantom gradient. First, the authors mention that their memory footprint is O(1), which is wrong. That would be the case only if a root-finding algorithm is used and the computationally intensive calculation of such inverse jacobian is performed. The authors use the more efficient phantom gradient approach, which provides an estimate of the inverse jacobian term. The computation of this phantom gradient, however, requires K unrolling steps (as per Eq. (7)) for the estimation, meaning that the memory footprint is rather O(K). Further, the accuracy of the phantom gradient depends on the number of unrolling steps K, which by the way, is not mentioned in the implementation details. A discussion on this memory footprint and its influence on training time and performance will be greatly appreciated.

  • 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 has provided an anonymized link to the source code, dataset, or any other dependencies.

  • 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
    • The image loss in Eq. (1) should be a measure of dissimilarity, as the algorithm maximises similarity
    • In Eq. (4), what is the weight w_t ?
    • In Eq. (4), I believe the upper bound for the summation is T-1
    • In Eq. (7), I believe the number of iterations of the sequence is K-1
    • In Table 1., it would be informative to report the current BEST result (regardless of the method) for both datasets
  • 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.

    (6) Strong Accept — must be accepted due to excellence

  • Please justify your recommendation. What were the major factors that led you to your overall score for this paper?

    The proposed DEQReg framework provides sound theoretical guarantees of convergence. The reported evaluation is convincing, both in terms of accuracy and performance, and in terms of superiority with respect to state-of-the-art unrolling methods. I believe this paper represents a contribution of exceptional quality, and has the potential for high impact.

  • Reviewer confidence

    Confident but not absolutely certain (3)

  • [Post rebuttal] After reading the authors’ rebuttal, please state your final opinion of the paper.

    Accept

  • [Post rebuttal] Please justify your final decision from above.

    As mentioned before, I believe this manuscript exhibits exceptional scientific rigor and quality. The authors have addressed all reviewer feedback while providing answers to raised concerns. While the inclusion of state-of-the-art performance on the investigated datasets would have provided additional context, the authors’ decision to omit these, because of methodological differences, is understandable, and more importantly, does not detract from the paper’s contributions.



Author Feedback

We thank all reviewers for their constructive feedback. We appreciate R1’s comments that our paper represents a “contribution of exceptional quality” and has “potential for high impact”. We appreciate the recognition of our work as offering a “fresh perspective” with convergence guarantees, improved stability, and memory efficiency. We address comments in detail below. R2 expressed concerns on network design novelty and performance w.r.t. Dice. Orthogonal to proposing new network architectures, our core contribution is reformulating image registration as an unsupervised fixed-point problem with guaranteed convergence and limited memory cost, which are unattainable with current unrolled methods. Therefore, we do not claim novelty of network design but adopt the established U-Net backbone [6,23,27] for fair benchmarking. We note our network is structurally different from the 5-layer CNN in [22]. On performance, Dice alone is insufficient for evaluation [8,15]. Our method maintains Dice with smoother deformations and stable convergence (Fig. 3), both highly desirable in image registration. R2 asked for an intuitive explanation of the degradation in unrolled models and regularity of DEQReg. The degradation arises from the training mechanism: unrolled models are trained for a fixed number of steps without enforcing convergence. Limited steps induce large updates, making them unstable when further unrolled. This has also been observed empirically (Fig. 1,3,[12]). In contrast, DEQReg solves for a fixed point, and empirically yields smoother deformations over longer trajectories. We appreciate R1’s insightful comment on memory and training trade-offs. R1 is correct that memory and backward time scale as O(K). Larger K adds higher-order Neumann terms, improving estimation accuracy, while K=1 recovers Jacobian-free backpropagation (Fung et al., AAAI 2022), with O(1) but less stability. We set K=2, balancing accuracy and efficiency. We will revise “O(1)” to “O(K)” and clarify this trade-off. We thank R4 for referring us to the latest DIO paper (Jena et al., MedIA 2025). We were not aware of this work and DEQReg was developed in parallel. The two works differ in motivation and formulation of implicit differentiation. DIO uses a bilevel semi-supervised setup with label losses: a network extracts features, then displacements are iteratively optimized via the feature MSE. DEQReg instead imposes an unsupervised fixed-point formulation on the entire network. We will cite DIO and clarify the distinction. R4 asked about the inverse Jacobian computation: how φ* is obtained, whether ground truths are used, and loss. We initialize φ₀ = 0 and iteratively apply the network until convergence or T steps, yielding φ* w/o ground truth. Only in training, φ^p, an unroll beyond φ* (φ^0), is used to compute the phantom gradient [11] for the first term in the loss (Eq. 9). The second term in Eq. 9 applies the same estimation (Eq.8) to φ_t (the intermediate states before φ*) for training stability [9,10,27]. SOTA Comparisons (R4): While SOTA methods like LapIRN use instance optimization (IO), our work focuses on unrolled baselines w/o IO. LapIRN’s challenge entry, confirmed with the authors, included IO and expanded feature pyramids. corrField generated NLST keypoints for optimization and evaluation. In comparison, DEQReg and all baselines are image-based. DEQReg matches SOTA among non-IO methods. Inference cost (R2): In clinical settings, processing and I/O dominate runtime, making DEQReg’s ~0.5s overhead (3D) minor. This modest cost leads to DEQReg’s stable convergence. In inference, solver acceleration (e.g., Anderson) can reduce the iterations by ~3×. We will add discussion on this. Paper Organization (R2): We will move Sec. 2.1’s background into a preliminary section. Notations & Figures (R1 & R4): We will: (i) use “iterative CNN” instead of RNN; (ii) clarify Eq. 1 as dissimilarity; (iii) define weight w_t, fix sum bounds; (iv) enlarge vector overlays.



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’

    The authors have addressed most of the concerns and the reviewers agree on the importance of the topic treated.



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’

    This paper leverages an under-used method, and its presentation is extremely clear. Consensus acceptance has not been reached because experiments are limited, as is improvement over SOTA. It is nonetheless likely to generate debate and discussion and I recommend acceptance.



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