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Abstract
Endpoint-only unsupervised 4D medical image interpolation synthesizes intermediate volumes from sparsely sampled sequences with only the start and end volumes available for training; however, this weakly constrained setting often yields intermediates with unstable boundaries and non-physiological motion, limiting interpretability and downstream analysis. We propose low-rank velocity fields as a structural prior, constraining motion to a structured Tucker low-rank velocity field space that decomposes motion into globally shared spatial bases and a compact sample-specific core, thereby encouraging spatially correlated, anatomy-consistent deformation while suppressing voxel-wise high-frequency artifacts. To capture global coordination and local non-rigid details, we model motion in a coarse-to-fine multi-scale scheme and compose scale-wise deformations at inference to synthesize volumes at arbitrary times. We further provide a theoretical analysis showing that, under Tucker parameterization, bounded low-rank parameters control the smoothness energy of the velocity field, offering an energy-based interpretation of smoother motion. Experiments on ACDC and 4D-Lung demonstrate state-of-the-art performance, remaining competitive with methods trained with intermediate-frame supervision, and producing intermediates with improved structural coherence and more stable anatomical contours. The code is available at https://github.com/JingHuaMan/LRVF.
Links to Paper and Supplementary Materials
Main Paper (Open Access Version): https://papers.miccai.org/miccai-2026/paper/0472_paper.pdf
SharedIt Link: Not yet available
SpringerLink (DOI): Not yet available
Supplementary Material: Not Submitted
Link to the Code Repository
https://github.com/JingHuaMan/LRVF
Link to the Dataset(s)
ACDC dataset: https://www.creatis.insa-lyon.fr/Challenge/acdc/
4D-Lung dataset: https://www.cancerimagingarchive.net/collection/4d-lung/
BibTex
@InProceedings{LiHao_LowRank_MICCAI2026,
author = { Li, Haojin AND Wang, Hengzhuo AND Liu, Chang AND Ma, Zhiheng AND Li, Heng AND Liu, Jiang},
title = { { Low-Rank Velocity Fields as a Structural Prior for Unsupervised 4D Medical Image Interpolation } },
booktitle = {Medical Image Computing and Computer Assisted Intervention -- MICCAI 2026},
year = {2026},
publisher = {Springer Nature Switzerland},
volume = {LNCS 16890},
month = {September},
page = {pending}
}
Reviews
Review #1
- Please describe the contribution of the paper
The paper proposes a low-rank velocity field prior for endpoint-only unsupervised 4D medical image interpolation. Motion is parameterized using a Tucker low-rank decomposition with shared spatial bases and a compact core, combined with a multi-scale coarse-to-fine framework. A theoretical analysis links low-rank modeling to smoothness control. Experiments on ACDC and 4D-Lung show strong performance, achieving competitive results with supervised methods
- 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.
Well-motivated structural prior: Low-rank motion modeling aligns with the low-dimensional, spatially correlated nature of anatomical motion. Theoretical support: Provides analysis linking low-rank parameterization to smoothness energy. Effective multi-scale design: Coarse-to-fine modeling is appropriate and well-integrated. Strong empirical results: Achieves leading structure-oriented metrics under endpoint-only supervision. Clear presentation and ablation studies.
- 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.
Incremental novelty: Mainly integrates existing ideas (low-rank priors, tensor decomposition, multi-scale motion). Limited clinical validation: No evaluation on downstream or clinically relevant tasks (e. g. , tracking, RT workflows). Intrinsic limitation of interpolation: Endpoint-only setting cannot capture irregular motion or excursions; the low-rank prior further biases toward smooth trajectories, potentially suppressing clinically relevant extremes. Limited evaluation scope: Only standard datasets and metrics are used.
- 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.
- Based on your review and your understanding of the MICCAI Scientific Code of Ethics, do you believe this submission may involve a potential ethics concern or violation?
N/A
- 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/2026/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 — marginally above the acceptance threshold, but would not mind if 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 is technically sound and well-motivated, with a clear low-rank motion formulation and strong results under endpoint-only supervision. However, the novelty is incremental, and evaluation is limited to interpolation metrics without downstream or clinical validation. Additionally, the endpoint-only setting and low-rank prior inherently favor smooth motion, limiting the ability to capture irregular dynamics. These factors lead to a borderline/weak accept score.
- 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
This paper proposes using Tucker-decomposed low-rank velocity fields as a structural prior for endpoint-only unsupervised 4D medical image interpolation. It is motivated by that anatomical motion is spatially correlated and compressible, so constraining velocity fields to a low-rank subspace should suppress high-frequency artifacts.
- 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.
1.The core idea is well-motivated and technically sound. 2.The observation that anatomical motion is dominated by coordinated global patterns with residual local detail is well-established in biomechanics, and encoding this as a Tucker low-rank constraint on velocity fields is a clean, principled translation of domain knowledge into model design. 3.Quantitative results look good.
- 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.
1.The theoretical analysis is limited in novelty and insight. The energy bound in Eq. 8 is loose, which holds for any linear operator and says nothing specific about the relationship between low-rank structure and the type of smoothness encouraged. A more insightful analysis would show, for instance, that the DCT-based parameterization preferentially suppresses high-frequency components of the velocity field, or that the Tucker structure induces a specific spectral decay. 2.The percentage of voxels with negative Jacobian determinant (folding) is the standard measure of deformation plausibility. The paper claims smoother, more physically plausible deformations but never reports this. 3.The chosen ranks follow a decreasing pattern from coarse to fine, which is counterintuitive. One might expect finer scales to need more degrees of freedom to capture local detail. 4.No training time, inference time, or parameter count comparisons are provided. Tucker decomposition and multi-scale processing will add complexity. What is the inference time per volume compared to the fastest and slowest baselines?
- 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.
- Based on your review and your understanding of the MICCAI Scientific Code of Ethics, do you believe this submission may involve a potential ethics concern or violation?
N/A
- 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/2026/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 — marginally above the acceptance threshold, but would not mind if rejected, dependent on rebuttal
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
See weakness
- Reviewer confidence
Confident but not absolutely certain (3)
- [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 paper tackles the challenging endpoint-only unsupervised 4D medical image interpolation problem by introducing low-rank velocity fields through Tucker decomposition as a structural prior. The main idea is to regularize motion in this highly under-constrained setting, enforcing anatomical coherence via shared spatial bases and multi-scale refinement. Overall, the paper is well-written and technically sound, with a clean formulation and reasonable experimental validation. However, the conceptual novelty is moderate, as low-rank or band-limited priors for motion and deformation modeling have a substantial precedent in registration and dynamic reconstruction literature.
- 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.
1.The paper addresses a concrete and under-constrained clinical scenario; the setting and motivation are well articulated. 2.The Tucker decomposition is applied systematically within the 3D velocity field context; the multi-scale hierarchy is logically consistent and well implemented. 3.The low-rank prior successfully improves anatomical stability, which is crucial for physically meaningful 4D interpolation. 4.The energy bounding analysis, while simple, contributes interpretability and mathematical clarity to the model behavior. 5.Clear implementation details and consistent reporting across datasets (ACDC, 4D-Lung) support reproducibility.
- 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.
1.The core idea—imposing low-rank or band-limited priors on motion fields—has been introduced in earlier works ([Fourier-Net 2023], [Deep-SLR 2020], etc.). The paper’s advancement mainly lies in integrating this prior within endpoint-only unsupervised interpolation, which feels primarily an application-specific adaptation rather than a conceptual leap. 2.Results show numerical improvement, but no diagnostic insights are given into why the method outperforms recent state-of-the-art diffusion or transformer-based interpolation approaches. 3.The paper does not discuss the inherent indeterminacy of endpoint-only interpolation — multiple valid motion trajectories may exist between endpoints. A deterministic velocity field assumes a single plausible path, which limits the model’s uncertainty representation. 4.Evaluation on only ACDC and 4D-Lung provides insufficient evidence of generalization across modalities or organ systems. There is no downstream validation (e.g., segmentation, registration accuracy post-interpolation). 5.Choices such as Tucker ranks, scale weights, and frequency truncation parameters (K=128) appear heuristic. Table 2’s ablation provides insights but lacks principled analysis. 6.Equation (8) and (9) provide an upper bound on smoothness energy. How tight is this bound in practice? Is there quantitative evidence correlating the theoretical bound with empirical smoothness metrics of the velocity fields? 7.Tucker decomposition introduces additional parameterization overhead. How does this compare computationally (in FLOPs and memory) with standard voxel-wise velocity networks? Does multi-scale inference significantly increase latency? 8.The paper uses truncated DCT bases. Why choose DCT over learned bases (e.g., via autoencoder or PCA initialization)? Would learned bases within the Tucker form yield better anatomical alignment? 9.The coarse-to-fine motion composition (Eq. 5) might accumulate integration error across scales. Have the authors assessed numerical stability or diffeomorphism preservation post-composition? 10.Does the low-rank velocity field correspond to interpretable anatomical modes (e.g., cardiac contraction vs. expansion)? If so, can this representation aid clinical understanding beyond interpolation? 11.Modern diffusion-based temporal interpolation (FB-Diff 2025, TMSDF 2025) implicitly models multi-modal motion distributions. How does the proposed deterministic approach relate conceptually—can low-rank parametrization be combined with diffusion sampling for uncertainty-aware interpolation?
- 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.
- Based on your review and your understanding of the MICCAI Scientific Code of Ethics, do you believe this submission may involve a potential ethics concern or violation?
N/A
- 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/2026/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 — marginally above the acceptance threshold, but would not mind if rejected, dependent on rebuttal
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
This submission presents a novel and well-motivated methodological contribution to 4D medical image interpolation. The proposed Tucker-based low-rank velocity representation introduces a meaningful structural prior that enables efficient, high-fidelity motion modeling under endpoint supervision. The idea is original, theoretically sound, and supported by clear experiments on public datasets (ACDC and 4D-Lung). The method improves image quality and motion plausibility compared to relevant baselines.
However, certain aspects limit the overall impact:
The clinical relevance remains relatively indirect—the paper focuses on technical methodology rather than explicit clinical application. Ablation studies and quantitative comparisons could be expanded to more clearly demonstrate the advantages of the Tucker decomposition over other low-rank or flow regularization approaches. Code or data availability would strengthen reproducibility. Overall, the paper is solid from a methodological standpoint, with good writing quality, sufficient experiments, and convincing results. These factors collectively position the submission slightly above the acceptance threshold. With some clarifications or additional experiments, it could become a strong contribution to the field.
- Reviewer confidence
Confident but not absolutely certain (3)
- [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 reviewers and AC for the comments and for recognizing the motivation, soundness, and empirical strength of our work. We clarify the points.
1.Novelty and relation to prior work (R1, R2, R3): We appreciate the reviewers’ observation that low-rank/band-limited priors have precedents. We view the novelty not as the isolated use of “low rank”, but as where and how the compact prior is imposed. In Fourier-domain registration such as Fourier-Net(AAAI 2023), compactness mainly parameterizes pairwise deformation efficiently; in structured low-rank reconstruction such as Deep-SLR(TMI 2020), the prior acts on image reconstruction; PCA/statistical respiratory models compress motion fields after estimation or from population statistics. These lines do not target endpoint-only interpolation ambiguity. Even within 4D interpolation, frequency-guided generative methods such as FB-Diff use Fourier bases to guide diffusion-based synthesis, while endpoint-only methods often use dense deformation/image prediction. Our formulation is different because the velocity tensor itself is factorized into shared spatial bases and a sample-specific Tucker core, making motion capacity explicit and controllable during endpoint-only training. Thus, the contribution is a structural velocity-field prior for stable interpolation, not a simple transfer of an existing compact representation. 2.Smoothness, folding, and deformation plausibility (AC, R2, R3): We thank the reviewers for pointing out the value of Jacobian-based folding analysis. Negative Jacobian percentage is a useful direct measure and would complement the current evaluation. Our smoothness claim is supported by components already in the paper: the truncated DCT span restricts bases to low-frequency components, the H1 regularizer penalizes velocity magnitude and spatial gradients, and the Tucker low-rank form limits motion-field degrees of freedom. Fig. 3(b) reports Effective Rank as a proxy for motion-field complexity. While not a substitute for folding statistics, it supports the intended effect of the prior: limiting overly flexible voxel-wise motion while retaining capacity for anatomical deformation. 3.Rank schedule, DCT basis, and theory (R2, R3): We appreciate these questions, as they help clarify our design. The decreasing rank schedule is intentional: coarse scales model globally coordinated anatomical motion and receive higher modal capacity, while fine scales serve as constrained residual refinements to reduce high-frequency overfitting. This matches Fig. 3(a,b), where increasing rank helps in low-capacity regimes but gains saturate as motion complexity grows. Eq. 8/9 is an interpretability result showing how bounded factors and cores control velocity energy. The spectral bias comes from the truncated DCT basis, while the Tucker form provides compact sample-dependent coefficients. 4.Computational complexity (R2, R3): We thank the reviewers for raising this practical aspect. Tucker parameterization and multi-scale processing introduce structured operations, but they also replace unrestricted voxel-wise velocity prediction with compact cores and separable spatial bases. The goal is to trade dense motion freedom for a controlled low-dimensional representation better suited to endpoint-only learning. We will clarify this trade-off. 5.Clinical validation, endpoint-only ambiguity, and uncertainty (R1, R3): We appreciate this perspective. The paper is primarily methodological: it studies endpoint-only interpolation and proposes a deterministic structural prior for coherent anatomical motion. We do not claim that a single velocity field represents all possible irregular or multi-modal trajectories. Rather, our objective is stable, structurally consistent interpolation under endpoint supervision. ACDC and 4D-Lung cover two distinct motion regimes and support this claim. Downstream validation and combination with uncertainty modeling are valuable extensions.
Meta-Review
Meta-review #1
- Your recommendation
Provisional Accept
- Please justify your decision. In case you deviate from the reviewers’ recommendations, explain in detail the reasons why. In case of an invitation for rebuttal, clarify which points are important to address in the rebuttal.
The reviewers agree that this work is technically sound, well-motivated, and shows strong empirical results. Meanwhile, all reviewers shared concerns about limitations in the evaluation, including the lack of clinical validation and smoothness metrics.
I recommend accepting this paper based on its consistently highlighted merits.