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Low-Rank Velocity Fields as a Structural Prior for Unsupervised 4D Medical Image Interpolation
Haojin Li, Hengzhuo Wang, Chang Liu, Zhiheng Ma, Heng Li, Jiang Liu
TL;DR
Endpoint-only unsupervised 4D interpolation must generate anatomically coherent intermediate volumes from only two endpoints, despite weak supervision and sparse sampling. The paper addresses this with Tucker-parameterized low-rank velocity fields and coarse-to-fine multi-scale motion modeling. On ACDC and 4D-Lung, it achieves the best structure-oriented scores with competitive reconstruction accuracy.
Problem
Endpoint-only interpolation has only two endpoint volumes for training, creating challenges in learning compact, multi-scale motion under weak supervision.
Method
The framework models motion with Tucker-parameterized low-rank velocity fields using shared spatial bases and a compact sample-dependent core, refined across multiple scales.
Results
The method achieves the best structure-oriented performance on ACDC and 4D-Lung while maintaining competitive interpolation error.
Takeaways & Limitations
Low-rank multi-scale motion modeling favors spatially correlated, anatomy-consistent deformation for endpoint-only interpolation.
Abstract
from arXiv · showhide
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, low-rank parameters control the smoothness energy of the velocity field, explaining why low-rank modeling promotes 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.
1 Introduction
Endpoint-only unsupervised interpolation must recover anatomically continuous motion from only two endpoint volumes, while sparse sampling and voxel-wise freedom can destabilize boundaries and deformation. The proposed framework uses low-rank, multi-scale velocity modeling to favor coordinated motion and reports strong structure-oriented results with competitive reconstruction accuracy.
- Motivation: Endpoint-only training uses only two endpoint volumes to synthesize intermediate volumes for sparse or irregularly sampled 4D medical sequences.The resulting intermediates support downstream tasks such as segmentation and reconstruction.
- Motivation: Voxel-wise high-dimensional motion can overfit locally inconsistent high-frequency motion, causing boundary instability, local misalignment, and non-physiological warping.Plausible medical-image intermediates must preserve anatomical continuity and topology while evolving smoothly.
- Motivation: Anatomical motion is spatially correlated and partially compressible, motivating compact structured motion modeling that suppresses spurious high-frequency motion.The representation emphasizes coordinated global patterns while retaining local non-rigid variation.
- Proposed framework: The framework introduces low-rank velocity fields as a structural prior that separates shared coordinated motion from sample-specific variation under endpoint-only learning.Its Tucker representation uses shared spatial bases and a compact sample-dependent core.
- Proposed framework: A coarse-to-fine multi-scale strategy allocates modeling capacity across resolutions to capture global coordination and local non-rigid deformation.The framework projects multi-scale endpoint features into structured low-rank velocity representations.
- Results: Experiments on ACDC and 4D-Lung achieve the best structure-oriented scores while maintaining competitive reconstruction accuracy.The reported results position structural quality as the strongest performance advantage.
2 Method
The method learns endpoint-to-endpoint deformations using Tucker low-rank velocity fields, whose shared spatial bases and sample-specific cores constrain motion while retaining variation. A coarse-to-fine multi-scale design composes deformations for arbitrary-time interpolation, while the theory links low-rank parameters to smoothness-energy control.
- Problem definition: Endpoint-only training predicts a velocity field, converts it to a deformation with Exp(·), and reconstructs the target endpoint by warping the source.The model supports both ordered endpoint directions and uses endpoint similarity together with velocity regularization.
- Structured low-rank motion representation: Tucker parameterization decomposes each voxel-grid velocity tensor into shared spatial bases and a compact core of sample-dependent coefficients.The bases absorb globally shared spatial correlations, while the core preserves instance-specific motion variation.
- Structured low-rank motion representation: Low-frequency DCT spans constrain the learned spatial bases, discouraging voxel-wise high-frequency oscillations in the induced motion.Each basis mixes fixed truncated DCT directions through a learnable transformation within the same low-frequency span.
- Multi-scale motion modeling: Three scale-specific low-rank velocity fields at resolutions 1/4, 1/2, and 1 are jointly trained and progressively composed from coarse to fine.Each scale has its own bases and core; scale-wise endpoint reconstruction and smoothness losses are aggregated during training.
- Continuous-time inference: At inference, scale-wise deformations are integrated, resampled to a common grid, composed, and time-scaled in both endpoint directions to synthesize an intermediate at arbitrary α.The two warped endpoint predictions are fused with time-based weights after travel-time scaling proportional to α and 1 − α.
- Theoretical analysis: Tucker structure upper-bounds velocity smoothness energy using the smoothing operator, spatial-basis norms, and core magnitude, with rank-dependent bounds applying at every scale.The rank-scaling corollary states that, under bounded factors and core amplitudes, the worst-case bound grows at most cubically with rank.
3 Experiment
Experiments on ACDC and 4D-Lung compare the method with unsupervised and intermediate-supervision baselines, then examine rank, scale-weight, and loss-term choices. The method leads on NMI/SSIM, while ablations show that moderate motion complexity, balanced fine-leaning supervision, and regularization support reliable interpolation.
- Comparison Experiment: ACDC and 4D-Lung evaluations compare against classical deformation models, unsupervised interpolation networks, and methods trained with intermediate-frame supervision.
- Comparison Experiment: The method achieves leading NMI/SSIM on both datasets, particularly on 4D-Lung, while maintaining competitive reconstruction error.Despite endpoint-only supervision, it reaches supervised-level overall quality.
- Comparison Experiment: Qualitatively, the method produces more regular contours and stable anatomy on ACDC and better preserves coherent global organization on 4D-Lung.Baselines show local inconsistencies, while diffusion-style methods can improve appearance but compromise structural stability.
- Ablation Study: Increasing Tucker rank improves NMI in low-capacity regimes, but gains saturate at large fine-scale ranks as Effective Rank and motion complexity increase.Very small ranks underfit, while the best performance occurs at low-to-moderate Effective Ranks.
- Ablation Study: Fine-leaning yet balanced scale weights perform best, whereas extreme fine dominance degrades NMI and mid/coarse guidance helps preserve global coherence.Fine-scale supervision supports refinement; mid/coarse contributions reduce locally driven inconsistencies.
- Ablation Study: Regularization stabilizes low-rank core tensors, curbs excessive parameter growth, and yields smoother, more robust motion than omitting it.Removing Charbonnier can slightly improve NMSE/SSIM but may destabilize optimization.
4 Conclusion
The paper concludes that Tucker low-rank velocity fields combined with coarse-to-fine multi-scale modeling favor anatomy-consistent deformation under endpoint-only training. Experiments show leading structure-oriented performance with competitive interpolation error, while the smoothness analysis supports the structural prior.
- Tucker low-rank velocity fields and coarse-to-fine multi-scale modeling favor spatially correlated, anatomy-consistent deformation under endpoint-only training.
- The method achieves best performance on structure-oriented metrics with competitive interpolation error on ACDC and 4D-Lung.
- Theoretical analysis links velocity-field smoothness energy to low-rank parameters, providing support for the structural prior.