Low-Rank Velocity Fields as a Structural Prior for Unsupervised 4D Medical Image Interpolation

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.Source: §1 · p. 2
• 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.Source: §1 · p. 2
• 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.Source: §1 · p. 2

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.Source: §1 · p. 2
• 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.Source: §1 · p. 2, Figure 1 · p. 3

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.Source: §1 · p. 2

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.Source: §2.1 · p. 3

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.Source: §2.2 · pp. 3–4
• 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.Source: §2.2 · p. 4

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.Source: §2.2 · p. 4

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.Source: §2.3 · p. 5

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.Source: §2.4 · p. 5

Comparison Experiment

• ACDC and 4D-Lung evaluations compare against classical deformation models, unsupervised interpolation networks, and methods trained with intermediate-frame supervision.Source: §3.1 · p. 6, §3.2 · p. 6
• 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.Source: §3.2 · pp. 6–7
• 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.Source: §3.2 · p. 7

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.Source: §3.3 · pp. 7–8
• 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.Source: §3.3 · p. 8
• 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.Source: §3.3 · p. 8

• Tucker low-rank velocity fields and coarse-to-fine multi-scale modeling favor spatially correlated, anatomy-consistent deformation under endpoint-only training.Source: §4 · p. 9
• The method achieves best performance on structure-oriented metrics with competitive interpolation error on ACDC and 4D-Lung.Source: §4 · p. 9
• Theoretical analysis links velocity-field smoothness energy to low-rank parameters, providing support for the structural prior.Source: §4 · p. 9