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

cs.CVAug 25, 2026

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

Authors: Haojin Li, Hengzhuo Wang, Chang Liu, Zhiheng Ma, Heng Li, Jiang Liu

Organizations: Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, China · Faculty of Biomedical Engineering, Shenzhen University of Advanced Technology, Shenzhen, China · School of Biomedical Engineering, Shenzhen University Medical School, Shenzhen University, Shenzhen, China · Faculty of Computility Microelectronics, Shenzhen University of Advanced Technology, Shenzhen, China ·

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, 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.

Figures & tables

Figure 1: Overview of the proposed framework. (a) Multi-scale endpoint features are projected into a structured low-rank velocity representation across resolutions via Tucker decomposition, aligning motion modeling with the coordinated and compact nature of anatomical dynamics. (b) Inference performs bi-directional continuous-time interpolation to produce structurally consistent intermediate volumes.

Method	Year	ACDC			4D-Lung
Method	Year	NMI ${}_{\times 10^{-2}}^{\uparrow}$	NMSE ${}_{\times 10^{-3}}^{\downarrow}$	SSIM ${}_{\times 10^{-2}}^{\uparrow}$	NMI ${}_{\times 10^{-2}}^{\uparrow}$	NMSE ${}_{\times 10^{-3}}^{\downarrow}$	SSIM ${}_{\times 10^{-2}}^{\uparrow}$
VM	2019	53.54 ±0.37	19.18 ±0.34	94.68 ±0.10	43.98 ±0.36	39.02 ±0.60	86.06 ±0.29
TM	2022	64.63 ±0.67	14.43 ±0.62	96.64 ±0.12	48.95 ±0.10	34.96 ±0.40	84.78 ±0.12
SVIN	2020	60.18 ±0.33	9.85 ±0.89	97.35 ±0.15	43.17 ±0.39	30.28 ±2.03	85.44 ±0.29
UVINet	2024	72.25 ±0.17	4.81 ±0.12	98.62 ±0.03	55.16 ±0.84	20.93 ±1.27	88.75 ±0.67
DDM	2022	64.14 ±1.35	10.36 ±0.30	97.24 ±0.87	42.54 ±0.89	33.19 ±0.73	85.23 ±0.27

Table 1: Quantitative comparison on ACDC and 4D-Lung datasets.

Figure 2: Qualitative comparison on ACDC and 4D-Lung datasets.

Figure 3: Ablation on ACDC. (a) NMI under different Tucker rank settings. (b) Velocity field Effective Rank [ 16 ] (log 10 -scaled). (c) NMI vs. scale weights (ternary diagram).

NCC	Charb	Reg	NMI ${}_{\times 10^{-2}}^{\uparrow}$	NMSE ${}_{\times 10^{-3}}^{\downarrow}$	SSIM ${}_{\times 10^{-2}}^{\uparrow}$
$\checkmark$		$\checkmark$	72.22	4.69	98.63
	$\checkmark$	$\checkmark$	72.35	4.81	98.59
$\checkmark$	$\checkmark$		71.45	5.18	98.49
$\checkmark$	$\checkmark$	$\checkmark$	72.61	4.74	98.62

Table 2: Loss term ablation on ACDC.

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Phase-Aligned Finite-Fourier Periodic Deformation for 4D Medical Image Interpolation

Haojin Li, Hengzhuo Wang, Zhiheng Ma, Mingyang Ou+2

School of Computer Science and Engineering Beihang University Beijing, China

4D Reconstruction Deformable Image Registration

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4D medical image interpolation aims to recover missing volumes from sparsely observed time points and is important for dynamic anatomical analysis in applications such as cardiac MRI and thoracic CT, where motion is often repetitive or near-periodic over clinically relevant intervals. A key challenge is that this structure is not always encoded directly in deformation representations for interpolation. In addition, physiological motion is often non-uniform, so equal temporal intervals do not necessarily correspond to equal amounts of anatomical change. To address these issues, we formulate interpolation as learning a continuous deformation process with a phase-structured prior. Given two endpoint volumes, we parameterize a phase-conditioned velocity field with a finite Fourier basis, which embeds near-periodic motion patterns directly into the deformation space and supports continuous querying at arbitrary target times. We further introduce a phase-aligned temporal reparameterization that maps normalized within-interval time to a latent motion phase according to deformation variation intensity, thereby better modeling non-uniform motion progression. Intermediate volumes are then synthesized by continuously warping both endpoints, followed by bidirectional fusion and lightweight residual refinement. Experiments on ACDC and 4D-Lung show that the proposed method achieves state-of-the-art performance over existing baselines while producing anatomically plausible and coherent intermediate volumes from sparse observations.

Abstract

Medical Image Synthesis

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