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Phase-Aligned Finite-Fourier Periodic Deformation for 4D Medical Image Interpolation
Haojin Li, Hengzhuo Wang, Zhiheng Ma, Mingyang Ou, Heng Li, Jiang Liu
TL;DR
Sparse and irregular observations make it difficult to reconstruct anatomically consistent intermediate volumes when motion is near-periodic but progresses non-uniformly. The paper learns a continuously queryable deformation process with finite-Fourier phase conditioning and phase-aligned temporal reparameterization, then synthesizes volumes through bidirectional endpoint warping. Experiments on ACDC and 4D-Lung report state-of-the-art interpolation performance and anatomically plausible, coherent intermediate volumes.
Problem
4D interpolation must recover missing volumes from sparse observations, but periodic motion is not directly encoded in many deformation representations and physiological progression is non-uniform.
Method
The method learns an endpoint-conditioned continuous deformation process using a finite-Fourier phase-conditioned velocity field, phase-aligned time reparameterization, bidirectional warping, fusion, and refinement.
Results
The method achieves state-of-the-art interpolation performance on ACDC and 4D-Lung while producing anatomically plausible and coherent intermediate volumes.
Takeaways & Limitations
Phase-structured deformation modeling supports anatomically plausible synthesis at arbitrary queried times from sparse endpoint observations.
Takeaways & Limitations
The phase-aligned map's well-posedness assumes continuous variation intensity, with additional bounds required for the stated stronger properties.
Abstract
from arXiv · showhide
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.
1 Introduction
4D medical image interpolation must recover missing volumes while preserving continuous, anatomically consistent motion from sparse observations. The proposed framework directly encodes near-periodic deformation and aligns time with non-uniform motion progression for continuous interpolation.
- Sparse or irregular acquisitions leave substantial portions of dynamic anatomical motion unobserved, requiring recovery of missing volumes from few time points.
- Existing methods use deformation estimation or direct synthesis, with temporal regularity typically added through conditioning or auxiliary guidance.
- The paper identifies two challenges: directly encoding structured motion in a continuous deformation representation and modeling uneven anatomical change within physiological cycles.
- The framework parameterizes a phase-conditioned continuous velocity field with a finite Fourier basis, embedding repetitive or near-periodic structure directly into deformation.
- Phase-aligned temporal reparameterization maps queried time to motion phase according to deformation variation, while bidirectional endpoint warping, fusion, and refinement synthesize intermediate volumes.
- Experiments on ACDC and 4D-Lung report state-of-the-art interpolation performance, with ablations verifying contributions from phase-structured deformation and temporal reparameterization.
2 Related Works
Related work spans deformation-based interpolation, direct synthesis, temporal regularization, and continuous-time motion modeling. The paper distinguishes its approach by encoding periodic structure directly in a continuously queryable deformation representation.
- Natural-scene video interpolation commonly estimates deformation or correspondences for warping and refinement, or uses conditional generation for flexible synthesis.
- 4D medical interpolation primarily estimates volumetric motion or voxel flow between observed scans and reconstructs intermediate volumes through warping.
- Medical methods also combine temporal context, periodic cues, deformation modeling, diffusion generation, multi-scale fusion, and temporal modulation for complex dynamics.
- Continuous spatiotemporal models extend interpolation from isolated timestamps to arbitrary-time querying by representing dynamics as continuous processes.
- The proposed framework uses endpoint-conditioned finite-Fourier deformation to encode repetitive motion directly, rather than relying mainly on temporal conditioning or auxiliary guidance.
3 Methodology
The method learns a continuously queryable deformation process from endpoint volumes using a finite-Fourier phase representation and a deformation-aware time-to-phase mapping. It synthesizes intermediate volumes by bidirectionally warping the endpoints and supervising the resulting candidates before refinement.
- 3.1 Task Formulation and Framework Overview: Given two endpoint volumes, the model learns a continuous interpolation function queried at any time between the observations.Intermediate observations may be used only for supervision and are not provided to the network.
- 3.2 Finite-Fourier Periodic Deformation: The phase-conditioned velocity field is represented with a finite Fourier expansion whose periodicity is intrinsic to the deformation representation.The zeroth-order field and sine/cosine coefficient fields encode coarse and fine phase-dependent motion, respectively.
- 3.2 Finite-Fourier Periodic Deformation: Frequency-aware regularization penalizes higher-order Fourier components to control spectral complexity and stabilize finite-order continuous interpolation.The regularizer uses a coefficient penalty with strength γ, set to 1.5 in practice.
- 3.3 Time-to-Phase Reparameterization: A deformation-induced temporal reparameterization assigns finer temporal resolution to phases with greater deformation variation while preserving the underlying trajectory.The mapping is endpoint-aligned and changes traversal speed rather than the trajectory itself.
- 3.4 Bidirectional Continuous Synthesis and Training Objective: At a queried time, both endpoints are transported along the phase-conditioned flow, and the warped candidates are directly supervised with similarity and Charbonnier losses.The flow is numerically integrated with a discretized Euler scheme; the similarity term combines local normalized cross-correlation with a weighted Charbonnier term.
4 Experiments and Results
Experiments on ACDC and 4D-Lung evaluate the proposed interpolation framework against established baselines and through targeted ablations. Results indicate consistent quantitative and qualitative improvements, with periodic deformation modeling and temporal reparameterization contributing to motion quality.
- Experimental Settings: The evaluation uses ACDC and 4D-Lung, reporting PSNR, NMI, and SSIM, with endpoint-conditioned interpolation and standardized training protocols.ACDC uses cardiac MRI endpoints, while 4D-Lung uses respiratory phases; intermediate frames provide supervision and evaluation.
- Comparison Study: Our method achieves the most consistent overall performance across both benchmarks and the reported metrics.Compared methods include deformation-based, volumetric interpolation, natural video, and generative baselines.
- Comparison Study: Predicted intermediates better preserve thin intracavitary structures, regular boundaries, global organization, and perihilar vessel-bundle-like structures.These improvements are reported qualitatively on ACDC and 4D-Lung, alongside improved continuity and plausibility of recovered motion.
- Ablation on Periodic Deformation Modeling: A compact low-order periodic representation performs best, while regularization concentrates spectral energy in low Fourier orders and suppresses higher-order components.The truncation-order study reports improvement from a small number of phase-varying terms before saturation or slight decline; spectral regularization particularly emphasizes the first harmonic.
- Ablation on Temporal Reparameterization: Temporal reparameterization provides a clearer benefit for high-motion and large-motion cases than for low-motion groups.The ablation compares motion-intensity groups on both datasets and reports weaker interpolation and structural recovery when reparameterization is removed.
- Additional Analysis: Each added loss term improves the reported metrics, and the full objective achieves the best overall results.The cycle-related constraint improves temporal consistency, while regularization stabilizes the learned deformation representation.
5 Conclusion
The paper presents a phase-aware framework that explicitly models structured and non-uniform physiological motion for continuous 4D medical image interpolation. It combines Fourier-based deformation modeling, phase alignment, endpoint warping, and refinement to synthesize anatomically plausible volumes at arbitrary times.
- The framework explicitly models structured and non-uniform physiological motion rather than treating interpolation as direct intensity prediction.
- A finite Fourier parameterization encodes periodic deformation patterns in a compact and interpretable representation.
- Phase-aligned temporal reparameterization links physical time with motion progression when physiological dynamics evolve unevenly.
- Bidirectional endpoint warping and lightweight refinement enable anatomically plausible synthesis at arbitrary queried times.
- Experiments on ACDC and 4D-Lung demonstrate the effectiveness of explicitly modeling physiological motion in deformation space.