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

This paper proposes a novel 4D medical image interpolation method that learns a continuous deformation process using a phase-conditioned velocity field with a finite Fourier basis and a phase-aligned temporal reparameterization to effectively model near-periodic and non-uniform physiological motion, achieving state-of-the-art performance in generating anatomically plausible intermediate volumes from sparse observations.

Original authors: Haojin Li, Hengzhuo Wang, Zhiheng Ma, Mingyang Ou, Heng Li, Jiang Liu

Published 2026-08-26

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CC BY 4.0

Original authors: Haojin Li, Hengzhuo Wang, Zhiheng Ma, Mingyang Ou, Heng Li, Jiang Liu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of medical imaging, doctors often rely on four-dimensional scans to watch the body in motion. While a standard three-dimensional image captures a static snapshot of anatomy, a four-dimensional scan adds the dimension of time, revealing how organs change shape and position as they function. This is crucial for understanding complex biological processes, such as the rhythmic beating of a heart or the expansion and contraction of lungs during breathing. However, capturing these movements in high detail is difficult. Medical scanners are often limited by the speed of acquisition or the need to minimize radiation exposure, resulting in data that is sparse or "gappy." Instead of a smooth, continuous movie of an organ's journey, doctors might only see a few isolated frames, leaving the moments in between a mystery. Filling in these missing moments is not just a matter of guessing; it requires understanding the specific, often repetitive, patterns of biological motion to ensure the reconstructed images remain anatomically accurate and safe for diagnosis.

Researchers have long tried to solve this problem by creating computer models that can predict the missing frames between two known images. Traditional methods often treat the time between scans as a simple, straight line, assuming that the organ moves at a steady pace from one moment to the next. This approach works well for some things, but it fails to capture the reality of human physiology. A heart does not beat with a perfectly uniform rhythm; the speed of its contraction and relaxation varies, and the amount of tissue movement changes throughout the cycle. Similarly, breathing is not a linear process. When a patient inhales, the chest expands quickly at first and then slows down, meaning that equal slices of time do not correspond to equal amounts of physical change. If a computer model ignores these nuances, the resulting images can look distorted or anatomically impossible, potentially hiding critical details or creating false structures that could mislead a physician.

To address these limitations, a team of researchers has developed a new method that treats medical image interpolation not as a simple guessing game, but as the learning of a continuous, structured motion process. Instead of trying to predict each missing frame independently, their approach builds a single, unified model of how the organ moves over time. They realized that many biological motions, like the heartbeat, are naturally repetitive. By recognizing this pattern, they designed a system that encodes this repetitive nature directly into the mathematical description of the movement. Imagine the motion as a wave that repeats itself; the researchers found a way to describe this wave using a specific set of building blocks that naturally fit a repeating cycle. This allows the computer to understand that the movement at one point in the cycle is related to the movement at another, creating a much more stable and realistic prediction of the missing moments.

A key innovation in this work is how the researchers handle the timing of these movements. They recognized that the clock time on a scanner does not always match the "phase" of the biological motion. To fix this, they introduced a way to map the actual time of a scan to a "motion phase" based on how much the organ is actually changing at that moment. If the organ is moving rapidly, the system allocates more detail to that part of the cycle; if it is moving slowly, it allocates less. This ensures that the computer does not waste effort trying to predict smooth, unchanging moments with high precision, nor does it miss the rapid, complex shifts that happen in a split second. By aligning the time of the scan with the actual intensity of the movement, the model can generate intermediate images that follow the true, uneven rhythm of the human body.

The researchers tested their method on two major sets of medical data: cardiac magnetic resonance images of the heart and four-dimensional computed tomography scans of the lungs. In these experiments, they provided the computer with only the start and end points of a motion cycle—such as the heart at its fullest and the heart at its emptiest—and asked it to generate the images in between. The results showed that their new approach produced significantly clearer and more accurate images than existing methods. The generated heart images preserved the delicate, thin structures inside the heart chambers, which often get blurred or lost in other techniques. Similarly, the lung images maintained the clear organization of blood vessels and airways throughout the breathing cycle. The method was particularly effective at handling the non-uniform speed of breathing and heartbeats, creating a smooth, continuous flow of images that looked like a real, high-quality movie rather than a series of disjointed snapshots.

Beyond just filling in the gaps, the study demonstrated that this approach could even predict what might happen slightly beyond the observed time, a capability known as extrapolation. While other methods tended to break down or produce unrealistic shapes when asked to guess beyond the known data, this new system maintained its accuracy for a short distance into the future. This suggests that the model had truly learned the underlying rules of the motion, rather than just memorizing the specific frames it was shown. The researchers found that the most successful results came from combining their motion-based model with a lightweight refinement step, which acted as a final polish to correct any small visual errors. This combination allowed them to achieve state-of-the-art performance, setting a new standard for how medical images can be reconstructed from limited data.

The implications of this work extend beyond just better pictures. By creating a more reliable way to visualize the body in motion from sparse data, this technology could help doctors analyze dynamic conditions with greater precision, potentially leading to better treatment plans for patients with heart or lung diseases. The study confirms that explicitly modeling the repetitive and uneven nature of physiological motion is a powerful strategy for medical imaging. It moves the field away from simple linear assumptions and toward a deeper understanding of how the body actually moves, ensuring that the digital reconstruction of our internal organs remains faithful to the complex reality of human life.

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

1. Problem Statement

4D medical image interpolation aims to recover missing volumetric data from sparsely observed time points, a task critical for dynamic anatomical analysis in applications like cardiac MRI and thoracic CT. While physiological motion in these domains is often repetitive or near-periodic, existing interpolation methods face two primary limitations:

Deformation Representation: Current approaches often treat temporal structure (e.g., periodicity) as external conditioning or auxiliary guidance rather than encoding it directly into the deformation representation itself. This makes it difficult to learn a continuous deformation process that remains reliable when queried at arbitrary target times.
Non-Uniform Motion: Physiological motion does not evolve at a uniform pace within a cycle. Equal temporal intervals do not necessarily correspond to equal amounts of anatomical change. Consequently, using linear time as a coordinate for continuous interpolation is suboptimal, as it fails to account for varying deformation intensities across the cycle.

2. Methodology

The authors propose a framework that reformulates 4D medical image interpolation as learning a structured, continuous deformation process conditioned on endpoint volumes. The method consists of three core components:

A. Finite-Fourier Periodic Deformation

Instead of predicting intermediate states independently, the method models the motion as a phase-conditioned continuous velocity field, $v(x, u)$ , where $x$ is the spatial coordinate and $u \in [0, 1]$ is a normalized phase coordinate.

Parameterization: The velocity field is parameterized using a finite Fourier basis:
$v(x, u) = c(x) + \sum_{n=1}^{N} [a_n(x) \cos(2\pi n u) + b_n(x) \sin(2\pi n u)]$
This formulation explicitly encodes near-periodic motion patterns directly into the deformation space. The network predicts the coefficient fields $\{c, a_n, b_n\}$ from the endpoint pair $(I_s, I_e)$ only once.
Regularization: A frequency-aware regularization term is applied to the coefficient fields to control spectral complexity and ensure stability under finite-order truncation. This regularizer penalizes higher-order terms based on their frequency squared, encouraging the model to capture dominant motion cycles with compact low-order representations.

B. Phase-Aligned Temporal Reparameterization

To address non-uniform motion progression, the authors introduce a deterministic mechanism to map queried physical time $t$ to a latent motion phase $u$ .

Mechanism: The mapping is derived from the learned deformation process itself. The method quantifies phase-wise deformation variation intensity, $s(u)$ , and defines a normalized phase density $\rho(u)$ .
Alignment: A cumulative phase measure $\Psi(u)$ is constructed, and the queried time is mapped to the aligned phase coordinate via the inverse function $\tau(t) = \Psi^{-1}(t)$ .
Effect: This reparameterization allocates finer temporal resolution to phases with high deformation variation (rapid motion) and coarser resolution to phases with low variation, ensuring that the interpolation step size aligns with the actual anatomical change rather than physical time.

C. Bidirectional Synthesis and Refinement

Intermediate volumes are synthesized by transporting both endpoint volumes to the target time along the learned velocity field:

Bidirectional Warping: The source volume $I_s$ is warped forward, and the target volume $I_e$ is warped backward to the queried time using the phase-aligned flow.
Fusion: The two warped candidates are fused using distance-aware weighting based on their proximity to the endpoints.
Residual Refinement: A lightweight residual network processes the fused volume to correct appearance errors not resolved by deformation alone.
Training Objective: The model is trained using a combination of morphing loss (supervising warped candidates), cycle consistency loss (ensuring full phase traversal returns to the start), refinement loss (image-level fidelity), and spectral regularization.

3. Key Contributions

The paper outlines four specific contributions:

Phase-Structured Framework: A novel framework for 4D interpolation based on phase-structured continuous motion modeling, enabling anatomically coherent synthesis from sparse endpoints at arbitrary query times.
Finite-Fourier Parameterization: The introduction of a finite-Fourier parameterization for phase-conditioned velocity fields, which encodes repetitive or near-periodic motion structures directly into the deformation representation.
Time-to-Phase Reparameterization: A mechanism that aligns queried time with motion phase progression based on deformation variation intensity, effectively capturing non-uniform motion evolution.
Empirical Validation: Comprehensive validation on ACDC (cardiac MRI) and 4D-Lung (thoracic CT) benchmarks, demonstrating state-of-the-art performance and verifying the distinct contributions of the proposed components through ablation studies.

4. Experimental Results

The proposed method was evaluated on the ACDC and 4D-Lung datasets against various baselines, including deformation-based methods (e.g., VoxelMorph, TransMorph), direct synthesis methods (e.g., IFRNet, AMT), and generative models (e.g., Diffusion-based approaches).

Quantitative Performance: The method achieved state-of-the-art results across all metrics (PSNR, NMI, SSIM) on both datasets. For instance, on ACDC, it achieved a PSNR of 32.503 dB and SSIM of 98.765, outperforming the next best methods. On 4D-Lung, it reached a PSNR of 30.460 dB and SSIM of 92.229.
Qualitative Improvements: Visual comparisons showed superior preservation of thin anatomical structures (e.g., intracavitary low-signal structures in the heart and perihilar vessel bundles in the lungs) compared to baselines, which often exhibited local inconsistencies or unstable structural arrangements.
Ablation Studies:
- Fourier Order: Performance peaked at a moderate truncation order ( $N=4$ ), confirming that a compact low-order periodic subspace is sufficient for dominant motion dynamics.
- Basis Comparison: The Fourier basis outperformed non-periodic alternatives (Chebyshev polynomials, cubic splines), validating the importance of periodic inductive bias.
- Temporal Reparameterization: The benefits of phase-aligned reparameterization were most pronounced in high-motion scenarios, where linear time failed to approximate motion progression accurately.
- Extrapolation: The method demonstrated slower performance decay during zero-shot extrapolation beyond the observed interval compared to baselines, suggesting the learned deformation captures extensible motion patterns.

5. Significance and Claims

The authors claim that their work addresses a critical gap in 4D medical imaging by moving beyond treating temporal structure as mere conditioning. By encoding periodicity directly into the deformation representation and aligning time with motion intensity, the method provides a more anatomically plausible and continuous interpolation process.

The paper emphasizes that this approach is particularly valuable for clinical applications where motion is repetitive but non-uniform. The ability to query arbitrary time points continuously, while preserving anatomical consistency and structural coherence, offers a significant improvement over existing discrete or linear-time-based interpolation techniques. The results suggest that explicitly modeling physiological motion in the deformation space is a promising direction for future continuous 4D medical image analysis.