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

Haojin Li; Hengzhuo Wang; Zhiheng Ma; Mingyang Ou; Heng Li; Jiang Liu

cs.CVAug 25, 2026

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

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

Organizations: School of Computer Science and Engineering Beihang University Beijing, China

Abstract

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.

Figures & tables

Figure 1. Overview of the proposed framework. (a) The endpoint pair $(I_{\mathrm{s}},I_{\mathrm{e}})$ is used to predict Fourier coefficient fields that parameterize a phase-conditioned periodic velocity field $v(x,u)$ . (b) The queried time $t$ is reparameterized to phase $u=\tau(t)$ according to deformation variation. (c) The volume at time $t$ is synthesized by bidirectional deformation integration, followed by fusion and refinement. The figure illustrates the overall framework for continuous 4D medical image interpolation, consisting of three main components: (a) finite-Fourier periodic deformation, (b) time-to-phase reparameterization, and (c) bidirectional continuous synthesis. \parOn the left, the inputs include an endpoint pair of volumes, denoted as the start frame $I_s$ and end frame $I_e$, together with an observation time $t \in[t_s, t_e]$. These inputs are fed into a coefficient prediction network, which outputs spatially varying Fourier coefficients $a_n(x)$, $b_n(x)$, and a constant term $c(x)$. \parIn module (a), these coefficients are combined with cosine and sine basis functions, $\cos(2\pin u)$ and $\sin(2\pin u)$, where $u$ denotes a normalized motion phase. Through element-wise multiplication and summation, a time-varying periodic velocity field $v(x,u)$ is constructed using a finite Fourier basis. This field represents continuous deformation dynamics over the phase domain. \parIn module (b), a time-to-phase reparameterization is performed to handle non-uniform motion progression. First, a variation metric $\|\partialv(x,u)/\partialu\|$ is computed to measure deformation intensity over phase. This is normalized to obtain a phase density function $\rho(u)$, where regions with larger deformation correspond to higher density. The density is then integrated and inverted to produce a mapping from observation time $t$ to phase $u=\tau(t)$, aligning physical time with deformation progression. \parIn module (c), bidirectional continuous synthesis is performed. The phase-aligned velocity field is integrated over time to obtain deformation fields. Both endpoint volumes $I_s$ and $I_e$ are warped toward the target time using deformation integration. The resulting intermediate estimates are combined and refined through a lightweight refinement network $R(\cdot)$. The final output is a sequence of continuous intermediate frames between $I_s$ and $I_e$. \parSymbols in the figure indicate operations: circles denote Hadamard (element-wise) products, circled plus signs denote element-wise addition, and crossed circles denote concatenation. — Figure 1. Overview of the proposed framework. (a) The endpoint pair $(I_{\mathrm{s}},I_{\mathrm{e}})$ is used to predict Fourier coefficient fields that parameterize a phase-conditioned periodic velocity field $v(x,u)$ . (b) The queried time $t$ is reparameterized to phase $u=\tau(t)$ according to deformation variation. (c) The volume at time $t$ is synthesized by bidirectional deformation integration, followed by fusion and refinement. The figure illustrates the overall framework for continuous 4D medical image interpolation, consisting of three main components: (a) finite-Fourier periodic deformation, (b) time-to-phase reparameterization, and (c) bidirectional continuous synthesis. \parOn the left, the inputs include an endpoint pair of volumes, denoted as the start frame $I_s$ and end frame $I_e$ , together with an observation time $t \in[t_s, t_e]$ . These inputs are fed into a coefficient prediction network, which outputs spatially varying Fourier coefficients $a_n(x)$ , $b_n(x)$ , and a constant term $c(x)$ . \parIn module (a), these coefficients are combined with cosine and sine basis functions, $\cos(2\pin u)$ and $\sin(2\pin u)$ , where $u$ denotes a normalized motion phase. Through element-wise multiplication and summation, a time-varying periodic velocity field $v(x,u)$ is constructed using a finite Fourier basis. This field represents continuous deformation dynamics over the phase domain. \parIn module (b), a time-to-phase reparameterization is performed to handle non-uniform motion progression. First, a variation metric $\|\partialv(x,u)/\partialu\|$ is computed to measure deformation intensity over phase. This is normalized to obtain a phase density function $\rho(u)$ , where regions with larger deformation correspond to higher density. The density is then integrated and inverted to produce a mapping from observation time $t$ to phase $u=\tau(t)$ , aligning physical time with deformation progression. \parIn module (c), bidirectional continuous synthesis is performed. The phase-aligned velocity field is integrated over time to obtain deformation fields. Both endpoint volumes $I_s$ and $I_e$ are warped toward the target time using deformation integration. The resulting intermediate estimates are combined and refined through a lightweight refinement network $R(\cdot)$ . The final output is a sequence of continuous intermediate frames between $I_s$ and $I_e$ . \parSymbols in the figure indicate operations: circles denote Hadamard (element-wise) products, circled plus signs denote element-wise addition, and crossed circles denote concatenation.

Method	Venue	ACDC			4D-Lung			#P (M)
Method		PSNR ${}_{\text{dB}}^{\uparrow}$	NMI ${}_{\times 10^{-2}}^{\uparrow}$	SSIM ${}_{\times 10^{-2}}^{\uparrow}$	PSNR ${}_{\text{dB}}^{\uparrow}$	NMI ${}_{\times 10^{-2}}^{\uparrow}$	SSIM ${}_{\times 10^{-2}}^{\uparrow}$
VM ( Balakrishnan et al., 2019 )	TMI’19	25.694 ±0.074	53.543 ±0.367	94.678 ±0.104	26.077 ±0.066	43.982 ±0.355	86.055 ±0.292	10.044
TM ( Chen et al., 2022 )	MedIA’22	27.125 ±0.200	64.628 ±0.671	96.644 ±0.123	26.683 ±0.051	48.945 ±0.104	84.779 ±0.125	10.804
MPVF ( Wei et al., 2023 )	JBHI’23	32.144 ±0.011	70.658 ±0.038	98.669 ±0.003	29.909 ±0.035	53.074 ±0.389	90.180 ±0.053	14.533
SVIN ( Guo et al., 2020 )	CVPR’20	28.434 ±0.443	60.178 ±0.335	97.352 ±0.152	27.312 ±0.280	43.171 ±0.392	85.442 ±0.286	12.393
IFRNet ( Kong et al., 2022 )	CVPR’22	32.447 ±0.070	70.736 ±0.149	98.741 ±0.023	29.825 ±0.058	51.567 ±0.539	90.370 ±0.092	24.520

Table 1. Quantitative comparison on the ACDC and 4D-Lung datasets in terms of PSNR, NMI and SSIM, with model size (#P) also reported.

Figure 2. Qualitative comparison on the ACDC and 4D-Lung datasets, showing better preservation of thin intracavitary low-signal structures in ACDC and clearer perihilar vessel-bundle-like structures in 4D-Lung. Qualitative comparison of interpolated 4D medical images on two datasets. The top row shows ACDC and the bottom row shows 4D-Lung. Each row contains nine columns labeled, from left to right, Ground Truth, MPVF, IFRNet, AMT, UVI-Net, PerVFI, LDDM, TMSDF, and Ours, with “Ours” shown in bold. Large vertical dataset labels appear on the left side. \parIn the ACDC row, each panel shows a cardiac MRI slice. A light pink square marks a small region near the ventricular cavity, and a larger zoomed inset at the lower left is connected to the marked region by pink dotted guide lines. The enlarged comparison highlights a thin low-signal intracavitary structure. In the ground-truth inset, this structure appears narrow and continuous. Several competing methods make this local pattern appear blurrier, thicker, or less continuous, while the proposed method preserves a thinner and more coherent structure that is visually closer to the ground truth. \parIn the 4D-Lung row, each panel shows a thoracic CT slice. A light yellow square marks a region near the lung hilum, and a larger zoomed inset at the lower left is connected by yellow dotted guide lines. The enlarged region emphasizes fine perihilar vessel-bundle-like anatomy. In the ground-truth inset, the local texture appears relatively clear and structured. Several competing methods look smoother or less distinct in this region, whereas the proposed method shows a sharper and more organized local pattern that is visually closer to the ground truth. \parOverall, the figure highlights local anatomical fidelity through matched zoomed insets, showing that the proposed method better preserves subtle intracavitary structures on ACDC and clearer perihilar vessel-bundle-like structures on 4D-Lung.

Figure 3. Ablation studies on periodic deformation field modeling. (a) Effect of the truncation order $N$ and the regularization term $\mathcal{L}_{\mathrm{reg}}$ . (b) Replacing the Fourier basis with Chebyshev polynomials and cubic splines to assess the role of periodic parameterization. (c) Spectral energy distribution across Fourier orders, with and without $\mathcal{L}_{\mathrm{reg}}$ . A figure with three subplots arranged horizontally, labeled (a), (b), and (c), presenting ablation studies on periodic deformation field modeling. \parSubplot (a), on the left, is a line chart showing the effect of the Fourier truncation order $N$ and the regularization term $\mathcal{L}_{\mathrm{reg}}$. The horizontal axis is the truncation order, increasing from left to right, and the vertical axis is interpolation performance. Two curves are shown: a red curve for the model with $\mathcal{L}_{\mathrm{reg}}$ and a blue curve for the model without $\mathcal{L}_{\mathrm{reg}}$. Both curves improve substantially as the order increases from low values to middle values, then level off or slightly decline at higher orders. Across most orders, the red curve stays above the blue curve, indicating better performance when the regularization term is used. \parSubplot (b), in the middle, is a grouped bar chart comparing three periodic parameterizations: Fourier (ours), Chebyshev, and Cubic Spline. Two groups of bars are shown, corresponding to two datasets or evaluation settings. In both groups, the red Fourier bars are the tallest, the blue Chebyshev bars are lower, and the green Cubic Spline bars are slightly lower still. Numerical values are printed above the bars, showing that Fourier achieves the best results in both groups. \parSubplot (c), on the right, shows the spectral energy distribution across Fourier orders, comparing the settings with and without $\mathcal{L}_{\mathrm{reg}}$. The horizontal axis is the Fourier order, and the vertical axis is the normalized energy magnitude. Red markers and shaded shapes indicate the model with $\mathcal{L}_{\mathrm{reg}}$, and blue markers and shaded shapes indicate the model without it. The displayed values show that lower orders contain the largest energy, while higher orders have progressively smaller energy. The two settings allocate energy differently across orders: with $\mathcal{L}_{\mathrm{reg}}$, the first order has noticeably higher energy, while some middle and higher orders show different relative magnitudes compared with the model without regularization. \parOverall, the figure shows that using a Fourier parameterization gives the strongest results among the tested basis choices, that moderate truncation orders perform best, and that the regularization term changes the spectral distribution while also improving overall interpolation performance. — Figure 3. Ablation studies on periodic deformation field modeling. (a) Effect of the truncation order $N$ and the regularization term $\mathcal{L}_{\mathrm{reg}}$ . (b) Replacing the Fourier basis with Chebyshev polynomials and cubic splines to assess the role of periodic parameterization. (c) Spectral energy distribution across Fourier orders, with and without $\mathcal{L}_{\mathrm{reg}}$ . A figure with three subplots arranged horizontally, labeled (a), (b), and (c), presenting ablation studies on periodic deformation field modeling. \parSubplot (a), on the left, is a line chart showing the effect of the Fourier truncation order $N$ and the regularization term $\mathcal{L}_{\mathrm{reg}}$ . The horizontal axis is the truncation order, increasing from left to right, and the vertical axis is interpolation performance. Two curves are shown: a red curve for the model with $\mathcal{L}_{\mathrm{reg}}$ and a blue curve for the model without $\mathcal{L}_{\mathrm{reg}}$ . Both curves improve substantially as the order increases from low values to middle values, then level off or slightly decline at higher orders. Across most orders, the red curve stays above the blue curve, indicating better performance when the regularization term is used. \parSubplot (b), in the middle, is a grouped bar chart comparing three periodic parameterizations: Fourier (ours), Chebyshev, and Cubic Spline. Two groups of bars are shown, corresponding to two datasets or evaluation settings. In both groups, the red Fourier bars are the tallest, the blue Chebyshev bars are lower, and the green Cubic Spline bars are slightly lower still. Numerical values are printed above the bars, showing that Fourier achieves the best results in both groups. \parSubplot (c), on the right, shows the spectral energy distribution across Fourier orders, comparing the settings with and without $\mathcal{L}_{\mathrm{reg}}$ . The horizontal axis is the Fourier order, and the vertical axis is the normalized energy magnitude. Red markers and shaded shapes indicate the model with $\mathcal{L}_{\mathrm{reg}}$ , and blue markers and shaded shapes indicate the model without it. The displayed values show that lower orders contain the largest energy, while higher orders have progressively smaller energy. The two settings allocate energy differently across orders: with $\mathcal{L}_{\mathrm{reg}}$ , the first order has noticeably higher energy, while some middle and higher orders show different relative magnitudes compared with the model without regularization. \parOverall, the figure shows that using a Fourier parameterization gives the strongest results among the tested basis choices, that moderate truncation orders perform best, and that the regularization term changes the spectral distribution while also improving overall interpolation performance.

Figure 4. Ablation on temporal reparameterization. (a) Analysis of interpolation performance across frame groups with different motion intensities on ACDC and 4D-Lung. (b) Qualitative and quantitative comparison with and without temporal reparameterization, including visualization examples under large motion. Ablation figure on temporal reparameterization, consisting of two subfigures labeled (a) and (b). \parSubfigure (a) shows four split violin plots comparing interpolation performance with and without temporal reparameterization under different motion intensities on ACDC and 4D-Lung. The first and second plots correspond to ACDC, and the third and fourth plots correspond to 4D-Lung. In each dataset, the first plot is for low-motion and the second plot is for high-motion. Red denotes results with temporal reparameterization, and blue denotes results without temporal reparameterization. Each violin includes a colored dot indicating the average performance. In the low-motion groups, the red and blue distributions are close, indicating that temporal reparameterization brings only a small difference. In the high-motion groups, the red distributions and mean markers are clearly higher than the blue ones, showing that temporal reparameterization provides a more obvious improvement when motion is larger. \parSubfigure (b) presents qualitative and quantitative comparisons under large motion, with ACDC on the left and 4D-Lung on the right, separated by a vertical dashed divider. For each dataset, the image with temporal reparameterization and the image without temporal reparameterization are shown on the top row, and the ground-truth image is shown below. Small square boxes mark regions of interest, and enlarged insets connected by dotted guide lines highlight local details. \parFor ACDC, the result without temporal reparameterization is noticeably blurrier in the enlarged region, while the version with temporal reparameterization preserves clearer local structure and is visually closer to the ground truth. For 4D-Lung, the result without temporal reparameterization shows abnormal artifacts in the highlighted region, whereas the version with temporal reparameterization appears cleaner and more anatomically plausible, again closer to the ground truth. \parText annotations in subfigure (b) report the PSNR drop after removing temporal reparameterization: from 32.503 to 31.300 on ACDC, and from 30.460 to 29.448 on 4D-Lung. Overall, the figure shows that temporal reparameterization has limited effect in low-motion cases but yields clear quantitative and visual benefits in high-motion cases.

$\mathcal{L}_{\mathrm{refine}}$	$\mathcal{L}_{\mathrm{morph}}$	$\mathcal{L}_{\mathrm{cycle}}$	$\mathcal{L}_{\mathrm{reg}}$	PSNR ${}_{\text{dB}}^{\uparrow}$	NMI ${}_{\times 10^{-2}}^{\uparrow}$	SSIM ${}_{\times 10^{-2}}^{\uparrow}$
$\checkmark$				30.700	63.753	98.206
$\checkmark$	$\checkmark$			31.786	66.766	98.528
$\checkmark$	$\checkmark$	$\checkmark$		31.927	67.812	98.583
$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	32.503	72.408	98.765

Table 2. Loss term ablation on ACDC.

Figure 5. Zero-shot extrapolation beyond the observed interval on the ACDC dataset. Models are trained using only frames within $[t_{s},t_{e}]$ and directly evaluated on unseen future queries beyond the observed endpoint. (a) PSNR at different extrapolation offsets for our method and baselines. (b) Visual comparison at two increasing future query offsets. A figure with two vertically stacked parts illustrating unsupervised extrapolation beyond the observed interval on the ACDC dataset. \parThe upper part, labeled as subplot (a) by the caption, is a line chart of PSNR versus extrapolation offset. The horizontal axis represents increasingly larger future query offsets beyond the observed endpoint, and the vertical axis represents PSNR. Three methods are compared: Ours in red, LDDM in blue, and VM in green. The red curve remains the highest across all extrapolation offsets and decreases only slightly as the extrapolation horizon increases. The blue curve starts relatively high but drops steadily and substantially with increasing extrapolation distance. The green curve is consistently lower than the red curve and changes more gradually, staying close to the blue curve at later offsets. Overall, the plot shows that the proposed method maintains the strongest and most stable extrapolation performance over the full range of future query offsets. \parThe lower part, labeled as subplot (b) by the caption, is a qualitative comparison grid with four columns and two rows. The columns are Ground Truth, VM, LDDM, and Ours. The two rows correspond to two unseen future query offsets, marked on a vertical arrow at the left as $+0.1$ for the upper row and $+0.35$ for the lower row. Each image is a cardiac MRI slice. In every panel, a small square marks a local region of interest near the ventricular cavity, and a larger zoomed inset connected by dotted guide lines shows this region in detail. \parAt both extrapolation offsets, the Ground Truth images show a compact bright cardiac structure with a surrounding boundary that remains smooth and anatomically coherent. The proposed method produces zoomed regions that visually remain closest to the Ground Truth in both shape and local intensity pattern. VM appears blurrier and less well defined, especially in the magnified views. LDDM is closer to the Ground Truth at the smaller extrapolation offset but shows stronger distortion, artifact, and local structural degradation at the larger extrapolation offset $+0.35$. The visual comparison therefore matches the quantitative trend in the line plot, showing that the proposed method preserves local anatomy more reliably as extrapolation moves farther beyond the observed interval. — Figure 5. Zero-shot extrapolation beyond the observed interval on the ACDC dataset. Models are trained using only frames within $[t_{s},t_{e}]$ and directly evaluated on unseen future queries beyond the observed endpoint. (a) PSNR at different extrapolation offsets for our method and baselines. (b) Visual comparison at two increasing future query offsets. A figure with two vertically stacked parts illustrating unsupervised extrapolation beyond the observed interval on the ACDC dataset. \parThe upper part, labeled as subplot (a) by the caption, is a line chart of PSNR versus extrapolation offset. The horizontal axis represents increasingly larger future query offsets beyond the observed endpoint, and the vertical axis represents PSNR. Three methods are compared: Ours in red, LDDM in blue, and VM in green. The red curve remains the highest across all extrapolation offsets and decreases only slightly as the extrapolation horizon increases. The blue curve starts relatively high but drops steadily and substantially with increasing extrapolation distance. The green curve is consistently lower than the red curve and changes more gradually, staying close to the blue curve at later offsets. Overall, the plot shows that the proposed method maintains the strongest and most stable extrapolation performance over the full range of future query offsets. \parThe lower part, labeled as subplot (b) by the caption, is a qualitative comparison grid with four columns and two rows. The columns are Ground Truth, VM, LDDM, and Ours. The two rows correspond to two unseen future query offsets, marked on a vertical arrow at the left as $+0.1$ for the upper row and $+0.35$ for the lower row. Each image is a cardiac MRI slice. In every panel, a small square marks a local region of interest near the ventricular cavity, and a larger zoomed inset connected by dotted guide lines shows this region in detail. \parAt both extrapolation offsets, the Ground Truth images show a compact bright cardiac structure with a surrounding boundary that remains smooth and anatomically coherent. The proposed method produces zoomed regions that visually remain closest to the Ground Truth in both shape and local intensity pattern. VM appears blurrier and less well defined, especially in the magnified views. LDDM is closer to the Ground Truth at the smaller extrapolation offset but shows stronger distortion, artifact, and local structural degradation at the larger extrapolation offset $+0.35$ . The visual comparison therefore matches the quantitative trend in the line plot, showing that the proposed method preserves local anatomy more reliably as extrapolation moves farther beyond the observed interval.

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Aug 25, 2026J/K move · Enter open · S save

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

Haojin Li, Hengzhuo Wang, Chang Liu, Zhiheng Ma+2

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+2

Medical Image Synthesis 4D Reconstruction Deformable Image Registration

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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.

Abstract

4D Reconstruction

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