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

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.