X$^{2}$-Gaussian: 4D Radiative Gaussian Splatting for Continuous-time Tomographic Reconstruction

Four-dimensional computed tomography ($4 D C T$) reconstruction is crucial for capturing dynamic anatomical changes but faces inherent limitations from conventional phase-binning workflows. Current methods discretize temporal resolution into fixed phases with respiratory gating devices, introducing motion misalignment and restricting clinical practicality. In this paper, We propose $X^{2}$ Gaussian, a novel framework that enables continuoustime 4D-CT reconstruction by integrating dynamic radiative Gaussian splatting with self-supervised respiratory motion learning. Our approach models anatomical dynamics through a spatiotemporal encoder-decoder architecture that predicts time-varying Gaussian deformations, eliminating phase discretization. To remove dependency on external gating devices, we introduce a physiology-driven periodic consistency loss that learns patient-specific breathing cycles directly from projections via differentiable optimization. Extensive experiments demonstrate state-of-the-art performance, achieving a $9.93 d B$ PSNR gain over traditional methods and $2.25 d B$ improvement against prior Gaussian splatting techniques. By unifying continuous motion modeling with hardware-free period learning, $X^{2}$ Gaussian advances high-fidelity 4D CT reconstruction for dynamic clinical imaging. Code will be publicly available at: https://github.com/CUHK-AIM-Group/X2-Gaussian.