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

🔼 Figure 3 visualizes the periodic nature of respiratory motion. It shows snapshots of a 3-second respiratory cycle (T=3s), highlighting the cyclical movement of specific anatomical structures. The colored boxes track the same anatomical region across different time points (t) within the cycle and at corresponding points in subsequent cycles (t+nT), demonstrating the repetitive pattern of breathing.

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Figure 2: Periodic display of respiratory motion (T=3⁢s𝑇3𝑠T=3sitalic_T = 3 italic_s). A specific anatomical structure (framed by boxes of the same color) at time t𝑡titalic_t has the same position at time t+n⁢T𝑡𝑛𝑇t+nTitalic_t + italic_n italic_T.

🔼 This figure shows the impact of two key techniques, Bounded Cycle Shifts and Log-Space Parameterization, on the learning process of the respiratory cycle period (T). The x-axis represents the number of iterations during training, and the y-axis represents the estimated period (T). The blue line (‘Ours’) demonstrates the stable and accurate convergence to the true respiratory period when both techniques are used. The orange line (‘w/o Bounded Cycle Shifts’) shows the instability where the estimated period oscillates and approaches half the true value. The green line (‘w/o Log-Space Parameterization’) shows significant oscillations in the estimated period. This visualization clearly illustrates the importance of these two methods in ensuring stable and accurate learning of the breathing cycle.

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Figure 3: Convergence behavior of the learnable period T^^𝑇\hat{T}over^ start_ARG italic_T end_ARG. Without Bounded Cycle Shifts, T^^𝑇\hat{T}over^ start_ARG italic_T end_ARG undergoes wide-ranging oscillations approaching half the true period. Without Log-Space Parameterization, the optimization curve exhibits large oscillations. With both techniques implemented, T^^𝑇\hat{T}over^ start_ARG italic_T end_ARG converges stably and accurately to the correct breathing cycle.

🔼 Figure 5 presents a qualitative comparison of 4D CT reconstruction results obtained using different methods across coronal, sagittal, and axial views. The images showcase the ability of each method to model dynamic anatomical structures, such as the diaphragm and airways, during respiration. X2-Gaussian, the proposed method, demonstrates superior performance in capturing the continuous movement of these structures, resulting in clearer and more detailed representations than existing methods. This superiority is particularly evident in the accurate depiction of finer anatomical details, which are often blurred or lost in other reconstruction approaches. The comparison highlights X2-Gaussian’s enhanced ability to resolve the complexities of dynamic imaging.

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Figure 4: Qualitative comparison of reconstruction results across coronal, sagittal, and axial planes. Our method shows superior performance in modeling dynamic regions (e.g. diaphragmatic motion and airway deformation) while preserving finer anatomical details compared to existing approaches.

🔼 Figure 6 presents a two-part visualization of the X2-Gaussian model’s performance. Subfigure (a) shows a graph illustrating the relationship between the number of X-ray projections used for reconstruction and the resulting 3D PSNR (Peak Signal-to-Noise Ratio) and 3D SSIM (Structural Similarity Index). The graph demonstrates the improvement in reconstruction quality as the number of projections increases. Subfigure (b) displays the temporal changes in lung volume over a respiratory cycle, as obtained from the 4D CT scan reconstructed by X2-Gaussian. The plot shows the volume variations of both the right and left lungs over time, highlighting the model’s ability to capture the dynamic nature of respiratory motion.

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Figure 5: (a) Reconstruction results of X2-Gaussian using different numbers of projections. (b) Temporal variations of lung volume in 4D CT reconstructed by X2-Gaussian.

🔼 Table 2 presents a quantitative comparison of the proposed X2-Gaussian method against several state-of-the-art 4D CT reconstruction techniques on two publicly available datasets: 4DLung and SPARE. The table shows the Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) metrics, which are common image quality assessment measures. Higher PSNR and SSIM values indicate better reconstruction quality. By comparing the performance of X2-Gaussian to other methods across these datasets, the table demonstrates the superior reconstruction capabilities of the proposed approach in terms of both quantitative metrics.

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Table 2: Comparison of our X2-Gaussian with different methods on the 4DLung and SPARE datasets.

🔼 This table presents a ablation study on the impact of different optimization techniques on the accuracy of respiratory cycle estimation using the DIR dataset. It compares the performance of the proposed method with and without key optimization components such as log-space parameterization and bounded cycle shifts, highlighting their importance in achieving accurate and stable period estimation.

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Table 3: Results of respiratory cycle estimation and different optimization techniques used on DIR dataset.

🔼 This table presents the results of ablation studies conducted to analyze the impact of different components and hyperparameters on the performance of the X2-Gaussian model. It specifically examines the contributions of dynamic Gaussian motion modeling (DGMM), self-supervised respiratory motion learning (SSRML), and the weight (α) assigned to the periodic consistency loss (Equation 12). By comparing the performance metrics (PSNR and SSIM) across various configurations, the table helps to quantify the individual contributions of each component and the optimal setting for the hyperparameter α.

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Table 4: Ablation studies on components and hyperparameters. DGMM denotes dynamic gaussian motion modeling in Sec. 4.2, and SSRML is self-supervised respiratory motion learning in Sec. 4.3. α𝛼\alphaitalic_α is the weight of periodic consistency loss in Eq. 12.

🔼 This table presents a quantitative comparison of the proposed X2-Gaussian method against several state-of-the-art techniques for 4D CT reconstruction. The evaluation is performed on the 4DLung dataset, a publicly available dataset containing 4D CT scans of patients with lung cancer. The comparison metrics used are PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity Index), both common measures of image quality. Results are provided for each of five patients in the dataset, as well as an overall average, allowing for a comprehensive assessment of the performance differences between the methods. The methods compared encompass traditional analytical approaches, novel neural radiance field (NeRF)-based methods, and other Gaussian splatting techniques.

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Table 5: Comparison of our X2-Gaussian with different methods on the 4DLung dataset.

🔼 This table presents a quantitative comparison of the proposed X2-Gaussian method against several state-of-the-art techniques for 4D CT reconstruction. The comparison is performed on the SPARE dataset, a publicly available benchmark dataset for 4D lung CT scans. The metrics used for comparison are PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity Index), which measure the image quality and structural similarity respectively. The results for each patient in the dataset are shown, alongside average values across all patients. This allows for a comprehensive evaluation of the relative performance of different methods in reconstructing 4D CT data from sparse projections.

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Table 6: Comparison of our X2-Gaussian with different methods on the SPARE dataset.