Computed tomography (CT) has been a powerful diagnostic tool since its emergence in the 1970s. Using CT data, three-dimensional (3D) structures of human internal organs and tissues, such as blood vessels, can be reconstructed using professional software. This 3D reconstruction is crucial for surgical operations and can serve as a vivid medical teaching example. However, traditional 3D reconstruction heavily relies on manual operations, which are time-consuming, subjective, and require substantial experience. To address this problem, we develop a novel semiparametric Gaussian mixture model tailored for the 3D reconstruction of blood vessels. This model extends the classical Gaussian mixture model by enabling nonparametric variations in the component-wise parameters of interest according to voxel positions. We develop a kernel-based expectation-maximization algorithm for estimating the model parameters, accompanied by a supporting asymptotic theory. Furthermore, we propose a novel regression method for optimal bandwidth selection. Compared to the conventional cross-validation-based (CV) method, the regression method outperforms the CV method in terms of computational and statistical efficiency. In application, this methodology facilitates the fully automated reconstruction of 3D blood vessel structures with remarkable accuracy.

翻译：计算机断层扫描（CT）自20世纪70年代问世以来，已成为一种强大的诊断工具。利用CT数据，可通过专业软件重建人体内部器官和组织（如血管）的三维结构。这种三维重建对于外科手术至关重要，并可作为生动的医学教学示例。然而，传统三维重建严重依赖人工操作，不仅耗时、主观性强，还需要丰富的经验。为解决这一问题，我们开发了一种专用于血管三维重建的新型半参数高斯混合模型。该模型通过允许感兴趣的分量参数根据体素位置进行非参数变化，扩展了经典高斯混合模型。我们开发了一种基于核的期望最大化算法来估计模型参数，并建立了相应的渐近理论。此外，我们提出了一种用于最优带宽选择的新颖回归方法。与传统的基于交叉验证（CV）的方法相比，该回归方法在计算效率和统计效率方面均更优。应用表明，该方法能够以卓越的精度实现三维血管结构的全自动重建。