  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.
