Mathematical Models For Natural Tooth Morphology Representations

This study aims to establish unified mathematical models for natural tooth morphology to address challenges in morphological analysis, classification, and simulation due to irregularity and inter-individual variations. Three approaches were proposed: NURBS-based reconstruction, surface parameterization (SP), and template-based non-rigid deformation (TBNR). Ninety right maxillary central incisor models (80 training, 10 validation) were preprocessed. For NURBS, 6×6 curvature-sensitive points were used to invert control points and reconstruct surfaces (100×100 resolution). SP utilized conformal mapping via discrete Dirichlet energy to project meshes onto a 2D domain, followed by 100×100 resampling and inverse mapping. TBNR employed a deep learning model (ToothDIT) to generate an implicit template from SDF-sampled teeth, with non-rigid deformation guided by 6×6 paired points. Fitting accuracy was evaluated using RMS, with paired t-tests comparing methods. Results showed average RMS values of 0.237 mm (NURBS), 0.0016 mm (SP), and 0.034 mm (TBNR). Significant differences existed between NURBS and SP/TBNR (p<0.05), but not between SP and TBNR. The study demonstrates that SP and TBNR achieve superior accuracy over NURBS, offering versatile tools for tooth morphology analysis. Researchers can select models based on project-specific needs, balancing precision, computational complexity, and interpretability. These methods advance foundational frameworks for applications in dental morphology studies, feature extraction, and clinical simulations.