Deep learning-based tooth axis estimation from 3D tooth crowns using quaternion representation and multi-loss optimization

Abstract

Tooth axis estimation plays a key role in various digital dental workflows, including orthodontic diagnosis and prosthetic design. In clinical practice, tooth axes are typically determined by jointly considering crown morphology and radiographic information; however, this process is subjective and time-consuming. In particular, anatomical tooth axes are defined based on root information, and cone-beam computed tomography is commonly used for this purpose. However, its routine use is limited by radiation exposure, cost, and workflow complexity. As an alternative, crown-based references such as the facial axis of the clinical crown have been used as auxiliary indicators of tooth orientation in clinical settings. In this study, we propose a deep learning–based framework for automatically estimating crown-based tooth axes using only geometric information from 3D tooth crown point clouds. The proposed method first translates each tooth to its center and applies absolute orientation alignment to normalize all samples into a common coordinate system. Rotation-based data augmentation is then applied to incorporate diverse pose variations. During inference, medial–distal orientation alignment is performed to ensure consistent directional alignment of each tooth. A quaternion-based rotation regression model is used to estimate the tooth axis by rotating a predefined initial axis, and a composite loss function is employed to jointly enforce numerical, geometric, and directional consistency. Experimental results on a controlled patient-level split show that the proposed method achieves an average angular error of 3.23$^{\circ }$ (standard deviation: 2.06$^{\circ }$), along with relatively consistent Chamfer distance measures. In addition, qualitative analysis indicates that the predicted tooth axes exhibit relatively consistent directional patterns at the full-arch level. These findings suggest that crown surface geometry, combined with alignment-based preprocessing, may be useful for approximating crown-based tooth axes. Rather than replacing anatomical tooth axes, the proposed framework aims to explore the feasibility of estimating clinically meaningful reference axes under limited information settings, and may serve as a supportive geometric reference for crown-based digital dentistry workflows.