Revisiting 3D Curve to Surface Registration using Tangent and Normal Vectors for Computer-Assisted Orthopedic Surgery

Abstract

In this paper, we present a novel curve-to-surface registration method, termed Bi-directional Hybrid Mixture Model Registration based on Dual-constrained Tangent and Normal Vectors (BiHMM-DTN), where two different tangent vectors at the intraoperative point are simultaneously used with the normal vector at the corresponding preoperative point to construct the geometric constraints. While hybrid registration models incorporating tangent and normal vectors (HMM-TN) demonstrate success, their geometric constraints prove inadequate or inappropriate for sparse intraoperative point sets, frequently yielding suboptimal optimization outcomes. By critically revisiting the geometric constraints of HMM-TN, we propose a dual-constraints-based hybrid mixture model registration framework with enhanced intraoperative point set acquisition protocols. To deal with noise and outliers in preoperative and intraoperative point sets—caused by reconstruction inaccuracies and tracking errors, respectively—our approach employs a bi-directional registration mechanism for curve-to-surface registration. We provide rigorous proofs validating the geometric completeness of the dual constraints within this mechanism. The BiHMM-DTN framework is formulated as a maximum likelihood estimation (MLE) problem and optimized using an expectation-maximization (EM) algorithm. Furthermore, to enhance convergence stability and accelerate optimization, the rotation matrix is updated iteratively through successive incremental steps. Extensive experiments on human femur and hip models demonstrate that our method outperforms state-of-the-art approaches, including both traditional optimization and deep learning methods, under various noise and outlier conditions. Furthermore, real-world phantom experiments highlight the potential clinical value of our method for surgical navigation applications. The codes and data are available at https://github.com/sam-zyzhang/BiHMM-DTN.git.