A generalized closed-form solution for 3D registration of two-point sets under isotropic and anisotropic scaling

Abstract

Fitting of two-point sets under non-uniform scaling involves several applications, including ultrasound-guided surgery, where ultrasound images generally show anisotropy between its lateral and depth resolution. However, the current state-of-the-art only accomplishes the closed-form solution under the assumption of isotropic scaling, and only the iterative form of the solution is available for registration regardless of scaling type. A generalized closed-form solution, including anisotropic scaling, could robust the computation in the applications of multi-modality image fusion and surgical navigation based on ultrasound images. In this paper, a generalized closed-form solution (GCFS) that accounts for isotropic scaling and anisotropic scaling is proposed. GCFS finds the least-squares solution of rotation, translation, and scaling based on the singular value decomposition (SVD). The method was demonstrated to 1) register two 3D point sets under the effect of anisotropic scaling and 2) register two coplanar 3D point sets under the effect of anisotropic scaling. This study evaluates 3D registration between the preoperative model and intraoperative ultrasonic scanning of a femur and 2D registration of ultrasound images to the phantom for ultrasound calibration. In addition, the conditions that affect the transformation estimations like estimating reflection and noise degeneracy were also considered in the evaluation.