Generalized 3-D Point Set Registration With Hybrid Mixture Models for Computer-Assisted Orthopedic Surgery: From Isotropic to Anisotropic Positional Error

Abstract

Registering two point sets (PSs) is an essential problem in medical robotics and computer-assisted surgery (CAS). As one typical example, in computer-assisted orthopedic surgery (CAOS), the preoperative scan has to be aligned with the intraoperative scan accurately. In this article, we first formally formulate the generalized PS registration problem in a probabilistic manner. Especially, not only positional but also orientational information is incorporated into the registration. Notably, the positional error is assumed to obey a multivariate Gaussian distribution to accommodate the anisotropic noise. The expectation–maximization (EM) framework is utilized to solve the maximum likelihood (ML) problem. In the E-step, the correspondence probabilities between points in two generalized PSs are computed. In the M-step, the constrained optimization problem with respect to the rigid transformation matrix is reformulated as an unconstrained one. This is achieved by utilizing the Rodrigues parameterization to represent the rotation matrix. Both extensive simulated and real experiments are conducted to validate the proposed algorithm by comparing it with state-of-the-art registration methods. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article was motivated by considering the anisotropic positional uncertainty into the rigid point set (PS) registration in the application of preoperative-to-intraoperative registration within image-guided surgery. We provide iterative solutions that compute the rotation and translation vector that aligns two 3-D PSs. The correspondences between points in two PSs are not known and regarded as hidden variables in the optimization process. Expectation–maximization technique is utilized to solve the maximum likelihood problem. We have demonstrated through experiments that our proposed approach can achieve lower registration error values than the compared state-of-the-art registration methods on various data sets. The readers should note that the proposed method is particularly suitable for cases that anisotropic noise is involved.