Robust and Accurate Nonrigid Point Set Registration Algorithm to Accommodate Anisotropic Positional Localization Error Based on Coherent Point Drift

Nonrigid point set (PS) registration is an outstanding and fundamental problem in the fields of robotics, computer vision, medical image analysis, and image-guided surgery (IGS). The aim of a nonrigid registration problem is to align together two point sets where one has been deformed. The assumption of isotropic localization error is shared in the previous nonrigid registration algorithms. In this article, we have derived and presented a novel nonrigid registration algorithm, where the position localization error (PLE) is generalized to be anisotropic, which means that the error distribution is not the same in different spatial directions. The motivation of considering the anisotropic characteristic is that the PLE is actually different in three spatial directions in real applications of registrations, such as IGS. Mathematically, the difficulty in dealing with the anisotropic error case comes from the change from a standard deviation that is a scalar to a covariance matrix. The formulas for updating the parameters in both expectation and maximization steps are derived. More specifically, in the expectation step, we compute the posterior probabilities that represent the correspondences between points in two PSs. In the maximization step, given the current posteriors, the covariance matrix of the PLE and the nonrigid transformation are updated. To further speed up the proposed algorithm, the low-rank approximation variation of our method is also presented. We have demonstrated through experiments on both general and medical data sets (corrupted with noise) that the proposed algorithm outperforms the state-of-the-art ones in terms of registration accuracy and robustness to noise. More specifically, all the experimental results have passed the statistical tests at the 5% significance level. Note to Practitioners—This article was motivated by solving the problem of nonrigidly registering two point sets where one has been deformed and corrupted with anisotropic noise. Most existing registration methods generally assume the positional error to be the same in all directions, which in fact is not the case in real scenarios. This article presents a new robust method that assumes the positional error to be anisotropic, which is the case in point sets coming from the stereo reconstruction. The nonrigid registration problem is formulated as a maximum-likelihood (ML) problem and solved with the expectation–maximization (EM) technique. We have demonstrated through extensive experiments on both general and medical data sets that the proposed registration algorithm achieves significantly improved accuracy, robustness to noise, and outliers compared with state-of-the-art algorithms. The algorithm is particularly suitable for biomedical applications involving the registration, such as medical imaging and image-guided surgery (IGS).

I. Introduction

Point set registration (PSR) is an important and fundamental problem in the fields of robotics [1], [2], medical image analysis [3]–[5], computer vision [1], [6]–[10], and image-guided surgery (IGS) [11], [11]–[23]. Broadly speaking, PSR can be divided into rigid PSR and nonrigid PSR [9]. In terms of the rigid PSR, the parameters are a few (such as a rotation matrix, a translation vector, and possibly a scaling) and the problem is rather simple compared to the nonrigid registration one [24], [25]. Recently, the normal vectors are adopted in the rigid registration problem and have been validated to significantly improve the registration’s accuracy and robustness [20], [24], [26]–[28]. On the other hand, nonrigid PSR is still a challenging problem because the transformation between the two point sets (PSs) is nonrigid and unknown. The simplest approximations of the true nonrigid transformation, including piecewise affine and polynomial models, are often inaccurate for correct nonrigid transformation.