Joint Rigid Registration of Multiple Generalized Point Sets With Anisotropic Positional Uncertainties in Image-Guided Surgery

Abstract

In medical image analysis (MIA) and computer-assisted surgery (CAS), aligning two multiple point sets (PSs) together is an essential but also a challenging problem. For example, rigidly aligning multiple point sets into one common coordinate frame is a prerequisite for statistical shape modelling (SSM). Accurately aligning the pre-operative space with the intra-operative space in CAS is very crucial to successful interventions. In this article, we formally formulate the multiple generalized point set registration problem (MGPSR) in a probabilistic manner, where both the positional and the normal vectors are used. The six-dimensional vectors consisting of both positional and normal vectors are called as generalized points. In the formulated model, all the generalized PSs to be registered are considered to be the realizations of underlying unknown hybrid mixture models (HMMs). By assuming the independence of the positional and orientational vectors (i.e., the normal vectors), the probability density function (PDF) of an observed generalized point is computed as the product of Gaussian and Fisher distributions. Furthermore, to consider the anisotropic noise in surgical navigation, the positional error is assumed to obey a multi-variate Gaussian distribution. Finally, registering PSs is formulated as a maximum likelihood (ML) problem, and solved under the expectation maximization (EM) technique. By using more enriched information (i.e., the normal vectors), our algorithm is more robust to outliers. By treating all PSs equally, our algorithm does not bias towards any PS. To validate the proposed approach, extensive experiments have been conducted on surface points extracted from CT images of (i) a human femur bone model; (ii) a human pelvis bone model. Results demonstrate our algorithm’s high accuracy, robustness to noise and outliers. Note to Practitioners—This paper was motivated by solving the problem of registering two or more PSs. Most existing registration approaches use only the positional information associated with each point, and thus lacks robustness to noise and outliers. Three significant improvements are brought by our proposed approach. First, the normal vectors that can be extracted from the point sets are utilized in the registration. Second, the positional error distribution is assumed to be anisotropic and inhomogeneous. Third, all the PSs to be registered are treated equally that means no PS is considered as the model one. The registration problem is cast into a maximum likelihood (ML) problem and solved under the expectation maximization (EM) framework. We have demonstrated through extensive experiments that the proposed registration approach achieves significantly improved accuracy, robustness to noise and outliers. The algorithm is particularly suitable for biomedical applications involving the registration procedures, such as image-guided surgery.