Joint Alignment of Multiple Generalized Point Sets with Anisotropic Positional Uncertainty Based on Expectation Maximization

Abstract

Alignment of multiple point sets is an essential problem in medical imaging and computer-assisted surgery. For example, aligning multiple point sets into one common coordinate frame is a prerequisite for statistical shape modelling (SSM). In this paper, we first formally formulate the multiple generalized point cloud registration problem in a probabilistic manner. Not only positional but also the orientational information is utilized in the registration. All the observed generalized point sets to be registered are considered to be realizations of underlyinng unknown hybrid mixture models (HMMs). By (i) utilizing more enriched information, i.e. orientational information or normal vectors (ii) treating all point sets equally, our registration algorithm is more robust to outliers and does not bias towards any point set. Assuming that the positional and orientational data are co-independent, the probability density function (PDF) of an observed hybrid point is the multiplication of Gaussian and Fisher distributions. Notably, the positional error vector is assumed to obey a multivariate Gaussian distribution to accommodate anisotropic noise. Expectation maxmization (EM) framework is utilized to jointly estimate the parameters. In the E-step, the posteriors between points and underlying mixture model components are computed. In the M-step, the constrained optimization problem of the rigid transformation matrix is re-formulated as an unconstrained one using the Rodrigues Formula of a rotation matrix. Extensive experiments are conducted on CT data of a femur bone model to compare the proposed algorithm with the state-of-the-art registration methods. The experimental results demonstrate the algorithm's better accuracy, robustness to noise and outliers and faster convergence speed.