Affine registration for multidimensional point sets under the framework of Lie group

Abstract

An affine registration algorithm for multidimensional point sets under the framework of Lie group is proposed. This algorithm studies the affine registration between two data sets, and puts the expectation maximization-iterative closest point (EM-ICP) algorithm into the framework of Lie group, since all affine transformations form a Lie transformation group. The registration is carried out via minimizing an energy functional depending on elements of the affine transformation Lie group. The key point for applying the idea of Lie group is that, during the minimization via iteration, we must guarantee the next iteration step of the transformation is still an element in the same group, starting from an element in a Lie group. Our solution is utilizing the element of Lie algebra to represent that of Lie group near the identity via the exponential map, i.e., we use the first canonical coordinate representation of Lie group. Several comparative experiments between the proposed Lie-EM-ICP algorithm and the Lie-ICP algorithm are performed, showing that the proposed algorithm is more accurate and robust, especially in the presence of outliers. This algorithm can also be generalized to other registration problems in general, provided that desired transformations are within certain Lie group.