An Algebraic Approach to Affine Registration of Point Sets

Abstract

This paper proposes a new affine registration algorithm for matching two point sets in ℝ2 or ℝ3. The input point sets are represented as probability density functions, using either Gaussian mixture models or discrete density models, and the problem of registering the point sets is treated as aligning the two distributions. Since polynomials transform as symmetric tensors under an affine transformation, the distributions' moments, which are the expected values of polynomials, also transform accordingly. Therefore, instead of solving the harder problem of aligning the two distributions directly, we solve the softer problem of matching the distributions' moments. By formulating a least-squares problem for matching moments of the two distributions up to degree three, the resulting cost function is a polynomial that can be efficiently optimized using techniques originated from algebraic geometry: the global minimum of this polynomial can be determined by solving a system of polynomial equations. The algorithm is robust in the presence of noises and outliers, and we validate the proposed algorithm on a variety of point sets with varying degrees of deformation and noise.