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Point set registration under affine transformation is an important problem in computer vision because not only it has many direct applications, but it is also often used as an initial step for non-rigid registration. This problem is challenging when no correspondences between the two point sets are known, and most existing methods start from an initial pose and find a local optimal transformation. This paper presents a deterministic global optimization method for affine point set registration, which is called GO-APSR. We model the two sets to be registered with Gaussian Mixture Models (GMMs) and minimize the L2 distance between the two GMMs under affine transformation. Branch-and-Bound (BnB) is employed to search the transformation parameter space, and we propose a convex quadratic function as the under-estimator of the objective function in each branch. Therefore, calculation of the lower bound in each branch is casted into a bound-constrained convex quadratic programming problem, which can be solved globally and efficiently. Experiment results verify the global optimality of the proposed method and its robustness to noise and outliers. Furthermore, it works very well in the challenging partially-overlap scenarios.

GO-APSR: A Globally Optimal Affine Point Set Registration Method