HBSP: a hybrid bilinear and semidefinite programming approach for aligning partially overlapping point clouds

In many applications, there is a need for algorithms that can align partially overlapping point clouds while remaining invariant to corresponding transformations. This research presents a method that achieves these goals by minimizing a binary linear assignment-least squares (BLALS) energy function. First, we reformulate the BLALS problem as the minimization of a quadratic function with quadratic and linear constraints through variable substitution. By utilizing semidefinite relaxation and the convex envelope of bilinear monomials, we relax the problem to create a lower bound that can be solved using linear assignment and low-dimensional semidefinite programming. Additionally, we develop a branch-and-bound (BnB) algorithm that only branches over the transformation variable, which enhances convergence. Experimental results show that, compared to state-of-the-art approaches, the proposed method is robust against non-rigid deformation and outliers when the outliers are separate from the inliers. However, its robustness decreases when outliers are mixed with inliers. The run time of our method is relatively high due to the need to solve a semidefinite program in each iteration of the BnB algorithm.