In many applications, the demand arises for algorithms capable of aligning partially overlapping point sets while remaining invariant to the corresponding transformations. This research presents a method designed to meet such requirements through minimization of the objective function of the robust point matching (RPM) algorithm. First, we show that the RPM objective is a cubic polynomial. Then, through variable substitution, we transform the RPM objective to a quadratic function. Leveraging the convex envelope of bilinear monomials, we proceed to relax the resulting objective function, thus obtaining a lower bound problem that can be conveniently decomposed into distinct linear assignment and low-dimensional convex quadratic program components, both amenable to efficient optimization. Furthermore, a branch-and-bound (BnB) algorithm is devised, which solely branches over the transformation parameters, thereby boosting convergence rate. Empirical evaluations demonstrate better robustness of the proposed methodology against non-rigid deformation, positional noise, and outliers, particularly in scenarios where outliers remain distinct from inliers, when compared with prevailing state-of-the-art approaches.

翻译：在众多应用场景中，需要能够在保持变换不变性的同时实现部分重叠点集配准的算法。本研究提出一种通过最小化鲁棒点匹配（RPM）算法目标函数来满足上述需求的方法。首先我们证明RPM目标函数为三次多项式，进而通过变量替换将其转化为二次函数。利用双线性单项式的凸包络，我们对所得目标函数进行松弛处理，从而获得一个可便捷分解为独立线性分配问题与低维凸二次规划问题的下界问题，两者均能高效优化。此外，我们设计了仅对变换参数进行分支限界（BnB）的算法，从而显著提升收敛速度。实验评估表明，与现有最优方法相比，本方法在非刚性形变、位置噪声和离群点干扰下具有更优的鲁棒性，尤其在离群点与内点存在显著差异的场景中优势更为突出。