Variable Substitution and Bilinear Programming for Aligning Partially Overlapping Point Sets

• URL: http://arxiv.org/abs/2405.08589v1
• Date: Tue, 14 May 2024 13:28:57 GMT
• Title: Variable Substitution and Bilinear Programming for Aligning Partially Overlapping Point Sets
• Authors: Wei Lian, Zhesen Cui, Fei Ma, Hang Pan, Wangmeng Zuo,
• Abstract summary: This research presents a method to meet requirements through the minimization objective function of the RPM algorithm. A branch-and-bound (BnB) algorithm is devised, which solely branches over the parameters, thereby boosting convergence rate. Empirical evaluations demonstrate better robustness of the proposed methodology against non-rigid deformation, positional noise, and outliers, when compared with prevailing state-of-the-art transformations.
• Score: 48.1015832267945
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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