Robust estimation is essential in computer vision, robotics, and navigation, aiming to minimize the impact of outlier measurements for improved accuracy. We present a fast algorithm for Geman-McClure robust estimation, FracGM, leveraging fractional programming techniques. This solver reformulates the original non-convex fractional problem to a convex dual problem and a linear equation system, iteratively solving them in an alternating optimization pattern. Compared to graduated non-convexity approaches, this strategy exhibits a faster convergence rate and better outlier rejection capability. In addition, the global optimality of the proposed solver can be guaranteed under given conditions. We demonstrate the proposed FracGM solver with Wahba's rotation problem and 3-D point-cloud registration along with relaxation pre-processing and projection post-processing. Compared to state-of-the-art algorithms, when the outlier rates increase from 20% to 80%, FracGM shows 53% and 88% lower rotation and translation increases. In real-world scenarios, FracGM achieves better results in 13 out of 18 outcomes, while having a 19.43% improvement in the computation time.

翻译：鲁棒估计在计算机视觉、机器人学和导航领域中至关重要，其目标在于最小化异常值测量的影响以提高精度。我们提出了一种用于Geman-McClure鲁棒估计的快速算法FracGM，该算法利用了分式规划技术。该求解器将原始非凸分式问题重构为凸对偶问题与线性方程组，并以交替优化模式迭代求解。与渐进非凸性方法相比，此策略展现出更快的收敛速度与更优的异常值剔除能力。此外，所提求解器在给定条件下可保证其全局最优性。我们通过Wahba旋转问题与三维点云配准任务，结合松弛预处理与投影后处理，验证了所提出的FracGM求解器。与现有先进算法相比，当异常值比率从20%增至80%时，FracGM的旋转误差与平移误差增幅分别降低了53%与88%。在实际场景中，FracGM在18组结果中的13组取得了更优性能，同时计算时间提升了19.43%。