提出快速求解鲁棒估计的分数规划算法，提升精度与效率。
FracGM: A Fast Fractional Programming Technique for Geman-McClure Robust Estimator
- 将非凸问题转为凸对偶与线性系统，交替优化加速收敛。
- 外点率20%到80%时，旋转/平移误差降低53%和88%。
- 适合高外点场景下的视觉、机器人定位任务，计算快19.43%。
鲁棒估计在计算机视觉、机器人与导航中至关重要，旨在降低异常值测量的影响以提升精度。本文提出一种针对Geman-McClure鲁棒估计的快速算法FracGM，利用分数规划技术。该求解器将原非凸分数问题重构为凸对偶问题与线性方程组，通过交替优化迭代求解。相比梯度非凸方法，该策略具有更快收敛速度与更强外点剔除能力。在给定条件下可保证全局最优。我们在Wahba旋转问题与三维点云配准中验证了FracGM，结合松弛预处理与投影后处理。相比当前最优算法，当外点率从20%增至80%时，旋转与平移误差分别降低53%和88%。在真实场景中，FracGM在18组结果中有13组表现更优，计算时间减少19.43%。
原文摘要 · Abstract (English)
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.
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