Geman-McClure 穩健估計演算法與空間感知應用

Abstract

穩健估計是一種在電腦視覺、機器人學和導航等多種真實世界應用中至關重要的技術。本論文提出了針對 Geman-McClure 穩健估計問題的兩種演算法：FracGM 和 QGM。FracGM 採用了 fractional program 技巧，重新審視並擴展了現有方法，以應對 Geman-McClure 穩健函數所帶來的挑戰。FracGM 保證了迭代過程中對偶問題的凸性，並發展條件性的全局最優性保證。QGM 則提供 quadratic program 的形式，實現了全局收斂性並提高計算效率。我們將這兩個演算法應用於空間感知問題，該問題因旋轉矩陣條件和離群值而具有挑戰性。我們探索了四種優化鬆弛技術，包括線性鬆弛、半正定鬆弛、黎曼梯度下降和 Stiefel 流形鬆弛。在模擬數據集和真實數據集上的大量實驗表明，即使在極端離群值條件下，FracGM 和 QGM 在準確性和穩健性方面均優於現有的最先進的方法。通過研究穩健估計的理論基礎和實際應用，本論文為機器人學和電腦視覺中的問題提供了更可靠的工具。

Robust estimation is a fundamental technique in various real-world applications, including computer vision, robotics, and navigation. This thesis introduces two novel solvers for the Geman-McClure robust estimation problem: FracGM and QGM. FracGM employs fractional programming techniques, revisiting and extending existing methods to handle the challenges posed by the Geman-McClure robust function. It provides theoretical guarantees for dual problem convexity during the iterative process and establishes conditional global optimality guarantees. QGM offers a quadratic programming reformulation, achieves globally convergent and computational efficiency.We evaluate both solvers on spatial perception problems, which poses challenges by the rotational constraints and outliers. We explore four relaxation techniques, including linear relaxation, semidefinite relaxation, Riemannian gradient descent, and Stiefel manifold relaxation. Extensive experiments on synthetic datasets and real-world datasets demonstrate that FracGM and QGM outperform state-of-the-art solvers in terms of accuracy and robustness, even under extreme outlier conditions. By advancing the theoretical foundations and practical capabilities of robust estimation, this work contributes to the development of more reliable tools for challenging tasks in robotics and computer vision.