We revisit the problem of assigning a score (a quality of fit) to candidate geometric models -- one of the key components of RANSAC for robust geometric fitting. In a non-robust setting, the ``gold standard'' scoring function, known as the geometric error, follows from a probabilistic model with Gaussian noises. We extend it to spherical noises. In a robust setting, we consider a mixture with uniformly distributed outliers and show that a threshold-based parameterization leads to a unified view of likelihood-based and robust M-estimators and associated local optimization schemes. Next we analyze MAGSAC++ which stands out for two reasons. First, it achieves the best results according to existing benchmarks. Second, it makes quite different modeling assumptions and derivation steps. We discovered, however that the derivation does not correspond to sound principles and the resulting score function is in fact numerically equivalent to a simple Gaussian-uniform likelihood, a basic model within the proposed framework. Finally, we propose an experimental methodology for evaluating scoring functions: assuming either a large validation set, or a small random validation set in expectation. We find that all scoring functions, including using a learned inlier distribution, perform identically. In particular, MAGSAC++ score is found to be neither better performing than simple contenders nor less sensitive to the choice of the threshold hyperparameter. Our theoretical and experimental analysis thus comprehensively revisit the state-of-the-art, which is critical for any future research seeking to improve the methods or apply them to other robust fitting problems.

翻译：我们重新审视了为候选几何模型分配评分（拟合质量）的问题——这是RANSAC鲁棒几何拟合的关键组成部分之一。在非鲁棒设置中，被称为几何误差的“黄金标准”评分函数源自高斯噪声的概率模型。我们将其推广至球形噪声。在鲁棒设置中，我们考虑包含均匀分布异常值的混合模型，并证明基于阈值的参数化能够统一基于似然的鲁棒M估计器及其相关局部优化方案。接着我们分析了MAGSAC++，其突出之处有两点：首先，根据现有基准测试，它取得了最佳结果；其次，它采用了截然不同的建模假设和推导步骤。然而我们发现，其推导过程并不符合可靠原则，且最终得到的评分函数在数值上等价于简单的高斯-均匀似然模型——这正是我们所提出框架中的基础模型。最后，我们提出了一种评估评分函数的实验方法：假设存在大型验证集，或期望中的小型随机验证集。研究发现，所有评分函数（包括使用学习到的内点分布）表现完全一致。特别地，MAGSAC++评分既未表现出优于简单对比方法的性能，也未显示出对阈值超参数选择的更低敏感性。因此，我们的理论与实验分析全面重新审视了现有技术现状，这对未来任何旨在改进方法或将其应用于其他鲁棒拟合问题的研究至关重要。