Point-set registration is a classical image processing problem that looks for
the optimal transformation between two sets of points. In this work, we analyze
the impact of outliers when finding the optimal rotation between two point
clouds. The presence of outliers motivates the use of least unsquared
deviation, which is a non-smooth minimization problem over non-convex domain.
We compare approaches based on non-convex optimization over special orthogonal
group and convex relaxations. We show that if the fraction of outliers is
larger than a certain threshold, any naive convex relaxation fails to recover
the ground truth rotation regardless of the sample size and dimension. In
contrast, minimizing the least unsquared deviation directly over the special
orthogonal group exactly recovers the ground truth rotation for any level of
corruption as long as the sample size is large enough. These theoretical
findings are supported by numerical simulations.

研究点集注册问题中的异常值如何影响两个点云之间的最优旋转问题，比较了基于非凸优化和凸松弛的方法，实验发现，特殊正交群上直接最小化最小不平方偏差能够更好地处理异常值和恢复旋转。

利用非凸性最大化点集配准的稳健性

Maximizing robustness of point-set registration by leveraging  non-convexity

This paper focuses on a stochastic formulation of Bayesian attitude
estimation on the special orthogonal group. In particular, an exponential
probability density model for random matrices, referred to as the matrix Fisher
distribution is used to represent the uncertainties of attitude estimates and
measurements in a global fashion. Various stochastic properties of the matrix
Fisher distribution are derived on the special orthogonal group, and based on
these, two types of intrinsic frameworks for Bayesian attitude estimation are
constructed. These avoid complexities or singularities of the attitude
estimators developed in terms of quaternions. The proposed approaches are
particularly useful to deal with large estimation errors or large uncertainties
for complex maneuvers to obtain accurate estimates of the attitude.

本文提出了一种基于矩阵菲舍尔分布的随机贝叶斯态度估计方法，避开四元数表示法的复杂性和奇异性，对于复杂机动引起的大误差或大不确定性具有显著的提高。

在 SO（3）上使用 Matrix Fisher 分布的贝叶斯态度估计

Bayesian Attitude Estimation with the Matrix Fisher Distribution on  SO(3)

The synchronization problem over the special orthogonal group $SO(d)$
consists of estimating a set of unknown rotations $R_1,R_2,...,R_n$ from noisy
measurements of a subset of their pairwise ratios $R_{i}^{-1}R_{j}$. The
problem has found applications in computer vision, computer graphics, and
sensor network localization, among others. Its least squares solution can be
approximated by either spectral relaxation or semidefinite programming followed
by a rounding procedure, analogous to the approximation algorithms of
\textsc{Max-Cut}. The contribution of this paper is three-fold: First, we
introduce a robust penalty function involving the sum of unsquared deviations
and derive a relaxation that leads to a convex optimization problem; Second, we
apply the alternating direction method to minimize the penalty function;
Finally, under a specific model of the measurement noise and for both complete
and random measurement graphs, we prove that the rotations are exactly and
stably recovered, exhibiting a phase transition behavior in terms of the
proportion of noisy measurements. Numerical simulations confirm the phase
transition behavior for our method as well as its improved accuracy compared to
existing methods.

该研究提出了一种基于特殊正交群上的同步问题，该问题包括从它们成对比率的噪声测量中估计一组未知的旋转。它的最小二乘解可以通过谱松弛或半定规划来近似，其具有类似于 Max-Cut 的近似算法。该研究通过提出偏差平方和的罚函数来弱化其次方项，并引出了一种求解该问题的凸优化方法，同时在特定噪声模型下，证明了其稳定性并得到了相位转变行为的模拟结果。