In this paper, we present a second order spatial compatibility (SC^2) measure
based method for efficient and robust point cloud registration (PCR), called
SC^2-PCR. Firstly, we propose a second order spatial compatibility (SC^2)
measure to compute the similarity between correspondences. It considers the
global compatibility instead of local consistency, allowing for more
distinctive clustering between inliers and outliers at early stage. Based on
this measure, our registration pipeline employs a global spectral technique to
find some reliable seeds from the initial correspondences. Then we design a
two-stage strategy to expand each seed to a consensus set based on the SC^2
measure matrix. Finally, we feed each consensus set to a weighted SVD algorithm
to generate a candidate rigid transformation and select the best model as the
final result. Our method can guarantee to find a certain number of outlier-free
consensus sets using fewer samplings, making the model estimation more
efficient and robust. In addition, the proposed SC^2 measure is general and can
be easily plugged into deep learning based frameworks. Extensive experiments
are carried out to investigate the performance of our method. Code will be
available at https://github.com/ZhiChen902/SC2-PCR.

本文提出了一种基于空间兼容性 (SC2) 的点云配准方法 (SC2-PCR)，使用二阶空间兼容性 (SC2) 度量来计算对应关系的相似度，运用全局光谱技术找到可靠种子点并使用二阶空间兼容性 (SC2) 矩阵来设计一个两阶段的策略扩展每个种子点的一致性集，然后使用加权奇异值分解算法将每个一致性集提供给生成候选刚性变换，并选择最佳模型作为最终结果，能够保证使用更少的采样找到一定数量的无离群值的一致性集，使模型估计更有效和健壮。

SC^2-PCR: 二阶空间兼容性方法用于高效、鲁棒的点云配准

SC^2-PCR: A Second Order Spatial Compatibility for Efficient and Robust  Point Cloud Registration

In recent literature, a general two step procedure has been formulated for
solving the problem of phase retrieval. First, a spectral technique is used to
obtain a constant-error initial estimate, following which, the estimate is
refined to arbitrary precision by first-order optimization of a non-convex loss
function. Numerical experiments, however, seem to suggest that simply running
the iterative schemes from a random initialization may also lead to
convergence, albeit at the cost of slightly higher sample complexity. In this
paper, we prove that, in fact, constant step size online stochastic gradient
descent (SGD) converges from arbitrary initializations for the non-smooth,
non-convex amplitude squared loss objective. In this setting, online SGD is
also equivalent to the randomized Kaczmarz algorithm from numerical analysis.
Our analysis can easily be generalized to other single index models. It also
makes use of new ideas from stochastic process theory, including the notion of
a summary state space, which we believe will be of use for the broader field of
non-convex optimization.

本文研究了相位恢复问题，并提出一种基于随机梯度下降的方法，可以从任意初始点收敛，证明了其有效性并应用于单指数模型的求解，并提出了新的随机过程理论中的概念，可以用于更广泛的非凸优化领域。