## 1 Introduction
### Problems Formulation  
Rotation search aims to find the best rotation that aligns two 3D vector sets in different coordinate frames, which can be formulated as follows.  
Given two 3D vector sets \(\mathcal{U}=\{\mathbf{u}_{i}\}_{i=1}^{N}\) and \(\mathcal{V}=\{\mathbf{v}_{i}\}_{i=1}^{N}\) where we consider tuple \((\mathbf{u}_{i},\mathbf{v}_{i})\) as a vector correspondence, the following relation is satisfied:  
\[\mathbf{R}\mathbf{u}_{i}+\mathbf{e}_{i}=\mathbf{v}_{i}, \tag{1}\]where \(\mathbf{R}\in SO(3)\) (the explicit properties and constraints of \(SO(3)\) can be found in [13; 14]) denotes the rotation matrix and \(\mathbf{e}_{i}\in\mathbb{R}^{3}\) represents the measurement of Gaussian noise with isotropic covariance \(\sigma^{2}\mathbf{I}_{3}\) (\(\sigma\) is the standard deviation and \(\mathbf{I}_{3}\in\mathbb{R}^{3\times 3}\) is the identity matrix) and mean \(\mu=0\) w.r.t. the \(i_{th}\) vector correspondence. Thus, the rotation search problem can be mathematically defined as follows:  
**Problem 1** (Rotation search).: _The rotation search problem under the perturbance of noise can be equivalently formulated as a minimization problem such that:_  
\[\min_{\mathbf{R}\in\mathbf{SO}3}\sum_{i=1}^{N}||\mathbf{R}\mathbf{u}_{i}-\mathbf{v}_{i }||^{2}, \tag{2}\]  
_where globally optimal solution \(\mathbf{R}\) is the best rotation desired._  
Point cloud registration consists of estimating the transformation (including scale, rotation, and translation) between two given 3D point clouds. Now, we introduce the correspondence-based point cloud registration problem.  
With two 3D point clouds \(\mathcal{P}=\{\mathbf{P}_{i}\}_{i=1}^{N}\) and \(\mathcal{Q}=\{\mathbf{Q}_{i}\}_{i=1}^{N}\), where tuple \((\mathbf{P}_{i},\mathbf{Q}_{i})\) represents a point correspondence, we can have the following relation:  
\[s\mathbf{R}\mathbf{P}_{i}+\mathbf{t}+\mathbf{e}_{i}=\mathbf{Q}_{i}, \tag{3}\]  
where \(s>0\) is the scale, \(\mathbf{R}\in SO(3)\) is the rotation matrix, \(\mathbf{t}\in\mathbb{R}^{3}\) is the translation vector, and \(\mathbf{e}_{i}\in\mathbb{R}^{3}\) represents the measurement of Gaussian noise similar to that in Problem 1. The generalized point cloud registration problem can be defined as follows.  
**Problem 2** (Generalized point cloud registration).: _Solving the generalized point cloud registration problem under the perturbance of noise is equivalent to the following minimization problem:_  
\[\min_{s>0,\mathbf{R}\in\mathbf{SO}3,\mathbf{t}\in\mathbb{R}^{3}}\sum_{i=1}^{N}||s\bm {R}\mathbf{P}_{i}+\mathbf{t}-\mathbf{Q}_{i}||^{2}, \tag{4}\]  
_where global minimizers \((s,\mathbf{\hat{R}},\mathbf{\hat{t}})\) are the optimal solutions to this registration problem._  
When the scale aligning the two point clouds is fixed, which means scale \(s=1\), we can directly omit \(s\) in Problem 2, called the _known-scale_ registration problem. However, the scale could vary, which, for instance, exists in the cross-source point cloud registration problems. In this case, the scale needs to be estimated additionally, and such problems are called _unknown-scale_ point cloud registration.  
Our work seeks to solve both the known-scale and the unknown-scale registration problems in the presence of both noise and a great number of outliers among the putative correspondences, providing a highly robust backend for real applications.