Brief Introduction | 3D Point Cloud Registration: Theory, Methods and Challenges

3D Point Cloud Registration: Theory, Methods and Challenges

Note: The "Brief Introduction" series only gives a very brief introduction to a certain field conceptually, and is not suitable for in-depth and detailed understanding

3D point cloud registration is an important topic in the field of computer vision and robotics, which involves the process of finding the best alignment between 3D point cloud data collected from different perspectives or time points. This article will introduce the background, principle derivation, research status, challenges and future prospects of 3D point cloud registration in detail.

1. Background introduction

With the development of sensor technology, 3D point cloud data has been widely used in fields such as autonomous driving , remote sensing mapping and augmented reality . To extract useful information from these data, it is often necessary to fuse multiple point clouds into a unified coordinate system. 3D point cloud registration is the key technology to solve this problem.

2. Principle introduction and derivation

2.1 Mathematical representation of the registration problem

Given two point clouds P = { p 1 , p 2 , … , pm } P=\{p_1, p_2, \dots, p_m\}P={ p1​,p2​,…,pm​}和 Q = { q 1 , q 2 , … , q n } Q=\{q_1, q_2, \dots, q_n\} Q={ q1​,q2​,…,qn​}，其中$p_i,q_j\in {\mathbb{R}}^3$ . The goal of 3D point cloud registration is to find arigid body transformation(including the rotation matrix$R$ and the translation vector$t$ ), such that$P$ andQQThe alignment error between$Q\; is\; the\; smallest.$Usually, we usethe point-to-point distanceas the error metric:

$$E(R,t)=\sum _{i=1}^m\sum _{j=1}^n{w}_{ij}\Vert (R{p}_i+t)-q_j{\Vert }^2\text{\hspace{1em}(1)}$$

Among them, $w_{ij}$represents the point $p_i$and $q_j$The weight between them reflects the similarity between them.

2.2 Classic algorithm: Iterative Closest Point (ICP)

The iterative closest point (ICP) algorithm is one of the earliest proposed 3D point cloud registration methods. Its core idea is to find the point cloud PP in each iteration.Each point in$P$$The\; nearest\; neighbor\; points\; in\; Q$ , and solve the optimal rigid body transformation based on these point pairs. The basic steps of the ICP algorithm are as follows:

1. Initialize the rigid body transformation $(R^{(0)},t^{(0)})$；
2. at $In\; k$ iterations, the point cloudPPTransform$P$$Q$ 's coordinate system:$P^{(k)}=R^{(k-1)}P+t^{(k-1)}$；
3. Find point cloud $P^{}$Each point in${}^{(}$${}^k$${}^{)}$ is associated with the point cloud QQnearest neighbors in$Q\; ;$
4. According to the found point pair, calculate the optimal rigid body transformation $(R^{(k)},t^{(k)})$；
5. If converged, return the final rigid body transformation; otherwise, return to step 2.

The key step of the ICP algorithm is to calculate the optimal rigid body transformation. To solve this problem, we can use the singular value decomposition (SVD) method. Specifically, first calculate the centroid between point pairs :

$$m_P=\frac{1}{m}\sum _{i=1}^m{p}_i^{(k)},m_Q=\frac{1}{n}\sum _{j=1}^n{q}_j\text{\hspace{1em}(2)}$$

Then calculate the covariance matrix :

$$H=\sum _{i=1}^m(p_i^{\left(k\right)}-m_P)(q_i-m_Q{)}^T\text{\hspace{1em}(3)}$$

to $H$ performs singular value decomposition to get$H=U\Sigma V^T$ , then the optimal rotation matrix and translation vector are:

$$R^{(k)}=V{U}^T,t^{\left(k\right)}=m_Q-R^{(k)}{\mu }_P\text{\hspace{1em}(4)}$$

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2.3 Feature-based registration method

In addition to the ICP algorithm, there are many feature -based registration methods. These methods first extract meaningful features in point clouds, and then find point pairs through feature matching. Feature-based methods usually have better robustness and computational efficiency , but may be affected by errors in feature extraction and matching.

3. Research Status

In recent years, deep learning based methods have achieved remarkable results in the field of point cloud registration. These methods mainly include supervised learning methods and unsupervised learning methods .

2.2.1 Supervised learning methods

Supervised learning methods learn local or global features between point clouds by training neural networks. Common ideas include

• The input point cloud is subjected to feature extraction through a multi-layer perceptron (MLP), and a maximum pooling layer is used to obtain global features. Finally, the global features are mapped to the Euclidean transformation matrix by the decoder.

• By learning a deep neural network, the local features of the source point cloud and the target point cloud are mapped to a common description space, and then the optimal transformation matrix is ​​calculated by minimizing the distance between descriptors.

2.2.2 Unsupervised learning methods

The application of unsupervised learning methods to point cloud registration is relatively new, but has achieved some results. Common ideas include:

• Local features of source and target point clouds are learned by iteratively optimizing a deep neural network. First, the input point cloud is converted into a graph structure, and then feature extraction is performed using Graph Convolutional Networks (GCN). Then, the best point correspondence is found through the maximum graph matching algorithm, and the optimal transformation matrix is ​​calculated.

• Predict rigid transformations between point clouds by learning a deep neural network. First, local features are extracted from the input point cloud using a convolutional neural network (CNN) based feature extractor. Then, the best transformation matrix is ​​found by a global optimization algorithm such as RANSAC or branch and bound.

For reference based on learning point cloud registration: https://zhuanlan.zhihu.com/p/501503130

3. Challenge

Although point cloud registration technology has made remarkable progress in the past few years, it still faces some challenges:

1. Noise and outliers : Point cloud data usually contain noise and outliers, which pose challenges for the accuracy and robustness of point cloud registration. Researchers need to design more robust algorithms to deal with this problem.

2. Partial overlap and occlusion : In practical applications, there are often partial overlaps and occlusions between the source point cloud and the target point cloud. This makes point cloud registration more difficult. Future research needs to focus on how to improve the performance of algorithms in this case.

3. Computational Efficiency : Although deep learning methods have achieved significant improvements in the accuracy of point cloud registration, they often require long runtimes. In order to meet the needs of real-time applications, future research needs to focus on how to improve the computational efficiency of point cloud registration algorithms.

4. Large-scale point cloud data : With the advancement of sensor technology, we can obtain more and more large-scale point cloud data. Handling these large-scale data challenges the computational efficiency and memory consumption of point cloud registration algorithms. Researchers need to develop more efficient algorithms to deal with this problem.

5. Multimodal data fusion : In many practical application scenarios, point cloud data often coexists with other types of data (such as images, lidar data, etc.). How to effectively fuse these multimodal data to improve the performance of point cloud registration is a research direction worthy of attention.

4. Summary

This paper reviews the current state of research on point cloud registration and introduces the main traditional methods and deep learning-based methods. Although point cloud registration techniques have made remarkable progress in the past few years, they still face challenges such as noise and outliers, partial overlap and occlusion, computational efficiency, etc. Future research needs to focus on how to overcome these challenges to meet the needs in practical applications.