Brief Introduction | Point Cloud Distance Metric: From Principle to Application

Note 1: This article is one of the "Brief Introduction" series. It only gives a very brief introduction to point cloud distance measurement conceptually, and is not suitable for in-depth and detailed understanding.

Point Cloud Distance Metrics: From Principle to Application

3D deep learning on Point cloud

1. Background introduction

Point cloud data has a wide range of applications in computer graphics, computer vision, autonomous driving and other fields. To efficiently process and analyze point cloud data in these applications, we need to measure the similarity between point clouds. This leads to the problem of distance metrics between point clouds. This paper will introduce the basic concepts, principles, research status and challenges of point cloud distance measurement, and discuss the future development direction.

2. Principle introduction and derivation

A point cloud is composed of a set of discrete points in three-dimensional space, each point having X, Y, and Z coordinates. Measuring the distance between two point clouds usually includes the following methods:

2.1 Euclidean distance

Euclidean distance is the most commonly used distance measurement method. For two point clouds P = { p 1 , p 2 , . . . , pn } P=\{p_1, p_2, ..., p_n\}P={ p1​,p2​,...,pn​}和 Q = { q 1 , q 2 , . . . , q n } Q=\{q_1, q_2, ..., q_n\} Q={ q1​,q2​,...,qn​} , the Euclidean distance between them is defined as:

$$d(P,Q)=\sqrt{\sum _{i=1}^n(p_i-q_i{)}^2}$$

2.2 Hausdorff distance

Hausdorff distance (Hausdorff distance) is a method for measuring the distance between two point sets, which can handle the non-one-to-one correspondence between point clouds. For two point clouds $P$ and$Q$ , the Hausdorff distance between them is defined as:

d H ( P , Q ) = max ⁡ { max ⁡ p ∈ P min ⁡ q ∈ Q ∥ p − q ∥ , max ⁡ q ∈ Q min ⁡ p ∈ P ∥ p − q ∥ } d_H(P, Q) = \max\{\max_{p \in P} \min_{q \in Q} \|p - q\|, \max_{q \in Q} \min_{p \in P} \|p - q\|\} dH​(P,Q)=max{ p∈Pmax​q∈Qmin​∥p−q∥,q∈Qmax​p∈Pmin​∥p−q∥}

2.3 Chamfer distance

Chamfer distance (Chamfer distance) is a method of approximating Hausdorff distance with lower computational complexity. For two point clouds $P$ and$Q$ , the Chamfer distance between them is defined as:

$$d_C(P,Q)=\sum _{p\in P}\underset{q\in Q}{\min }\Vert p-q{\Vert }^2+\sum _{q\in Q}\underset{p\in P}{\min }\Vert p-q{\Vert }^2$$

3. Research Status

In recent years, point cloud distance metrics have received extensive attention and applications in many fields, including point cloud registration, point cloud classification, and point cloud generation, etc. In these applications, researchers have proposed many improved distance metrics to improve computational efficiency and application performance.

3.1 Improved Hausdorff distance

Because the Hausdorff distance is very sensitive to noise and outliers, researchers have proposed many improved methods. For example, Partial Hausdorff distance reduces the effect of noise by ignoring the farthest part of the point pairs.

3.2 Deep Learning Methods

With the development of deep learning, some researchers began to use neural networks to learn distance metrics between point clouds. For example, network structures such as PointNet and Dynamic Graph CNN can directly process point cloud data, thereby learning more advanced and meaningful feature representations.

4. Challenge

Although point cloud distance metrics have achieved remarkable progress in many aspects, there are still some challenges, such as:

1. Computational complexity : Some distance measurement methods (such as Hausdorff distance) have high computational complexity in large-scale point cloud data, and more efficient algorithms and techniques are needed to solve them.

2. Robustness : The robustness of existing distance metrics to problems such as noise, outliers, and occlusions needs to be improved.

3. Application performance : In practical applications, it is necessary to select an appropriate distance measurement method according to different tasks and scenarios to improve application performance.

5. Future Outlook

In the future, the research and application of point cloud distance measurement will continue to develop in depth. Possible directions include:

1. New distance measurement method : study the distance measurement method that is more suitable for practical application requirements.

2. Algorithm optimization : Improve the performance of existing distance measurement methods in terms of computational efficiency, accuracy, and robustness.

3. Deep learning method : use deep learning technology to improve and optimize point cloud distance measurement to improve application performance.

DPDist : Comparing Point Clouds Using Deep Point Cloud Distance

6. Code example

Here is a simple example to calculate the Euclidean distance between two point clouds using Python and NumPy library:

import numpy as np

def euclidean_distance(P, Q):
    returnreturn np.sqrt(np.sum((P - Q) ** 2, axis=1))

# 示例点云
P = np.array([[1, 2, 3],
              [4, 5, 6],
              [7, 8, 9]])

Q = np.array([[1, 2, 3],
              [4, 5, 6],
              [8, 9, 10]])

# 计算欧氏距离
distances = euclidean_distance(P, Q)
print("Euclidean distances:", distances)