三维点云处理(2)——PCA

PCA

PCA is to find the dominant directions of the point cloud

Applications:

• Dimensionality reduction
• Surface normal estimation
• Canonical orientation
• Keypoint detection
• Feature description

Physical intuitions:

• Vector Dot Product
  
• Matrix-Vector Multiplication
  
• Singular Value Decomposition (SVD)
  

Spectral Theorem:

$$A=U\Lambda U^T=\sum _{i=1}^n{\lambda }_i{u}_i{u}_i^T,\Lambda =diag({\lambda }_1,\cdots ,{\lambda }_n)$$

Rayleigh Quotients:

$${\lambda }_{min}(A)\le \frac{x^{T}Ax}{x^{T}x}\le {\lambda }_{max}(A),\forall x\ne 0$$

Summary:

1. Normalized by the center:
   $$\tilde{X}=[{\tilde{x}}_1,\cdots ,{\tilde{x}}_m],{\tilde{x}}_i=x_i-\overline{x}$$
2. Compute SVD:
   $$H=\tilde{X}{\tilde{X}}^T=U_r{\Sigma }^2{U}_r^T$$
3. The principle vectors are the columns of U_r (Eigenvector of $X$= Eigenvector of $H$)

KPCA

$$\tilde{H}\tilde{z}=\tilde{\lambda }\tilde{z}\to \tilde{z}=\sum _{j=1}^N{\alpha }_j\varphi (x_j)\to K\alpha =\lambda \alpha$$
The normalization of $\tilde{z}$:
$${\alpha }_r^T{\lambda }_r{\alpha }_r=1\to {\alpha }_r=1/{\lambda }_r$$

Kernel:

• Linear $k(x_i,x_j)=x_i^T{x}_j$
• Polynomial $k(x_i,x_j)=(1+x_i^T{x}_j{)}^p$
• Gaussian $k(x_i,x_j)=e^{-\beta ||x_i-x_j|{|}_2}$
• Laplacian $k(x_i,x_j)=e^{-\beta ||x_i-x_j|{|}_1}$

Summary

1. Select a kernel $k(x_i,x_j)$,compute the Gram matrix $K(i,j)=k(x_i,x_j)$
2. Normalize $K$:
   $$\tilde{K}=K-2{\text{II}}_{\frac{1}{N}}K+{\text{II}}_{\frac{1}{N}}K{\text{II}}_{\frac{1}{N}}$$
3. Solve the eigenvector/eigenvalues of $\tilde{K}$:
   $$\tilde{K}{\alpha }_r={\lambda }_r{\alpha }_r$$
4. Normalize ${\alpha }_r$ to be ${\alpha }_r^T{\alpha }_r=\frac{1}{{\lambda }_r}$
5. For any data point $x\in R^n$, compute its projection onto $r^{th}$ principle component $y_r\in R$:
   $$y_r={\varphi }^T(x){\tilde{z}}_r=\sum _{j=1}^N{\alpha }_{rj}k(x,x_j)$$

Surface normal on surface

The vector perpendicular to the tangent plane of the surface at a point P

Applications:

• Segmentation/Clustering
• Plane detection
• Point cloud feature for applications like Deep Learning

Steps:

1. Select a point P
2. Find the neighborhood that defines the surface
3. PCA$\to \underset{n\in R^n}{\min }{n}^T\tilde{X}W{\tilde{X}}^{T}n,s.t.:||n|{|}_2=1$
4. Normal $\to$ the least significant vector
5. Curvature $\to$ ratio between eigen values ${\lambda }_3/({\lambda }_1+{\lambda }_2+{\lambda }_3)$

Noise:

1. Select neighbors according to problem
2. Weighted based on other features
3. RANSAC
4. Deep Learning¹²

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1. Predicting Depth, Surface Normal and Semantic Labels with a Common Multi-Scale Convolutional Architecture, ICCV 2015 ↩︎

2. Nesti-Net: Normal Estimation for Unstructured 3D Point Clouds using Convolutional Neural Networks, CVPR 2019 ↩︎