[3D Vision] Spectral decomposition and application of grid (GFT map Fourier transform)

The spectral decomposition of the grid is the frequency decomposition of the grid. We have learned the Fourier transform of the signal, which transforms the signal from the spatial domain to the frequency domain. The two-dimensional image is composed of discrete Fourier transform DFT (Discrete Fourier Transform). In the field of graph signals, there is also the graph Fourier transform GFT (Graph Fourier Transform). The grid can be regarded as a graph, so the spectral decomposition of the grid is to perform the graph Fourier transform.

grid spectral decomposition

GFT graph Fourier transform

• First construct the Laplacian matrix of the graph, Laplacian matrix = degree matrix-adjacency matrix

• Calculate the eigenvalues ​​and eigenvectors of the Laplacian matrix: Perform eigenvalue decomposition on the constructed Laplacian matrix to obtain its eigenvalues ​​and corresponding eigenvectors. The eigenvalues ​​represent the spectral information of the grid, while the eigenvectors represent the different spectral components of the grid.

• Then the Laplacian matrix of the graph can represent a linear combination of the basis vectors, and its coefficients represent the intensity of different frequency components.

• Among the eigenvalues, the spectral component corresponding to the larger eigenvalue usually contains the high-frequency information in the grid , while the spectral component corresponding to the smaller eigenvalue represents the low-frequency information in the grid. Therefore, high-frequency information can be determined by the magnitude of the feature value.

Laplacian matrix of graph

Laplacian matrix = degree matrix - adjacency matrix

Why is this matrix called the Laplacian matrix? From my limited understanding, it is possible to relate the Laplacian coordinates of the mesh vertices.
Assuming that the graph has n vertices, then the Laplacian matrix of the graph $L^{n\times n}$ , the vertex coordinates of the graph are also expressed as a matrix as$V^{n\times 3}$ , left multiply the Laplacian matrix by the vertex matrix, that is,$L\times V$ , what is obtained is the Laplacian coordinate matrix$L{v}^{n\times 3}$

give a chestnut

Laplacian matrix = degree matrix - adjacency matrix

Consider a simple 2D mesh consisting of 4 vertices and 4 edges as follows:

   1 -- 2
   |    |
   3 -- 4

We can use adjacency matrix and degree matrix to construct Laplacian matrix.

• Adjacency matrix : The adjacency matrix represents the connection relationship between vertices. In this example, the adjacency matrix is:

   [[0, 1, 0, 1],
    [1, 0, 1, 0],
    [0, 1, 0, 1],
    [1, 0, 1, 0]]

• Degree matrix : The degree matrix represents the degree of each vertex (that is, the number of edges connected to it). In this example, the degree matrix is:

   [[2, 0, 0, 0],
    [0, 2, 0, 0],
    [0, 0, 2, 0],
    [0, 0, 0, 2]]

• Laplacian Matrix : By subtracting the degree matrix from the adjacency matrix, we can get the Laplacian matrix. In this example, the Laplacian matrix is:

   [[2, -1, 0, -1],
    [-1, 2, -1, 0],
    [0, -1, 2, -1],
    [-1, 0, -1, 2]]

This Laplacian matrix describes the connection relationships and boundary conditions between vertices in the mesh. By performing eigenvalue decomposition on this matrix, spectral decomposition and spectral analysis of the grid can be performed.
For a given Laplacian matrix:

$$L=\left[\begin{array}{cccc}2& -1& 0& -1\\ -1& 2& -1& 0\\ 0& -1& 2& -1\\ -1& 0& -1& 2\end{array}\right]$$

• Do eigenvalue decomposition

Use an appropriate numerical calculation library or linear algebra software for eigenvalue decomposition. This can be done by calling the corresponding function or method.

The result of eigenvalue decomposition can be expressed as:
$$L=Q\Lambda Q^{-1}$$

Among them, $Q$ is the eigenvector matrix,$\Lambda$ is the eigenvalue diagonal matrix.

The result is as follows:

Eigenvalues:
$l_1=0$ ,$l_2=1$ ,$l_3=3$ ,$l_4=4$

Eigenvectors:
$$v_1=\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]$$
$$v_2=\left[\begin{array}{c}-1\\ 1\\ -1\\ 1\end{array}\right]$$
$$v_3=\left[\begin{array}{c}1\\ -1\\ -1\\ 1\end{array}\right]$$
$$v_4=\left[\begin{array}{c}-1\\ -1\\ 1\\ 1\end{array}\right]$$

Note that the order of the eigenvalues ​​is from small to large. The eigenvectors correspond to the order of the eigenvalues, i.e. $v_1$Corresponds to $l_1$，$v_2$Corresponds to $l_2$,So on and so forth.

Eigenvalue decomposition provides important information about the structure and properties of a matrix. In this example, eigenvalue decomposition reveals the eigenmodes and frequencies of the mesh, where eigenvalue 0 corresponds to a constant mode, eigenvalue 1 corresponds to a linear mode, and eigenvalues ​​3 and 4 correspond to higher frequency modes . . The feature vector represents the spatial shape of each feature pattern.

application

mesh compression

Spectral Compression of Mesh Geometry

Article overview:
This article focuses on a technique called "grid geometry spectral compression", which is used to compress grid geometry data. The method exploits the spectral nature of the grid and achieves efficient compression by preserving key frequency components. The authors demonstrate through experiments that the method can reduce the size of grid data while maintaining high geometric quality and visual effect.

Technical points:

1. Utilization of the spectral properties of the grid: This method utilizes the spectral properties of the grid, that is, the relationship between the eigenvectors and eigenvalues ​​of the grid. By spectrally decomposing the grid, a set of eigenvectors and corresponding eigenvalues ​​can be obtained, which represent the geometry of the grid. Based on these eigenvectors, the mesh can be reconstructed or compressed.

2. Preservation of key frequency components: In order to achieve efficient compression, this method chooses to preserve key frequency components in the grid eigenvectors, that is, eigenvectors corresponding to larger eigenvalues. These key frequency components contain the main geometric information of the grid, and ignoring the eigenvectors corresponding to the smaller eigenvalues ​​can effectively reduce the data size.

3. Compression algorithm: The article proposes a compression algorithm based on key frequency components. The algorithm first spectrally decomposes the grid, and then selects the number of eigenvectors to retain according to a preset compression ratio. The data size can be greatly reduced by keeping only key frequency components and discarding others. Finally, the original mesh is reconstructed using the preserved eigenvectors.

Improvements over previous methods:
Compared with previous methods, the grid geometry spectrum compression method proposed in this paper has the following improvements:

1. Efficiency: The method exploits the spectral properties of the grid to achieve efficient compression by preserving key frequency components. Compared with traditional methods, this method can reduce the data size while maintaining high geometric quality and visual effect.
2. Preservation of geometric quality: During the compression process, this method chooses to retain key frequency components, which contain the main geometric information of the grid. Therefore, the grid compressed by this method can maintain a high geometric quality, avoiding the deformation or distortion that may be introduced by the traditional compression method.

Insufficiency of the article:
Although the grid geometry spectrum compression method proposed in this article has certain advantages, there are also some shortcomings:

1. Limitation on compression ratio: This method relies on a preset compression ratio when choosing the number of feature vectors to keep. This may result in high compression ratio requirements not being achieved in some cases.
2. Computational complexity: The article does not discuss the computational complexity of this method in detail. Since operations such as spectral decomposition may be time-consuming, the computational efficiency of this method on large-scale grid data still requires further research and improvement.

In summary, this article introduces a method for geometric compression that exploits the spectral properties of the grid. Compared with previous methods, this method has the advantages of high efficiency and geometric quality preservation. However, there are still some shortcomings, including limitations in compression ratio and areas for improvement in computational complexity.

High-Pass Quantization for Mesh Encoding

Article Overview:
This article introduces a method called "High-Pass Quantization for Mesh Encoding" for encoding and compressing meshes. This method utilizes high-pass filters and quantization techniques to achieve efficient compression by retaining the high-frequency detail information of the grid. The author experimentally demonstrated that this method can reduce the size of grid data while maintaining high geometric quality and visual effects.

Technical points:

1. Application of high-pass filter: This method uses a high-pass filter to process the grid to retain high-frequency detailed information. The high-pass filter can filter out low-frequency components, allowing the encoding process to focus more on the details of the grid, thereby achieving more efficient compression.

2. Quantitative technique: The article adopts a coding strategy based on quantification. By quantizing high-frequency detailed information, continuous floating-point values ​​are converted into discrete integer values, thereby reducing data representation and storage overhead. At the same time, in order to maintain the quality of encoding, the article proposes an adaptive quantization strategy to adjust the quantization accuracy according to the local characteristics of the grid.

3. Coding algorithm: The article proposes a coding algorithm based on high-pass filtering and quantization. First, the grid is processed through a high-pass filter to extract high-frequency detailed information. Then, the extracted high-frequency detail information is adaptively quantized to obtain discrete integer values. Finally, the quantized data is encoded and compressed to reduce storage space.

Improvements compared to previous methods:
Compared with previous methods, the High-Pass Quantization for Mesh Encoding method proposed in this article has the following improvements:

1. High-frequency detail retention: Traditional trellis encoding methods usually only focus on low-frequency components, while this method focuses on retaining high-frequency detail information by using a high-pass filter. This can better capture the detailed features of the mesh, improving geometric quality and visual effects.

2. Adaptive quantization: The article proposes an adaptive quantization strategy to adjust the quantization accuracy according to the local characteristics of the grid. This adaptability can better balance compression ratio and geometric quality, providing better encoding results.

Insufficiencies of the article:
Although this method has certain advantages in grid encoding and compression, there are still some shortcomings:

1. Coding Complexity: The article does not explore the coding complexity of this method in detail. Due to the influence of operations such as high-pass filtering and adaptive quantization, the encoding process of this method may be complicated, and further research and improvement are needed.

2. Limitation of compression rate: This method mainly focuses on the preservation of high-frequency details, and relatively little processing of low-frequency components. This may result in high compression ratio requirements not being achieved in some cases.

In summary, this article introduces a method for trellis coding and compression using high-pass filters and quantization techniques. Compared with previous methods, this method improves geometric quality and visual effects by retaining high-frequency detail information and adaptive quantization strategies. However, there are still some shortcomings, including limitations in coding complexity and compression rate, which require further research and improvement.

mesh smoothing

Reference article

Introduction to the Laplacian Matrix of Graphs: Understanding the Laplacian Matrix of Graphs - Zhihu (zhihu.com)