# Blind Mesh Assessment Based on Graph Spectral Entropy and Spatial Features

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Most of the existing 3D MQA methods are not blind. We propose a new BMQA method based on Graph Spectral Entropy and spatial features, referred to as BMQA-GSES. New features in the graph spectral domain and spatial domain are defined and extracted for BMQA. The graph spectral features of 3D mesh can reveal the underlying shape information, while the spatial features of the 3D mesh can simulate the external information of the 3D mesh that can be directly perceived by human eyes.
- (2)
- Inspired by GSP, the Gaussian curvature signal is transformed from the spatial domain to the graph spectral domain by GFT in the proposed method. In addition, the signal smoothness and information entropy of amplitude features under different frequency components are then extracted in the graph spectral domain as underlying shape features of 3D mesh.
- (3)
- Considering that the concavity, convexity and the structural information of the distorted 3D mesh will change, four spatial features are combined on the base of extracting the graph spectral features.

## 2. Motivation

## 3. The Proposed BMQA-GSES Method

#### 3.1. Graph Spectral Features Analysis

**W**is weighted adjacency matrix of 3D mesh. If there is an edge e = (i, j) connecting vertices v

_{i}and v

_{j}, the entry W

_{i}

_{,j}represents the weight of the edge; otherwise, W

_{i}

_{,j}= 0. Figure 4a shows an example weighted graph, and its corresponding matrix is shown in Figure 4b.

_{i}

_{,j}, of an edge connecting vertices v

_{i}and v

_{j}is expressed by a threshold Gaussian kernel weighting function

_{i}and v

_{j}, and $\sigma $ is the variance of the distance.

_{GC}be the Gaussian curvature signal on the weighted graph, then the smoothness can be defined to describe the intrinsic structural characteristics of the graph signal. The edge derivative of a signal f

_{GC}with respect to edge $e=(i,j)$ at the vertex v

_{i}is defined as

_{GC}(v

_{i}) is the Gaussian curvature signal at v

_{i}, and is expressed as

_{max}(v

_{i}) and k

_{min}(v

_{i}) represent the maximum and minimum values of the principal curvature of v

_{i}, respectively.

_{i}can be computed by

_{S}of the Gaussian curvature signal f

_{GC}on the weighted graph G can be defined by

_{S}of the signal f

_{GC}is adopted to describe the distorted 3D mesh in the proposed method. In general, the irregular signals are denoted by the graphs of vertices, edges, and weights. In this research area, the conventional time-domain or spatial domain operators, theorems and tools are extended to the vertex domain, including Fourier transform, frequency-selective filter and vertex-frequency analysis etc. The theoretical and practical tools for this GSP analysis can be adopted to solve various irregular signal processing problems [23]. Therefore, in the proposed method, the GFT is used to transform the irregular Gaussian curvature signal from spatial domain into graph spectral domain, and then the relevant graph spectral features are extracted to reveal the underlying shape information of 3D mesh and describe the distortion of 3D mesh.

**L**is defined as follows

**W**is the weight matrix,

**D**is the degree matrix that can be calculated as follows

_{q}; q = 0, 1, …, q, …, $\tilde{q}$), and $0\le {\lambda}_{1}\le {\lambda}_{2}\le \cdots \le {\lambda}_{\tilde{q}}$. Thus, $\sigma (L)=\{{\lambda}_{0},{\lambda}_{1},\cdots ,{\lambda}_{\tilde{q}}\}$ represents the frequency component of the whole spectrum. The problem of eigenvalue decomposition can be solved by many methods. In our method, the Lanczos method [27] is adopted to compute the eigenvectors

**U**of sparse matrices

**L**, where $U=[{u}_{1},{u}_{2},{u}_{3},\cdots ,{u}_{\tilde{q}}]$.

_{GC}in the graph spectral domain is used as the graph spectral feature,

**F**, reflecting the distortion of the 3D mesh, expressed as follow

_{AM}_{GC}appearing in the spectral domain.

- (1)
- The Gaussian curvature on each vertex of the 3D mesh is firstly calculated, denoted as signal ${f}_{GC}({v}_{i})$;
- (2)
- The weighted graph G = {v,ε,
**W**} is then used to represent 3D mesh, where v represents vertices of 3D mesh, ε represents edges of 3D mesh, and**W**is weighted adjacency matrix of 3D mesh; - (3)
- The graph Laplacian matrix
**L**can be calculated from (6) and (7). The graph Laplacian matrix is sparse and has very large dimensions because the 3D mesh is composed of a lot of vertices. Therefore, the Lanczos method is used to calculate the eigenvectors**U**and eigenvalues**E**of the sparse matrix**L**; - (4)
- The Gaussian curvature signal is transformed by GFT based on the obtained eigenvector
**U**, i.e., (9). Since there is a one-to-one correspondence between the eigenvalues and the eigenvectors, the corresponding amplitude ${\widehat{f}}_{GC}$ of each eigenvalue can be obtained. The eigenvalues of the GFT replace the concept of frequency in the classical Fourier transform, so the eigenvalues and the corresponding amplitudes constitute the graph spectrum of 3D mesh. - (5)
- The information entropy of amplitudes under different frequency components is extracted to describe the characteristics of 3D mesh in the graph spectral domain.

#### 3.2. Spatial Features Analysis

#### 3.2.1. Concave and Convex Feature Analyses

_{max}, and the smallest eigenvalue, k

_{min}, can be obtained by constructing the curvature tensor, T(v). k

_{max}and k

_{min}, respectively, represent the maximum principal curvature and minimum principal curvature of the vertex. Then, the shape index, SI, and the curvedness, C, can be expressed as follows

_{µ}, SI

_{σ}and SI

_{α}represent the mean, variance and scale parameter of shape index, respectively.

_{μ}, C

_{σ}and C

_{α}represent the mean, variance and scale parameter of curvedness.

#### 3.2.2. Structural Feature Analyses

_{t}of each face of the 3D mesh is calculated, and then the dihedral angles of two adjacent faces t

_{1}, t

_{2}can be obtained by following formula.

_{1}, t

_{2}, respectively.

_{DA}, in spatial domain is expressed as

_{μ}and D

_{σ}denote the mean and variance parameters of the dihedral angles, respectively.

_{A}, is expressed as

_{μ}, A

_{σ}and A

_{α}represent the mean, variance and scale parameter of triangular topological area, respectively. In addition, the triangular topological area can be obtained by a common mathematical formula for calculating the area of a triangle.

#### 3.3. Pooling Strategy

## 4. Experimental Results and Discussion

_{p}) and the Spearman rank order correlation coefficient (r

_{s}) are selected as the criteria to measure the performance of the proposed method (BMQA-GSES). Among them, r

_{p}can reflect the linear correlation between subjective and objective scores, while r

_{s}can reflect the consistency between subjective and objective scores. For an excellent objective MQA method, r

_{p}and r

_{s}should all be close to 1. In this paper, we randomly divide the database randomly into two 3D mesh sets, 80% of data is used for training and the other 20% is used for testing. To eliminate performance bias, random selection of the training and testing sets is repeated 1000 times, and the median performance indices for cases are adopted as the final results.

#### 4.1. Graph Spectral and Spatial Features Analysis

_{S}, F

_{AM}], and shape index, curvedness, dihedral angle and the distribution triangular topological area constitute the spatial feature, i.e., [F

_{SI}, F

_{C}, F

_{DA}, F

_{A}]. Although the promising performance of the proposed quality evaluation model has been proved, the specific role of different feature is unkown. Therefore, it is necessary to analyze the performance of each feature separately in the paper. For this purpose, the performance results for every feature when used alone to learn the regression model are shown in Table 2.

_{S}and the information entropy of amplitude F

_{AM}. Signal smoothness can reveal the intrinsic structure of graph signal to some extent. The 3D mesh with higher smoothness tends to be rougher, while the 3D mesh with lower smoothness tends to be smoother. In addition, the experimental results also show that the signal smoothness feature has good performance and overcomes the defect that the spatial signal cannot reflect the correlation and similarity between the vertices of the 3D mesh. Secondly, the Gaussian curvature signal is transformed by GFT in this paper. The Gaussian curvature in spatial domain can reflect the concave and convex information of the 3D mesh, but it cannot reveal the frequency information. Therefore, the information entropy of amplitudes under different frequency components are extracted in the graph spectral domain which have been proved that can represent the distortion of the 3D mesh, i.e., the low-frequency signal can represent the distortion information of the smoother region of the 3D mesh, and the high-frequency signal can represent the distortion information of the rough region of the 3D mesh. The experimental results show that the information entropy of amplitude features extracted in this paper can better reflect the surface distortion of the model. In addition, the performance of the method is improved to a certain extent when combining another graph spectral feature. Thirdly, in the case of considering the graph spectral features, some excellent spatial features are also combined in this paper. Shape index and curvedness can simulate human subjective visual perception information in spatial domain, i.e., concave and convex information of 3D mesh, while the dihedral angle of 3D mesh and the distribution of the triangular topology area can reflect the structure of 3D mesh, i.e., dihedral angle information and the distribution of the triangular topology area can reflect the change of the topology. Although any of the spatial features can play an important role alone, the ingenious combination of the four features can achieve better performance in the spatial domain. Finally, we observed that although the quality can be predicted well by using the features of the graph spectral domain or the spatial domain alone, the performance is further improved when all features are combined. This makes us believe that the graph spectral features and the spatial features are complementary to each other and should be simultaneously considered for objective quality assessment of 3D meshes.

#### 4.2. Overall Performance Comparison

_{p}and r

_{s}values than the method in NR-SVR [16].

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Two types of distorted 3D meshes. (

**a**) Distorted 3D mesh with random noise, (

**b**) distorted 3D mesh smoothed by smoothing filter.

**Figure 4.**A weighted graph and its corresponding weighting matrix. (

**a**) An example of a weighted graph; (

**b**) Weighting matrix.

**Figure 6.**Four kinds of 3D mesh from the public LIRIS/EPFL general-purpose database [9].

**Figure 9.**Visual map of the shape index in the vertex domain. (

**a**) Distorted 3D mesh with the distortion of random noise, (

**b**) distorted 3D mesh with the distortion of smooth.

**Figure 10.**Visual map of the curvedness in the vertex domain. (

**a**) Distorted 3D mesh with the random noise distortion, (

**b**) distorted 3D mesh with the smooth distortion.

**Figure 12.**Scatter plots of each type the 3D mesh. (

**a**) Armadillo, (

**b**) Venus, (

**c**) Dinosaur, (

**d**) Rocker.

Noise | Smooth | |||||
---|---|---|---|---|---|---|

Level | Low | Medium | High | Low | Medium | High |

Armadillo | −0.2032 | −0.1828 | −0.1616 | −0.2480 | −0.2678 | −0.2798 |

Venus | −0.1742 | −0.1505 | −0.1318 | −0.2292 | −0.2505 | −0.2594 |

Dinosaur | −0.2655 | −0.2456 | −0.2233 | −0.2813 | −0.2815 | −0.2836 |

Rocker | −0.0691 | −0.0656 | −0.0622 | −0.0837 | −0.0859 | −0.0878 |

Feature Types | Features | Armadillo | Venus | Dinosaur | Rocker | Whole Database | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

r_{p} | r_{s} | r_{p} | r_{s} | r_{p} | r_{s} | r_{p} | r_{s} | r_{p} | r_{s} | ||

Graph spectral domain | F_{S} | 98.2 | 80.0 | 89.2 | 73.8 | 94.9 | 80.0 | 97.3 | 94.9 | 79.4 | 52.6 |

F_{AM} | 97.0 | 80.0 | 90.8 | 80.0 | 71.9 | 40.0 | 98.3 | 80.0 | 94.8 | 62.4 | |

Spatial domain | F_{SI} | 90.7 | 40.0 | 96.5 | 80.0 | 90.6 | 73.8 | 96.4 | 94.9 | 40.4 | 23.1 |

F_{C} | 98.1 | 80.0 | 95.2 | 80.1 | 96.7 | 80.0 | 97.5 | 80.0 | 83.9 | 57.3 | |

F_{DA} | 98.6 | 80.0 | 98.5 | 80.0 | 98.1 | 75.5 | 93.5 | 94.8 | 84.9 | 87.0 | |

F_{A} | 96.5 | 79.9 | 94.5 | 79.9 | 98.2 | 80.0 | 96.9 | 80.1 | 82.2 | 86.6 | |

All | 98.7 | 80.0 | 98.8 | 80.1 | 99.2 | 80.4 | 99.5 | 99.9 | 90.5 | 87.9 |

Type | Method | Principle |
---|---|---|

FR | HD [6]
| Hausdorff distance |

GL2 [8]
| vertex coordinate positions and the Geometrical Laplacian operator | |

MSDM [11]
| Local curvature, Contrast, and Structure | |

MSDM2 [12]
| Multiscale mesh structural distortion | |

DAME [13]
| Dihedral angles | |

RR | 3DWPM1 [9]
| Global roughness |

3DWPM2 [10]
| Global roughness | |

FMPD [14]
| Local roughness analysis and global roughness computation | |

KLD Gamma [15]
| Dihedral angles and statistical Gamma distributionmodel | |

KLD Weibull [15]
| Dihedral angles and statistical Weibull distribution model | |

NR | NR-SVR [16]
| Dihedral angles, visual masking modulation and parameters estimation |

The proposed | Graph spectral features and spatial features |

Type | Method | Armadillo | Venus | Dinosaur | Rocker | Whole Database | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

r_{p} | r_{s} | r_{p} | r_{s} | r_{p} | r_{s} | r_{p} | r_{s} | r_{p} | r_{s} | ||

FR | HD [6]
| 30.2 | 69.5 | 0.8 | 1.6 | 22.6 | 30.9 | 5.5 | 18.1 | 11.4 | 13.8 |

GL2 [8]
| 55.5 | 77.8 | 77.6 | 91.0 | 12.5 | 30.6 | 17.1 | 29.0 | 42.4 | 39.3 | |

MSDM [11]
| 70.0 | 84.8 | 72.3 | 87.6 | 56.8 | 73.0 | 75.0 | 89.8 | 75.0 | 73.9 | |

MSDM2 [12]
| 85.3 | 81.6 | 87.5 | 89.3 | 85.7 | 85.9 | 87.2 | 89.6 | 81.4 | 80.4 | |

DAME [13]
| 76.3 | 60.3 | 83.9 | 91.0 | 88.9 | 92.8 | 80.1 | 85.0 | 75.2 | 76.6 | |

RR | 3DWPM1 [9]
| 35.7 | 65.8 | 46.6 | 71.6 | 35.7 | 62.7 | 53.2 | 87.5 | 61.8 | 69.3 |

3DWPM2 [10]
| 43.1 | 74.1 | 16.4 | 34.8 | 19.9 | 52.4 | 29.9 | 37.8 | 49.6 | 49.0 | |

FMPD [14]
| 83.2 | 75.4 | 83.9 | 87.5 | 88.9 | 89.6 | 84.7 | 88.8 | 83.5 | 81.9 | |

KLD Gamma [15]
| 77.7 | 71.1 | 83.4 | 88.6 | 70.6 | 67.9 | 57.5 | 78.7 | 74.0 | 71.6 | |

KLD Weibull [15]
| 77.2 | 67.5 | 75.4 | 86.1 | 70.6 | 71.3 | 70.4 | 77.0 | 74.1 | 71.7 | |

NR | NR-SVR [16]
| 91.5 | 76.8 | 88.6 | 85.7 | 84.1 | 78.6 | 86.6 | 86.2 | 87.8 | 81.5 |

The proposed | 98.7 | 80.0 | 98.8 | 80.1 | 99.2 | 80.4 | 99.5 | 99.9 | 90.5 | 87.9 |

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## Share and Cite

**MDPI and ACS Style**

Lin, Y.; Yu, M.; Chen, K.; Jiang, G.; Chen, F.; Peng, Z.
Blind Mesh Assessment Based on Graph Spectral Entropy and Spatial Features. *Entropy* **2020**, *22*, 190.
https://doi.org/10.3390/e22020190

**AMA Style**

Lin Y, Yu M, Chen K, Jiang G, Chen F, Peng Z.
Blind Mesh Assessment Based on Graph Spectral Entropy and Spatial Features. *Entropy*. 2020; 22(2):190.
https://doi.org/10.3390/e22020190

**Chicago/Turabian Style**

Lin, Yaoyao, Mei Yu, Ken Chen, Gangyi Jiang, Fen Chen, and Zongju Peng.
2020. "Blind Mesh Assessment Based on Graph Spectral Entropy and Spatial Features" *Entropy* 22, no. 2: 190.
https://doi.org/10.3390/e22020190