# 3D Mesh Pre-Processing Method Based on Feature Point Classification and Anisotropic Vertex Denoising Considering Scene Structure Characteristics

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Isotropic Mesh Filtering

#### 2.2. Anisotropic Mesh Filtering

#### 2.3. Conclusions of the Literature Review

## 3. Methodology

#### 3.1. Facet Normal Filtering

#### 3.1.1. Guidance Normal Considering Corner Features

**b**)–(

**e**) and (

**a**) are ${21.0551}^{\circ}$, ${0.691454}^{\circ}$, ${2.1604}^{\circ}$, and ${0.652975}^{\circ}$, respectively. It is clear that our method can avoid losing corner features in the process of denoising, and the result is closer to the original model. If directly using the isotropic neighborhood of the faces to calculate the guidance normals, for the non-flute corner area, the calculation of the guidance normal will not consider the larger neighborhood; it might be very close to the face normal of interest, and the facet normals in the non-featured areas will have a lower degree of filtering and a slower filtering speed, which will make it difficult to quickly denoise a flat or less variable area.

#### 3.1.2. Joint Bilateral Normal Filtering

#### 3.2. Feature Point Classification

#### 3.2.1. Feature Vertex Detection

#### 3.2.2. Weak Feature Recognition and False Feature Elimination

**c**), we recognize this as a feature point.

#### 3.3. Anisotropic Vertex Update

#### 3.3.1. Non-Feature Vertex Update

#### 3.3.2. Feature Vertex Update

## 4. Results

#### 4.1. Parameter Tuning

#### 4.2. Qualitative Assessment Experiments

#### 4.2.1. Qualitative Comparison of the Feature Vertex Classification Results

#### 4.2.2. Qualitative Comparison of the Mesh Denoising Results

#### 4.3. Quantitative Evaluation Experiments

#### 4.3.1. Quantitative Comparison of the Normal Difference

#### 4.3.2. Quantitative Comparison Of Distance

#### 4.3.3. Quantitative Comparison of Running Time

#### 4.4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NVT | Normal voting tensor |

BM | The bilateral mesh denoising method [3] |

NoIter | The noniterative, feature-preserving mesh smoothing method [27] |

Fast | Fast and effective feature-preserving mesh denoising [28] |

BNF | The local scheme of bilateral normal filtering method [4] |

L0 | Mesh denoising via L0 minimization [23] |

GMNF | Guided mesh normal filtering [5] |

RoFi | Robust and high fidelity mesh denoising [32] |

CPD | Constraint-based point set denoising [9] |

ENVT | Mesh denoising based on normal voting tensor and binary optimization [45] |

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**Figure 2.**Patches construction and guidance normal calculation of the face of interest (red) located at the corner position. (

**a**–

**j**): The candidate patches of the GMNF method [5] and the consistency measure $\mathcal{H}$. (

**k**): We calculate the guidance normal in the selected patch (assumed to be (

**c**)) using our dual weight function.

**Figure 4.**Weak feature point recognition: (

**a**) initially recognized feature points; (

**b**) eigenvectors of the tensor matrix at each feature point; and (

**c**) weak feature points are identified.

**Figure 5.**Comparison results for non-feature point updating: (

**a**) noisy model; (

**b**) the result without isotropic neighborhoods; and (

**c**) our result with isotropic neighborhoods.

**Figure 6.**The eigenvectors and eigenvalues of the tensor matrix. The eigenvectors correspond to the principal directions of the ellipsoid/ellipse, and the eigenvalues encode the size and shape of the ellipsoid/ellipse.

**Figure 7.**Correcting the direction of the feature vector: (

**a**) original eigenvectors; (

**b**) corrected eigenvectors; and (

**c**) clustering results (green) after denoising the non-feature vertices.

**Figure 8.**Comparison of the denoising effects of the updated feature points using different facet normals: (

**a**) the noisy mesh; (

**b**) update feature points using the current facet normals; (

**c**) update feature points using the guidance normals; (

**d**) update feature points using the last filtered facet normals in Equation (7); (

**e**) update feature points using a approximate faltered normal ${\mathbf{n}}_{{f}_{i}}^{\u2033}$ in Equation (14), where ${\mathbf{n}}_{{f}_{i}}^{\prime}$ is replaced by the guidance normal ${\mathbf{g}}_{i}$; (

**f**) our result.

**Figure 9.**Variation in the local sharp edge ${e}_{ij}\left({V}_{i},{V}_{j}\right)$ during the denoising of feature point ${V}_{i}$: (

**a**) 2D display of the sharp edges; (

**b**)–(

**d**) 2D representation of the dihedral angle changes; (

**b**) the original state in (

**a**); (

**c**) the dihedral angle of edge ${e}_{ij}\left({V}_{i},{V}_{j}\right)$ becomes smaller and the characteristic becomes weaker; and (

**d**) the dihedral angle of the edge ${e}_{ij}\left({V}_{i},{V}_{j}\right)$ becomes larger and the feature is sharpened.

**Figure 10.**Schematic diagram of the tensile forces of the local support neighborhoods on the feature point. ${d}_{j}$ is the distance from the feature point ${v}_{i}$ to the support plane ${\mathbf{p}}_{j}$, and ${\mathbf{n}}_{{\mathbf{p}}_{j}}$ is the normal of the ${j}^{th}$ plane. The tensile forces of the support planes on the shape edge point ${v}_{i}$ are shown in Equation (17).

**Figure 12.**Comparison of feature point classification results for the City model reconstructed from oblique images.

**Figure 13.**Comparison of feature point classification results for the Village model reconstructed from oblique images.

**Figure 17.**Comparison of the denoising results for the Village model reconstructed from oblique images.

**Figure 18.**Comparison of the denoising results for the Villa model reconstructed from oblique images.

Symbols | Glossary |
---|---|

${f}_{i}$ | Face i, ${f}_{i}=\left\{{v}_{i0},{v}_{i1},{v}_{i2}\right\}$ |

${A}_{i}$ | Area of face i |

${c}_{i}$ | Centroid of face i |

${v}_{i}$ | Vertex i |

${v}_{i}^{(t+1)}$ | New coordinate of vertex i in the ${t}^{th}$ iteration |

${\mathbf{n}}_{i}$ | Normal vector for face i |

${\mathbf{n}}_{{f}_{i}}^{\prime}$ | Filtered normal of face i for the first time in each iteration |

${\mathbf{n}}_{{f}_{i}}^{\u2033}$ | Filtered normal of face i for the second time in each iteration |

${\mathbf{g}}_{i}$ | Guidedance normal vector for face i |

${P}_{{f}_{i}}^{m}$ | The ${m}^{th}$ patch of the face ${f}_{i}$ |

${T}_{ij}$ | Dual weight function |

$N{F}_{{f}_{i}}$ | Set of neighboring faces of face i |

$N{F}_{{v}_{i}}$ | Set of neighboring faces of vertex i |

$N{F}_{{v}_{i}}^{isot}$ | Set of isotropic neighbor faces of vertex i |

${\mathit{T}}_{{v}_{i}}$ | Tensor voting matrix based on the normals at vertex i |

${\mathit{V}}_{{f}_{j}}$ | Normal voting component of the face ${f}_{j}$ |

${\lambda}_{ij}$ | Eigenvalue of ${\mathit{T}}_{{v}_{i}}$, $j=1,2,3$ |

${\widehat{\mathit{e}}}_{ij}$ | Unit eigenvector of ${\mathit{T}}_{{v}_{i}}$, $j=1,2,3$ |

$\tau $ | Feature classification threshold |

${V}_{f}$ | Set of non-feature vertices |

${V}_{e}$ | Set of shape edge vertices |

${V}_{c}$ | Set of corner vertices |

$\rho $ | Normal angle threshold |

${C}_{ij}$ | The ${j}^{th}$ supported neighborhood of feature point ${v}_{i}$, $j=1,2,3$ |

${\mathbf{p}}_{j}$ | The ${j}^{th}$ supported plane of feature point |

${\mathbf{P}}_{{v}_{i}}$ | Set of supported planes of feature point ${v}_{i}$ |

${\beta}_{i}$ | Dihedral angle constraint threshold, $i=1,2$ |

${\alpha}_{i}$ | Coefficients of the constraint terms, $i=1,2$, ${\alpha}_{1}+{\alpha}_{2}=1$ |

${K}_{s}$ | The spatial kernel |

${K}_{r}$ | The range kernel |

${\sigma}_{s}$ | Variance parameter of ${K}_{s}$ |

${\sigma}_{r}$ | Variance parameter of ${K}_{r}$ |

${\mathbf{N}}_{mean}$ | Average facet normal difference between the denoised mesh and the ground-truth mesh |

${\mathbf{D}}_{max}$ | Maximum distance from the resulting mesh vertices to the ground-truth mesh surface |

${\mathbf{D}}_{mean}$ | Average distance from the resulting mesh vertices to the ground-truth mesh surface |

${n}_{filter}$ | Number of iteration filtering facet normals, total number of iterations |

${n}_{update}$ | Number of times the vertices are updated in each iteration |

**Table 2.**The average normal difference ${\mathbf{N}}_{mean}$ (in degrees) between the facet normals of the denoised mesh and the ground-truth facet normals. The lowest value per row is highlighted in bold.

Model | Noisy | BM [3] | NoIter [27] | Fast [28] | BNF [4] | L0 [23] | GMNF [5] | RoFi [32] | Ours |
---|---|---|---|---|---|---|---|---|---|

Block | 33.7185 | 14.4753 | 13.8501 | 14.4753 | 5.30624 | 4.97352 | 3.77320 | 5.22034 | 3.18393 |

Fandisk | 28.4211 | 11.1113 | 9.81795 | 4.04264 | 3.34944 | 5.52852 | 2.62181 | 2.44769 | 1.37350 |

SharpSphere | 33.0070 | 16.7373 | 17.3618 | 11.8920 | 6.70474 | 12.9565 | 9.53772 | 9.05684 | 8.43270 |

Julius | 23.9629 | 10.2910 | 7.63005 | 7.01017 | 6.00205 | 7.97741 | 6.50926 | 7.70099 | 6.42512 |

Cube | 21.0551 | 8.71710 | 7.48375 | 2.02613 | 1.38315 | 1.82864 | 0.69145 | 2.42332 | 0.593294 |

Pyramid | 19.2730 | 5.81730 | 4.66767 | 0.68221 | 0.84914 | 3.51391 | 0.51771 | 1.35993 | 0.504128 |

jointSharpEdges | 33.6741 | 13.7765 | 13.2518 | 10.5136 | 8.25214 | 2.13461 | 3.35048 | 2.47633 | 2.28357 |

**Table 3.**The maximum distance ${\mathbf{D}}_{max}$ from the resulting mesh vertices to the ground-truth mesh surface. The lowest value per row is highlighted in bold.

Model | Noisy | BM [3] | NoIter [27] | Fast [28] | BNF [4] | L0 [23] | GMNF [5] | RoFi [32] | Ours |
---|---|---|---|---|---|---|---|---|---|

Block | 1.24700 | 0.86231 | 0.68534 | 0.60524 | 0.60328 | 0.60257 | 0.30105 | 0.60313 | 0.30061 |

Fandisk | 0.12625 | 0.08283 | 0.08349 | 0.05911 | 0.05895 | 0.08276 | 0.04153 | 0.04161 | 0.04150 |

SharpSphere | 0.45456 | 0.44946 | 0.33997 | 0.44865 | 0.22433 | 0.33649 | 0.33760 | 0.39257 | 0.33257 |

Julius | 0.00792 | 0.00790 | 0.00790 | 0.00789 | 0.00790 | 0.00790 | 0.00790 | 0.00789 | 0.00788 |

Cube | 0.08521 | 0.06631 | 0.04955 | 0.03227 | 0.01602 | 0.06350 | 0.02119 | 0.03186 | 0.01589 |

Pyramid | 0.03197 | 0.02104 | 0.02125 | 0.01587 | 0.01587 | 0.03156 | 0.01590 | 0.01578 | 0.01583 |

jointSharpEdges | 0.02117 | 0.01793 | 0.01194 | 0.01181 | 0.01181 | 0.01177 | 0.01181 | 0.00884 | 0.01179 |

**Table 4.**The average distance ${\mathbf{D}}_{mean}$ from the resulting mesh vertices to the ground-truth mesh surface. The lowest value per row is highlighted in bold.

Model | Noisy | BM [3] | NoIter [27] | Fast [28] | BNF [4] | L0 [23] | GMNF [5] | RoFi [32] | Ours |
---|---|---|---|---|---|---|---|---|---|

Block | 0.19502 | 0.08608 | 0.08632 | 0.05385 | 0.03436 | 0.13168 | 0.03559 | 0.11360 | 0.01988 |

Fandisk | 0.02151 | 0.00842 | 0.01017 | 0.00424 | 0.00321 | 0.01345 | 0.00457 | 0.00447 | 0.00118 |

SharpSphere | 0.06948 | 0.02864 | 0.03005 | 0.03235 | 0.00773 | 0.06397 | 0.01811 | 0.02110 | 0.01455 |

Julius | 0.00022 | 0.00004 | 0.00005 | 0.00002 | 0.00004 | 0.00004 | 0.00002 | 0.00003 | 0.00001 |

Cube | 0.01892 | 0.00954 | 0.01109 | 0.00437 | 0.00177 | 0.01029 | 0.00483 | 0.00605 | 0.00526 |

Pyramid | 0.00292 | 0.00104 | 0.00203 | 0.00015 | 0.00015 | 0.00102 | 0.00001 | 0.00010 | 0.00000 |

jointSharpEdges | 0.00338 | 0.00091 | 0.00112 | 0.00075 | 0.00078 | 0.00122 | 0.00085 | 0.00029 | 0.00028 |

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**MDPI and ACS Style**

Liu, Y.; Guo, B.; Xiao, X.; Qiu, W.
3D Mesh Pre-Processing Method Based on Feature Point Classification and Anisotropic Vertex Denoising Considering Scene Structure Characteristics. *Remote Sens.* **2021**, *13*, 2145.
https://doi.org/10.3390/rs13112145

**AMA Style**

Liu Y, Guo B, Xiao X, Qiu W.
3D Mesh Pre-Processing Method Based on Feature Point Classification and Anisotropic Vertex Denoising Considering Scene Structure Characteristics. *Remote Sensing*. 2021; 13(11):2145.
https://doi.org/10.3390/rs13112145

**Chicago/Turabian Style**

Liu, Yawen, Bingxuan Guo, Xiongwu Xiao, and Wei Qiu.
2021. "3D Mesh Pre-Processing Method Based on Feature Point Classification and Anisotropic Vertex Denoising Considering Scene Structure Characteristics" *Remote Sensing* 13, no. 11: 2145.
https://doi.org/10.3390/rs13112145