三角网格去噪算法综述与展望
Triangular Mesh Denoising Algorithms: A Review and Prospects
DOI: 10.12677/csa.2025.1510251, PDF,    科研立项经费支持
作者: 资 政, 仲彦军*, 王潇旖：新疆师范大学数学科学学院，新疆 乌鲁木齐；新疆师范大学CAD&CG实验室，新疆 乌鲁木齐；曹航宾：新疆师范大学数学科学学院，新疆 乌鲁木齐
关键词: 三角网格；数字几何处理；特征保持；网格去噪；深度几何学习；Triangular Mesh； Digital Geometry Processing； Feature Preservation； Mesh Denoising； Deep Geometric Learning
摘要: 三角网格去噪是数字几何处理的基础和核心步骤，对提高模型的质量和保障后续处理的效果有着关键的作用。随着三维数字化的发展，三维数据的获取变得容易，但受限于扫描仪的精度、物体表面光线反射等问题，获取的三维数据往往包含各种噪声，对后续数据的使用造成影响，因此网格去噪显得尤为重要。根据三角网格去噪算法的类型，将算法分为了优化法、滤波法和数据驱动法，并从理论框架、技术演进和应用场景三个维度深入分析了优化法、滤波法和数据驱动法三类去噪方法。简述了三种常用的评价指标，通过对比实验，对优化法、滤波法和数据驱动法在是否具有保持特征能力、有无顶点漂移现象、是否保持体积、是否存在面片翻转现象等方面进行了阐述，并通过评价指标进行比较。评估了各类方法在不同模型上的去噪效果及其优缺点，探讨了各方法的适用场景和发展方向，并对实时去噪、自适应参数调整以及深度几何学习等前沿方向进行了展望。
Abstract: Triangular mesh denoising is fundamental and core step to digital geometry processing, playing a crucial role in enhancing model quality and ensuring the effectiveness of downstream processing. The widespread use of 3D digitization facilitates data acquisition, yet scanner inaccuracies and surface light reflections introduce noise into acquired 3D data. This contamination significantly compromises downstream processing tasks, establishing mesh denoising as a critical preprocessing step in 3D data analysis. Triangular mesh denoising algorithms are categorized into optimization methods, filtering methods, and data-driven methods. This study systematically examines these three categories through a tripartite analytical framework encompassing theoretical foundations, technological evolution, and application scenarios. Three evaluation metrics are introduced. Comparative experiments examine optimization methods, filtering methods, and data-driven methods regarding feature preservation capability, vertex drift phenomena, volume retention, and mesh flipping artifacts, with systematic comparisons conducted using the metrics. A comprehensive evaluation demonstrates the denoising performance of various methods across different models, along with their respective advantages and disadvantages, and the applicability scenarios and future development directions for each method are explored. The study proposes future directions emphasizing real-time processing, adaptive parameter optimization, and deep geometric learning.
文章引用：资政, 仲彦军, 王潇旖, 曹航宾. 三角网格去噪算法综述与展望[J]. 计算机科学与应用, 2025, 15(10): 67-84. https://doi.org/10.12677/csa.2025.1510251

参考文献

[1] Zheng, Y., Fu, H., Au, O.K. and Tai, C. (2011) Bilateral Normal Filtering for Mesh Denoising. IEEE Transactions on Visualization and Computer Graphics, 17, 1521-1530. [Google Scholar] [CrossRef] [PubMed]
[2] Fleishman, S., Drori, I. and Cohen-Or, D. (2003) Bilateral Mesh Denoising. ACM SIGGRAPH 2003 Papers, San Diego, 27-31 July 2003, 950-953. [Google Scholar] [CrossRef] 
[3] Zhang, W., Deng, B., Zhang, J., Bouaziz, S. and Liu, L. (2015) Guided Mesh Normal Filtering. Computer Graphics Forum, 34, 23-34. [Google Scholar] [CrossRef] 
[4] He, L. and Schaefer, S. (2013) Mesh denoising via L0 minimization. ACM Transactions on Graphics, 32, 1-8. [Google Scholar] [CrossRef] 
[5] 王鹏, 王胜法, 曹俊杰, 等. L1优化在网格去噪中的应用[J]. 中国图象图形学报, 2014, 19(4): 637-644.
[6] Hattori, S., Yatagawa, T., Ohtake, Y. and Suzuki, H. (2022) Learning Self-Prior for Mesh Denoising Using Dual Graph Convolutional Networks. In: Avidan, S., Brostow, G., Cissé, M., Farinella, G.M. and Hassner, T., Eds., Computer Vision—ECCV 2022, Springer, 363-379. [Google Scholar] [CrossRef] 
[7] Li, X., Li, R., Zhu, L., Fu, C. and Heng, P. (2021) DNF-Net: A Deep Normal Filtering Network for Mesh Denoising. IEEE Transactions on Visualization and Computer Graphics, 27, 4060-4072. [Google Scholar] [CrossRef] [PubMed]
[8] Field, D.A. (1988) Laplacian Smoothing and Delaunay Triangulations. Communications in Applied Numerical Methods, 4, 709-712. [Google Scholar] [CrossRef] 
[9] Taubin, G. (1995) A Signal Processing Approach to Fair Surface Design. Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques—SIGGRAPH’95, Los Angeles, CA, 6-11 August 1995, 351-358. [Google Scholar] [CrossRef] 
[10] Taubin, G. (1995) Curve and Surface Smoothing without Shrinkage. Proceedings of IEEE International Conference on Computer Vision, Cambridge, 20-23 June 1995, 852-857.
[11] Vollmer, J., Mencl, R. and Müller, H. (1999) Improved Laplacian Smoothing of Noisy Surface Meshes. Computer Graphics Forum, 18, 131-138. [Google Scholar] [CrossRef] 
[12] Desbrun, M., Meyer, M., Schröder, P. and Barr, A.H. (1999) Implicit Fairing of Irregular Meshes Using Diffusion and Curvature Flow. Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques—SIGGRAPH’99, Los Angeles, CA, 8-13 August 1999, 317-324. [Google Scholar] [CrossRef] 
[13] Meyer, M., Desbrun, M., Schröder, P. and Barr, A.H. (2003) Discrete Differential-Geometry Operators for Triangulated 2-manifolds. In: Mathematics and Visualization, Springer Berlin Heidelberg, 35-57. [Google Scholar] [CrossRef] 
[14] Zhang, Y. and Ben Hamza, A. (2007) Vertex-based Diffusion for 3-D Mesh Denoising. IEEE Transactions on Image Processing, 16, 1036-1045. [Google Scholar] [CrossRef] [PubMed]
[15] Liu, Z., Wang, W., Zhong, S., Zeng, B., Liu, J. and Wang, W. (2020) Mesh Denoising via a Novel Mumford-Shah Framework. Computer-Aided Design, 126, Article ID: 102858. [Google Scholar] [CrossRef] 
[16] Wang, W., Pan, W., Dai, C., Dazeley, R., Wei, L., Rolfe, B., et al. (2023) Segmentation-Driven Feature-Preserving Mesh Denoising. The Visual Computer, 40, 6201-6217. [Google Scholar] [CrossRef] 
[17] Xu, L., Lu, C., Xu, Y. and Jia, J. (2011) Image Smoothing via L0 Gradient Minimization. Proceedings of the 2011 SIGGRAPH Asia Conference, Hong Kong, 12-15 December 2011, 1-12. [Google Scholar] [CrossRef] 
[18] Cheng, X., Zeng, M. and Liu, X. (2014) Feature-Preserving Filtering with L0 Gradient Minimization. Computers & Graphics, 38, 150-157. [Google Scholar] [CrossRef] 
[19] Centin, M. and Signoroni, A. (2018) Mesh Denoising with (Geo)metric Fidelity. IEEE Transactions on Visualization and Computer Graphics, 24, 2380-2396. [Google Scholar] [CrossRef] [PubMed]
[20] 赵勇, 李玲, 单欣, 等. 三维几何模型的L0去噪算法[J]. 计算机辅助设计与图形学学报, 2018, 30(5): 772-777.
[21] Donoho, D.L., Elad, M. and Temlyakov, V.N. (2006) Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise. IEEE Transactions on Information Theory, 52, 6-18. [Google Scholar] [CrossRef] 
[22] Wang, R., Yang, Z., Liu, L., Deng, J. and Chen, F. (2014) Decoupling Noise and Features via Weighted ℓ1-Analysis Compressed Sensing. ACM Transactions on Graphics, 33, 1-12. [Google Scholar] [CrossRef] 
[23] Wu, X., Zheng, J., Cai, Y. and Fu, C. (2015) Mesh Denoising Using Extended ROF Model with L1 Fidelity. Computer Graphics Forum, 34, 35-45. [Google Scholar] [CrossRef] 
[24] Zhong, S., Xie, Z., Liu, J. and Liu, Z. (2019) Robust Mesh Denoising via Triple Sparsity. Sensors, 19, Article 1001. [Google Scholar] [CrossRef] [PubMed]
[25] Li, X., Zhu, L., Fu, C. and Heng, P. (2018) Non‐Local Low‐Rank Normal Filtering for Mesh Denoising. Computer Graphics Forum, 37, 155-166. [Google Scholar] [CrossRef] 
[26] Zhang, H., Wu, C., Zhang, J. and Deng, J. (2015) Variational Mesh Denoising Using Total Variation and Piecewise Constant Function Space. IEEE Transactions on Visualization and Computer Graphics, 21, 873-886. [Google Scholar] [CrossRef] [PubMed]
[27] Zhong, S., Xie, Z., Wang, W., Liu, Z. and Liu, L. (2018) Mesh Denoising via Total Variation and Weighted Laplacian Regularizations. Computer Animation and Virtual Worlds, 29, e1827. [Google Scholar] [CrossRef] 
[28] Wei, M., Huang, J., Xie, X., Liu, L., Wang, J. and Qin, J. (2019) Mesh Denoising Guided by Patch Normal Co-Filtering via Kernel Low-Rank Recovery. IEEE Transactions on Visualization and Computer Graphics, 25, 2910-2926. [Google Scholar] [CrossRef] [PubMed]
[29] Liu, Z., Lai, R., Zhang, H. and Wu, C. (2019) Triangulated Surface Denoising Using High Order Regularization with Dynamic Weights. SIAM Journal on Scientific Computing, 41, B1-B26. [Google Scholar] [CrossRef] 
[30] Liu, Z., Zhong, S., Xie, Z. and Wang, W. (2019) A Novel Anisotropic Second Order Regularization for Mesh Denoising. Computer Aided Geometric Design, 71, 190-201. [Google Scholar] [CrossRef] 
[31] Bredies, K., Kunisch, K. and Pock, T. (2010) Total Generalized Variation. SIAM Journal on Imaging Sciences, 3, 492-526. [Google Scholar] [CrossRef] 
[32] Ferstl, D., Reinbacher, C., Ranftl, R., Ruether, M. and Bischof, H. (2013) Image Guided Depth Upsampling Using Anisotropic Total Generalized Variation. 2013 IEEE International Conference on Computer Vision, Sydney, 1-8 December 2013, 993-1000. [Google Scholar] [CrossRef] 
[33] Jung, M. and Kang, M. (2015) Simultaneous Cartoon and Texture Image Restoration with Higher-Order Regularization. SIAM Journal on Imaging Sciences, 8, 721-756. [Google Scholar] [CrossRef] 
[34] Feng, W.S., Lei, H. and Gao, Y. (2014) Speckle Reduction via Higher Order Total Variation Approach. IEEE Transactions on Image Processing, 23, 1831-1843. [Google Scholar] [CrossRef] [PubMed]
[35] Knoll, F., Bredies, K., Pock, T. and Stollberger, R. (2010) Second Order Total Generalized Variation (TGV) for MRI. Magnetic Resonance in Medicine, 65, 480-491. [Google Scholar] [CrossRef] [PubMed]
[36] Niu, S., Gao, Y., Bian, Z., Huang, J., Chen, W., Yu, G., et al. (2014) Sparse-View X-Ray CT Reconstruction via Total Generalized Variation Regularization. Physics in Medicine and Biology, 59, 2997-3017. [Google Scholar] [CrossRef] [PubMed]
[37] Liu, Z., Li, Y., Wang, W., Liu, L. and Chen, R. (2022) Mesh Total Generalized Variation for Denoising. IEEE Transactions on Visualization and Computer Graphics, 28, 4418-4433. [Google Scholar] [CrossRef] [PubMed]
[38] Zhang, H., He, Z. and Wang, X. (2022) A Novel Mesh Denoising Method Based on Relaxed Second-Order Total Generalized Variation. SIAM Journal on Imaging Sciences, 15, 1-22. [Google Scholar] [CrossRef] 
[39] Wang, J., Han, Z., Jiang, W. and Kim, J. (2023) A Fast, Efficient, and Explicit Phase-Field Model for 3D Mesh Denoising. Applied Mathematics and Computation, 458, Article ID: 128239. [Google Scholar] [CrossRef] 
[40] Wang, J., Li, Y., Choi, Y., Lee, C. and Kim, J. (2020) Fast and Accurate Smoothing Method Using a Modified Allen-Cahn Equation. Computer-Aided Design, 120, Article ID: 102804. [Google Scholar] [CrossRef] 
[41] Pan, J., Bo, P., Li, Y. and Wang, Z. (2024) Mesh Denoising of Developable Surfaces with Curved Foldings. Computer-Aided Design, 177, Article ID: 103776. [Google Scholar] [CrossRef] 
[42] Tomasi, C. and Manduchi, R. (1998) Bilateral Filtering for Gray and Color Images. Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271), Bombay, 7 January 1998, 839-846. [Google Scholar] [CrossRef] 
[43] Jones, T.R., Durand, F. and Desbrun, M. (2003) Non-Iterative, Feature-Preserving Mesh Smoothing. ACM SIGGRAPH 2003 Papers, San Diego, 27-31 July 2003, 943-949. [Google Scholar] [CrossRef] 
[44] Vialaneix, G. and Boubekeur, T. (2011) SBL Mesh Filter: Fast Separable Approximation of Bilateral Mesh Filtering. ACM SIGGRAPH 2011 Talks, Vancouver, 7-11 August 2011, 1. [Google Scholar] [CrossRef] 
[45] Choudhury, P. and Tumblin, J. (2005) The Trilateral Filter for High Contrast Images and Meshes. ACM SIGGRAPH 2005 Courses on—SIGGRAPH’05, Los Angeles, 31 July-4 August 2005, 5. [Google Scholar] [CrossRef] 
[46] Wang, Y., Yang, Y. and Liu, Q. (2020) Feature-Aware Trilateral Filter with Energy Minimization for 3D Mesh Denoising. IEEE Access, 8, 52232-52244. [Google Scholar] [CrossRef] 
[47] Yagou, H., Ohtake, Y. and Belyaev, A. (2002) Mesh Smoothing via Mean and Median Filtering Applied to Face Normals. Proceedings of Geometric Modeling and Processing, Wako, 10-12 July 2002, 124-131.
[48] Sun, X., Rosin, P.L., Martin, R. and Langbein, F. (2007) Fast and Effective Feature-Preserving Mesh Denoising. IEEE Transactions on Visualization and Computer Graphics, 13, 925-938. [Google Scholar] [CrossRef] [PubMed]
[49] Wei, M., Yu, J., Pang, W., Wang, J., Qin, J., Liu, L., et al. (2015) Bi-Normal Filtering for Mesh Denoising. IEEE Transactions on Visualization and Computer Graphics, 21, 43-55. [Google Scholar] [CrossRef] [PubMed]
[50] Hurtado, J., Gattass, M., Raposo, A. and Lopez, C. (2024) Sharp Feature-Preserving Mesh Denoising. Multimedia Tools and Applications, 83, 69555-69580. [Google Scholar] [CrossRef] 
[51] Wang, P., Fu, X., Liu, Y., Tong, X., Liu, S. and Guo, B. (2015) Rolling Guidance Normal Filter for Geometric Processing. ACM Transactions on Graphics, 34, 1-9. [Google Scholar] [CrossRef] 
[52] Zhang, J., Deng, B., Hong, Y., Peng, Y., Qin, W. and Liu, L. (2019) Static/Dynamic Filtering for Mesh Geometry. IEEE Transactions on Visualization and Computer Graphics, 25, 1774-1787. [Google Scholar] [CrossRef] [PubMed]
[53] Zhao, W., Liu, X., Wang, S. and Zhao, D. (2018) Multi-Scale Similarity Enhanced Guided Normal Filtering. In: Zeng, B., Huang, Q., El Saddik, A., Li, H., Jiang, S. and Fan, X., Eds., Advances in Multimedia Information Processing—PCM 2017, Springer, 645-653. [Google Scholar] [CrossRef] 
[54] Liu, S., Rho, S., Wang, R. and Jiang, F. (2018) Feature-Preserving Mesh Denoising Based on Guided Normal Filtering. Multimedia Tools and Applications, 77, 23009-23021. [Google Scholar] [CrossRef] 
[55] Zhao, W., Liu, X., Wang, S., Fan, X. and Zhao, D. (2021) Graph-Based Feature-Preserving Mesh Normal Filtering. IEEE Transactions on Visualization and Computer Graphics, 27, 1937-1952. [Google Scholar] [CrossRef] [PubMed]
[56] Zhong, S., Song, Z., Liu, Z., Xie, Z., Chen, J., Liu, L., et al. (2021) Shape-Aware Mesh Normal Filtering. Computer-Aided Design, 140, Article ID: 103088. [Google Scholar] [CrossRef] 
[57] Guo, M., Song, Z., Han, C., Zhong, S., Lv, R. and Liu, Z. (2021) Mesh Denoising via Adaptive Consistent Neighborhood. Sensors, 21, Article 412. [Google Scholar] [CrossRef] [PubMed]
[58] Li, T., Liu, W., Liu, H., Wang, J. and Liu, L. (2019) Feature-Convinced Mesh Denoising. Graphical Models, 101, 17-26. [Google Scholar] [CrossRef] 
[59] Han, H. and Han, J. (2022) Modified Bilateral Filter for Feature Enhancement in Mesh Denoising. IEEE Access, 10, 56845-56862. [Google Scholar] [CrossRef] 
[60] Yadav, S.K., Reitebuch, U. and Polthier, K. (2019) Robust and High Fidelity Mesh Denoising. IEEE Transactions on Visualization and Computer Graphics, 25, 2304-2310. [Google Scholar] [CrossRef] [PubMed]
[61] Wang, P., Liu, Y. and Tong, X. (2016) Mesh Denoising via Cascaded Normal Regression. ACM Transactions on Graphics, 35, 1-12. [Google Scholar] [CrossRef] 
[62] Wang, J., Huang, J., Wang, F.L., Wei, M., Xie, H. and Qin, J. (2019) Data-Driven Geometry-Recovering Mesh Denoising. Computer-Aided Design, 114, 133-142. [Google Scholar] [CrossRef] 
[63] Zhao, W.B., Liu, X.M., Zhao, Y.S., et al. (2019) NormalNet: Learning Based Guided Normal Filtering for Mesh Denoising. arXiv: 1903.04015.
https://arXiv.org/abs/1903.04015
[64] Armando, M., Franco, J. and Boyer, E. (2022) Mesh Denoising with Facet Graph Convolutions. IEEE Transactions on Visualization and Computer Graphics, 28, 2999-3012. [Google Scholar] [CrossRef] [PubMed]
[65] Shen, Y., Fu, H., Du, Z., Chen, X., Burnaev, E., Zorin, D., et al. (2022) GCN-Denoiser: Mesh Denoising with Graph Convolutional Networks. ACM Transactions on Graphics, 41, 1-14. [Google Scholar] [CrossRef] 
[66] Xing, Y., Tan, J., Hong, P., He, Y. and Hu, M. (2022) Mesh Denoising Based on Recurrent Neural Networks. Symmetry, 14, Article 1233. [Google Scholar] [CrossRef] 
[67] Zhao, W., Liu, X., Jiang, J., Zhao, D., Li, G. and Ji, X. (2022) Local Surface Descriptor for Geometry and Feature Preserved Mesh Denoising. Proceedings of the AAAI Conference on Artificial Intelligence, 36, 3446-3453. [Google Scholar] [CrossRef] 
[68] Tang, W., Gong, Y. and Qiu, G. (2023) Feature Preserving 3D Mesh Denoising with a Dense Local Graph Neural Network. Computer Vision and Image Understanding, 233, Article ID: 103710. [Google Scholar] [CrossRef] 
[69] Wang, X., Wei, H., Fan, X. and Zhao, D. (2024) Hyper-MD: Mesh Denoising with Customized Parameters Aware of Noise Intensity and Geometric Characteristics. 2024 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), Seattle, 16-22 June 2024, 4651-4660. [Google Scholar] [CrossRef] 
[70] Lee, S., Heo, S. and Lee, S. (2025) DMESH: A Structure-Preserving Diffusion Model for 3-D Mesh Denoising. IEEE Transactions on Neural Networks and Learning Systems, 36, 4385-4399. [Google Scholar] [CrossRef] [PubMed]
[71] Wang, X., Zhang, X., Cui, W., Xiong, R., Fan, X. and Zhao, D. (2024) Mesh Denoising Using Filtering Coefficients Jointly Aware of Noise and Geometry. Proceedings of the 32nd ACM International Conference on Multimedia, Melbourne, 28 October-1 November 2024, 1791-1799. [Google Scholar] [CrossRef] 
[72] Zhao, W.B., Liu, X.M., Zhai, D.M., et al. (2024) Mesh Denoising Transformer. arXiv: 2405.06536.
https://arxiv.org/abs/2405.06536
[73] Zhou, Z., Yuan, M., Zhao, M., Guo, J. and Yan, D. (2025) Resgem: Multi-Scale Graph Embedding Network for Residual Mesh Denoising. IEEE Transactions on Visualization and Computer Graphics, 31, 2028-2044. [Google Scholar] [CrossRef] [PubMed]