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图学学报

图学学报 ›› 2022, Vol. 43 ›› Issue (6): 957-966.DOI: 10.11996/JG.j.2095-302X.2022060957

• 综述 • 上一篇    下一篇

计算机辅助拓扑设计 ——持续同调在几何设计和处理中的应用

  

  1. 1. 浙江大学数学科学学院，浙江 杭州 310027；  2. 浙江大学 CAD&CG 国家重点实验室，浙江 杭州 310058
  • 出版日期:2022-12-30 发布日期:2023-01-11
  • 基金资助:
    国家自然科学基金项目(61872316，61932018) 

Computer aided topological design ——survey on geometric design and processing based on persistent homology

  1. 1. School of Mathematical Sciences, Zhejiang University, Hangzhou Zhejiang 310027, China;  2. State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou Zhejiang 310058, China
  • Online:2022-12-30 Published:2023-01-11
  • Supported by:
    National Natural Science Foundation of China (61872316, 61932018) 

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摘要：

持续同调是一种计算不同尺度拓扑特征的有效方法。其从一簇向后包含的单纯复形序列中提取 出拓扑特征的出现和消失时刻，并使用拓扑特征的“生命周期”来量化地衡量该特征的几何尺度和重要程度。拓 扑特征的提取与应用在几何设计中扮演着重要角色，催生出了一些基于持续同调的几何设计研究。从持续同调 特征的提取与基于持续同调的建模和优化两方面进行综述，在持续同调特征的提取方面，介绍了从点云和三角 网格数据中提取拓扑特征的不同方法，总结了拓扑特征在部分几何设计问题中的应用路径。在建模和优化方面， 综述了基于拓扑变换的单纯复形重建方法、拓扑可感知的曲面重建方法与基于持续同调的拓扑去噪和优化 方法。

关键词: 持续同调, 拓扑特征提取, 形状重建, 去噪与优化, 几何设计, 拓扑设计

Abstract:

Persistent homology is an effective method to compute topological features with different scales. It captures the birth and death time from a nested sequence of simplicial complexes, and employs the life span of a topological feature to measure its geometric scale and significance. The extraction and application of topological features play an important role in geometric design, spawning some studies on geometric design based on persistent homology. In this paper, we introduced the application studies in two aspects, i.e., the feature extraction of persistent homology and the persistent homology-based modeling and optimization. In the feature extraction of persistent homology, we introduced various methods for topological feature extraction from point clouds and triangular meshes, respectively. Meanwhile, the pipeline for applying topological features to some geometric design problems was summarized. In the persistent homology-based modeling and optimization, we reviewed the simplicial complex reconstruction methods based on topology transform, topology-aware surface reconstruction methods, and topological denoising and optimization based on persistent homology. 

Key words: persistent homology, topological feature extraction, shape reconstruction, denoising and optimization; geometric design, topological design

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