Abstract: With the continuous development of 3D shape research technology, the issue of non-rigid 3D shape correspondence becomes increasingly important, with applications spanning multiple fields such as computer graphics, computer vision, and pattern recognition. The functional maps framework has achieved advanced results in non-rigid 3D shape correspondence, as it can capture complex relationships between shapes and exhibit robustness to the topological noise of non-rigid shapes. This paper first outlines the fundamental concepts and research directions of 3D shape correspondence, then it elaborates on the basic framework of functional maps. Building on this, it systematically reviews the classic works in the related field, including traditional functional maps methods and deep functional maps methods, highlighting the advantages and limitations of different approaches in addressing non-rigid shape correspondence. Subsequently, the paper introduces commonly used datasets in the field of non-rigid 3D shape correspondence, and conducts experimental comparisons and analysis of different methods. Finally, it provides an outlook on the future development trends of non-rigid 3D shape correspondence.

Key words: non-rigid 3D shape, shape correspondence, functional maps, Laplace-Beltrami operator

摘要： 随着三维形状研究的不断发展，非刚性三维形状对应的问题变得愈加重要，这一问题的应用涵盖了计算机图形学、计算机视觉和模式识别等多个领域。泛函映射框架在非刚性三维形状对应中取得了先进的效果，其能够捕捉形状之间的复杂关系，对非刚性形状的拓扑噪声具有鲁棒性。说明了三维形状对应的基本概念以及其研究方向；阐述了泛函映射的基本框架；在此基础上系统地梳理了相关领域内的经典工作，包括传统泛函映射方法和深度泛函映射方法，突出说明了不同方法在处理非刚性形状对应上的优势和局限性；介绍了非刚性三维形状对应领域常用的数据集，对不同方法进行了实验对比与分析；对非刚性三维形状对应的发展趋势进行了展望。

关键词: 非刚性三维形状, 形状对应, 泛函映射, 拉普拉斯-贝尔特拉米算子