中文摘要：

3D多尺度几何分析是图像、视频和几何模型等数字可视媒体处理的技术基础,其目的在于高效地表示这些媒体中存在的点、线、面奇异.为此,依据不同变换捕获奇异的能力演进及其非线性逼近效率的提高,从2D图像多尺度几何分析的研究进展切入,着重阐述视频3D多尺度几何分析的发展,并将其归纳为3类：由2D基函数直接扩展的3D多尺度几何分析、基于3D基函数的3D多尺度几何分析和基于时空非局部相关性的3D多尺度几何分析,深入探讨了各种典型变换方法的主要思想、非线性逼近效率、计算复杂度、优势和不足.同时,概要介绍了数字几何模型的3D多尺度几何分析研究进展.在此基础上,对3D多尺度几何分析的未来发展趋势进行了展望.

英文摘要：

Three-Dimensional （3D） multiscale geometrical analysis is the technological fundamental for the processing of digital visual media, such as images, videos, and geometrical models. Its objective is to efficiently represent the point singularity, curve singularity, as well as surface singularity presented in those visual media. This study first reviews the research advances in two-dimensional （2D） multiscale geometrical analysis. It then elaborates on the development of 3D multiscale geometrical analysis for video according to the capability evolution in capturing singularity and nonlinear approximation efficiency improvement of various transforms. State-of-the-Art 3D multiscale geometrical analysis is classified into three categories： the extended multiscale geometrical analysis from 2D basis functions, the multiscale geometrical analysis based on 3D basis function, and the multiscale geometrical analysis based on spatiotemporal non-local correlation. The basic ideas of typical transforms are thoroughly discussed subsequently, and so are their nonlinear approximation efficiency, computational complexity, advantages, and disadvantages. Meanwhile, this study also presents a general review on the development of the 3D multiscale geometrical analysis for geometrical models. Based on the study above, the development trend of the 3D multiscale geometrical analysis is forecast in the near future.