A spectral geometric framework for 3D shape analysis and applications

鉴于3D游戏和娱乐的巨大普及，大量的3D模型数据库在互联网上免费或商业地分发。这种存储库的可用性和广泛使用，再加上需要深入和广度探索3D形状，引发了搜索这些庞大的数据集，检索最相关的选择，并允许它们被有效地重用的需求。该研究项目旨在开发一个统一的理论和计算框架，用于设计新颖的不变形状描述符，以便于形状或形状部分之间的比较和区分。其主要思想是谱几何提供了流形的几何结构和正则定义的微分算子的谱之间的关系，而莫尔斯理论通过研究流形上的可微函数给出了一种非常直接的方法来分析流形的拓扑。通过结合这些理论，本项目将研究形状处理，签名提取和匹配算法如何相互作用，以确定整体形状检索系统的性能。此外，为了更深入地了解我自己在3D形状分析算法开发方面的专业知识，我将专注于将理论结果转化为现实的实际应用。这个项目的目标是研究新的形状描述符的设计和分析，并建立在我们现有的一些工作沿着这个方向。特别是，我们的目标是（一）通过引入更多的声音的概念，从频谱几何和莫尔斯理论的形状签名的设计和分析形式化;（二）开发一个理论上严格和计算上可行的方法，三维物体识别;和（三）探索使用这些签名在其他应用程序，包括动画和多媒体安全。在这个项目中，将调查的形状签名提供了一种新的方式来看待形状分析问题，通过了解三维形状的内在几何形状。其结果将对3D形状分析的最先进技术做出重要贡献，并将对一系列计算机视觉，图形和多媒体应用产生深远影响。

In light of the prodigious popularity of 3D gaming and entertainment, vast databases of 3D models are distributed freely or commercially on the Internet. The availability and widespread usage of such repositories, coupled with the need to explore 3D shapes in depth as well as in breadth, has sparked the need to search these vast data collections, retrieve the most relevant selections, and permit them to be effectively reused. This research program seeks to develop a unified theoretical and computational framework for the design of novel invariant shape descriptors to facilitate comparison and differentiation between shapes or parts of a shape. The main idea is that spectral geometry provides relationships between geometric structures of manifolds and spectra of canonically defined differential operators, while Morse theory gives a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. By combining these theories, this project will study how shape processing, signature extraction, and matching algorithms interact to determine overall shape retrieval system performance. Furthermore, to gain greater insight into my own expertise in algorithmic development for 3D shape analysis, I will focus on translating the theoretical results into realistic practical applications. The goal of this project is to investigate the design and analysis of novel shape descriptors, and also to build on some of our existing work along this direction. In particular, we aim to (i) Formalize both the design and analysis of shape signatures by introducing more sound concepts from spectral geometry and Morse theory; (ii) Develop a theoretically rigorous and computationally feasible methodology for 3D object recognition; and (iii) Explore the use of these signatures in other applications, including animation and multimedia security. The shape signatures that will be investigated in this project provide a new way to look at the shape analysis problem by understanding the intrinsic geometry of a 3D shape. The outcome will be important contributions to the state-of-the-art techniques for 3D shape analysis, and will also have a profound impact on a slew of computer vision, graphics, and multimedia applications.