A spectral geometric framework for 3D shape analysis and applications

鉴于3D游戏和娱乐的惊人流行，互联网上自由或商业地分发着大量的3D模型数据库。这类存储库的可用性和广泛使用，加上深入和广泛探索3D形状的需要，引发了搜索这些海量数据集合、检索最相关的选择并允许它们有效重复使用的需求。这项研究计划寻求开发一个统一的理论和计算框架来设计新的不变形状描述符，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规范定义的微分算子的谱之间的关系，而Morse理论通过研究流形上的可微函数提供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、特征提取和匹配算法如何相互作用来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，并在此方向上建立我们现有的一些工作。具体地说，我们的目标是(I)通过从谱几何和Morse理论中引入更多合理的概念来形式化形状签名的设计和分析；(Ii)开发一种理论上严格且计算上可行的3D对象识别方法；以及(Iii)探索这些签名在其他应用中的应用，包括动画和多媒体安全。将在本项目中研究的形状签名提供了一种通过理解3D形状的内在几何来看待形状分析问题的新方法。这一成果将是对最先进的3D形状分析技术的重要贡献，也将对许多计算机视觉、图形和多媒体应用产生深远的影响。

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.