A spectral geometric framework for 3D shape analysis and applications

鉴于3D游戏和娱乐的巨大流行，大量的3D模型数据库在互联网上免费或商业化分发。这种存储库的可用性和广泛使用，再加上在深度和广度上探索3D形状的需要，引发了搜索这些庞大的数据集合，检索最相关的选择，并允许它们被有效重用的需求。该研究计划旨在为设计新颖的不变形状描述符开发统一的理论和计算框架，以促进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微分算子谱之间的关系，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分析流形拓扑的方法。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，以确定整体形状检索系统的性能。此外，为了更深入地了解自己在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，同时也在这个方向上建立我们现有的一些工作。特别是，我们的目标是(i)通过引入更多来自光谱几何和莫尔斯理论的声音概念来形式化形状特征的设计和分析；发展一种理论上严谨、计算上可行的三维物体识别方法；（iii）探索这些签名在其他应用程序中的使用，包括动画和多媒体安全。该项目将研究的形状特征提供了一种新的方法，通过理解三维形状的内在几何形状来看待形状分析问题。该结果将对最先进的3D形状分析技术做出重要贡献，并将对大量计算机视觉，图形学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.