A spectral geometric framework for 3D shape analysis and applications

点击翻译按钮获取中文摘要

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.