大尺度变形的三维几何模型的对应关系和分割问题研究

项目名称： 大尺度变形的三维几何模型的对应关系和分割问题研究

项目编号： No.61462059

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 计算机科学学科

项目作者： 杨军

作者单位： 兰州交通大学

项目金额： 47万元

中文摘要： 大尺度非刚性变形的三维模型的对应关系是几何处理中的一个基础问题，也是一个难点问题。目前绝大多数应用中所涉及的对应关系都依赖于模型上标记特征点，并且许多研究工作都针对模型的刚性变换和等距或类似等距的非刚性变换，而对大尺度变形的模型间的对应关系的研究尚处在探索阶段。本研究拟开展对大尺度变形的三维模型的对应关系和分割算法的基础性研究，旨在计算准确的对应关系目标函数和完成一簇模型的协同分割。主要研究工作:(1)基于Riemann流形上的热扩散原理和量子粒子的波动理论，计算模型的本征描述符，推导对应关系的目标函数。（2）提出了无监督的协同分层分析理论，主要为了发现一簇模型的分层部件结构，揭示部件之间的关系，完成协同分割。(3)根据模型的Laplace-Beltrami微分算子的特征函数和特征值，将大尺度变形的模型嵌入到一个无群维的特征空间中，然后由曲面上Green函数值的分布计算出对应关并完成分割。

中文关键词： 对应关系；模型分割；波函数；热扩散；Laplace-Beltrami；微分算子

英文摘要： 3D shape correspondence for non-rigid large scale deformation is a fundamental and difficult problem in geometry processing.The existing shape correspondence methods in many applications depend strongly on initial feature markers, and also a large amount of research has been done in developing correspondence on rigid transformation and non-rigid isometric(or nearly isometric) deformation. However, study on shape correspondence of 3D geometric models differing by large-scale deformations is still at the exploratory stage. 3D shape correspondence and segmentation algorithms capable of handling large, non-rigid variations are going to be studied in this project so as to obtain some accuate objective functions of shape correspondence and to accomplish co-segmentation of a set of shapes. The main contributions of the research include: (1)Two state-of-art shape correspondence algorithms are presented respectively after finding intrinsic geometry descriptors of the shapes based on heat diffusion on Riemannian manifolds and wave functions of quantum particles.(2)An unsupervised co-hierarchical analysis of a set of shapes is introduced, aimed at discovering their hierarchical part structutes, revealing relations between geometrically dissimilar yet functionally equivalent shape parts across the set and accomplishing co-segmentation. (3)Shapes with large-scale deformations are embedded into infinite dimensional space by using the eigenvalues and eigenfunctions of the Laplace-Beltrami differential operator. Moreover, a deformation invariant shape descriptor related to the distribution of the Green's function's values on the surface is introduced to derive shape correspondence and finish co-segmentation.

英文关键词： shape correspondence;shape segmentation;wave function;heat diffusion;Laplace-Beltrami differential operator