AF: Small: Fundamental Geometry Processing

在本项目中，将开发基于低阶微分形式的简单直观离散化的新的离散几何处理算法，以及支持的理论基础，并且将表明所提出的方法统一并扩展了用于去噪，细分和交互式形状变形的许多现有网格松弛算法。在经典曲面理论中，曲面斑块是由两个参数的光滑三维值参数化函数定义的，在微分形式的语言中称为三维值微分0形式。其中一个0形式的两个偏导数是三维向量场，它定义了一个三维值的微分1形式。表面变形的一种简单方法是通过局部拉伸和旋转其两个分量向量场来修改该1-form，然后求解其偏导数与修改后的1-form的分量向量场匹配的参数化函数。这种方法的离散模拟图嵌入和多边形网格的变形将被开发。第一种基本形式测量光滑表面上的距离和角度，第二种基本形式测量表面法线的变化，即曲率。这两种基本形式对表面的刚体变换是不变的，并且满足gaas - codazzi - mainardi （CDM）方程。反之，给定两个二阶对称张量场同时满足CDM方程，曲面理论基本定理断言：1)在三维欧几里得空间中存在一个以这些场为其第一和第二基本形式的曲面；且2)表面是唯一模刚体变换。将阐述并证明多边形网格的模拟定理，包括扩展到任意拓扑的流形网格，带边界的网格，甚至非流形网格。利用重建算法和保持连通性的网格压缩方案之间的关系，将对网格压缩文献做出新的贡献。

In this project new discrete geometry processing algorithms based on simple and intuitive discretizations of low order differential forms will be developed, along with the supporting theoretical foundations, and it will be shown that the proposed approach unifies and extends a number of existing mesh relaxation algorithms used for denoising, subdivision, and interactive shape deformation. In the classical theory of surfaces, a surface patch is defined by a smooth 3D-valued parameterization function of two parameters, which in the language of differential forms is referred to as a 3D-valued differential 0-form. The two partial derivatives of one of these 0-forms are three dimensional vector fields which define a 3D-valued differential 1-form. A simple approach to surface deformations is to modify this 1-form by locally stretching and rotating its two component vector fields, and then solve for a parameterization function whose partial derivatives match the component vector fields of the modified 1-form. The discrete analog of this approach for deformations of graph embeddings and polygon meshes will be developed. The first fundamental form measures distances and angles on a smooth surface, and the second fundamental form measures how the surface normal varies, i.e., curvature. The two fundamental forms are invariant to rigid body transformations of the surface, and satisfy the Gauss-Codazzi-Mainardi (CDM) equations. Conversely, given two second order symmetric tensor fields satisfying together the CDM equations, the Fundamental Theorem of Surface Theory asserts that: 1) there exists a surface immersed in three-dimensional Euclidean space with these fields as its first and second fundamental forms; and 2) the surface is unique modulo rigid body transformations. The analog theorem for polygon meshes will be formulated and proven, including extensions to manifold meshes of arbitrary topology, meshes with border, and even non-manifold meshes. New contributions to the mesh compression literature will be made by exploiting the relationship between reconstruction algorithms and connectivity-preserving mesh compression schemes.