AF: Small: Fundamental Geometry Processing

在这个项目中，新的离散几何处理算法的基础上简单直观的离散化的低阶微分形式将开发，沿着支持的理论基础，它将被证明，所提出的方法统一和扩展了一些现有的网格松弛算法用于去噪，细分，和交互式形状变形。 在经典曲面理论中，曲面片由两个参数的光滑3D值参数化函数定义，在微分形式的语言中称为3D值微分0-形式。 这些0-形式之一的两个偏导数是定义3D值微分1-形式的三维向量场。 曲面变形的一个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此1-形式，然后求解其偏导数与修改后的1-形式的分量向量场相匹配的参数化函数。 这种方法的图形嵌入和多边形网格的变形的离散模拟将被开发。 第一基本形式测量光滑表面上的距离和角度，第二基本形式测量表面法线如何变化，即，曲率 这两种基本形式对曲面的刚体变换是不变的，并且满足Gauss-Codazzi-Mainardi（CDM）方程。 相反，给定两个二阶对称张量场同时满足CDM方程，曲面理论的基本定理断言：1）存在一个浸入三维欧氏空间的曲面，这些场作为其第一和第二基本形式; 2）曲面是唯一的模刚体变换。 多边形网格的类比定理将被公式化和证明，包括任意拓扑的流形网格、有边界的网格，甚至非流形网格的扩展。 新的贡献网格压缩文献将利用重建算法和连通性保持网格压缩方案之间的关系。

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.