高精度常平均曲率曲面建模理论与应用研究

In this proposal, we present a uniform optimization framework for constructing accurate constant mean curvature (CMC for short) surfaces. This framework can supply new solutions and thoughts for high quality discrete meshes modeling for approximating constant mean curvature surfaces which are natural surfaces. This kind of surfaces have wide application background in architecture modeling, physical simulation area, and so on. We will focus on the application expanding in these areas and study CMC surfaces modeling theories in this project. This project will take a systematic research to this argument, including: (1) Study the geometric properties of CMC surface, we will present a uniform framework for accurate and flexible Constant Mean Curvature surfaces discrete modeling; (2) Efficient optimizing method is studied to solve this constrained optimization energy function for constructing CMC surfaces; (3) The accurate volume control with physical property constraints CMC surface construction; (4) The applications of CMC surfaces with constraints. The research include theories, methods, technologies and prototype system. This project will give new theories and methods for accurate and flexible CMC surfaces modeling. The completion of the project will provide new theories and methods for solving the construction of high quality CMC surfaces with discrete meshes, give out a series of simple, efficient and robust technologies for the problems of 3D geometric modeling, with great theoretical and application value.

本项目提出了一种构造高精度常平均曲率（Constant Mean Curvature，简称CMC）曲面的统一优化框架，为构造高质量离散曲面模型尤其是自然曲面模型提供新的解决方案和思路。该类型曲面在建筑造型、物理模拟等领域具有广泛的应用背景，本项目将以以上领域的应用需求为中心，展开曲面模型的构造理论和算法研究。主要研究内容包括：1)研究CMC曲面的几何属性，建立一个统一的高精度、带约束CMC曲面构造及优化理论框架；2)研究多约束情况下CMC曲面构造目标函数的有效优化方法；3)研究带有物理属性约束及精确体积控制下的CMC曲面构造；4)带约束CMC曲面的应用。研究范围涉及理论与方法、技术和原型系统。项目主要目标是为解决高精度、带约束CMC曲面建模及应用问题提供新理论和方法，并提供一系列简单、高效和鲁棒的技术，为上述领域中的相关理论和算法问题的解决提供新的思路和手段，具有很好的理论意义和应用价值。

本项目主要围绕高精度离散常平均曲率曲面建模及相关的建模与数据处理这一问题展开研究，目标是建立一种统一的优化理论框架，构建带约束的高精度离散常平均曲率曲面模型。我们深入研究了极小曲面的性质，提出了一个统一的高精度、带约束常平均曲率曲面建模的统一框架，该框架能够构造网格质量高且平均曲率误差小的极小曲面和常平均曲率曲面，且可以在框架下增加新的约束。此外，我们还研究了高精度建模及图像处理中的其他相关问题，例如复杂拓扑的重新网格化问题、三维点云数据的骨架提取问题、基于测地距离的RGB和RGB-D图像的超像素分割问题。针对以上问题，提出了有效的解决方法，并有相关论文发表。为其关键和难点问题的解决提供了新思路和途径、新理论和方法。完成了项目的既定目标。