MSPA-MCS: Collaborative Research: Computer Graphics and Visualization Using Conformal Geometry

拟将保形曲面理论应用于各种几何表示来计算构象结构。利用计算的共形结构，可以开发新的几何表示和分析工具，这将为科学和工程的多个前沿领域的进步铺平道路。本文提出的工作将特别探索计算机图形学和可视化领域的这些潜力。构建上保角结构将许多三维(3D)几何问题转化为二维(2D)问题，并导致了许多基本几何问题的有效方法。这些方法可以立即受益于广泛的应用，如保证人脸分类、曲面匹配和形状分析、几何建模、仿真、图形绘制和可视化。进行保角参数化需要求解大的最小二乘问题。为了进一步将保角结构应用于交互或时间关键运算，研究小组将研究基于稀疏QR和不完全稀疏QR算法的最小二乘问题的新颖迭代方法。各种科学和工程应用涉及建模、设计、分析、仿真和图形绘制等关键操作。所有这些操作都建立在几何表示的基础之上。在这个项目中，科学家试图通过引入一种唯一表征几何表面的构象结构来彻底改变这一基础。许多物理定律都受保角结构的支配。例如，表面上的热扩散和电磁场分布、肥皂泡中的张力以及理论物理中的部分弦理论都是由共形表面结构决定的。受到现有成功的鼓舞，科学家们努力探索和揭示保形结构在计算机图形学、几何建模和许多科学计算中的潜力。为了说明这一潜力，考虑共形结构的一个方面，即将一个表面规范地展平成一个平面，从而产生看似复杂的三维几何的图像表示。总体而言，该项目开发的工具可以帮助促进有效技术的开发，以处理与科学模拟、数据探索以及监视和生物发现的身份匹配或形状分析相关的新出现的问题。

The proposed research is to apply conformal surface theory tovarious geometry representations to compute conformalstructures. Using computed conformal structures, new geometryrepresentation and analysis tools can be developed, which will pavethe road for advances in multiple fronts of science andengineering. The work proposed herein will especially explore thesepotentials in computer graphics and visualization. Building upconformal structures recasts many three dimensional (3D) geometricproblems into two dimensions (2D) and leads to efficient approachesfor a number of fundamental geometric problems. These approaches canthen immediately benefit a wide range of applications, such assurface classification, surface matching and shape analysis,geometric modeling, simulation, graphics rendering andvisualization. Performing conformal parameterization requiressolving large least squares problems. To further push theapplication of the conformal structure to interactive or timecritical operations, the research team will investigate noveliterative methods for least-squares problems based on sparse QR andincomplete sparse QR algorithms.Various scientific and engineering applications concern about keyoperations such as modeling, design, analysis, simulation, andgraphics rendering. All these operations are built on top afoundation of geometry representation. In this project, scientistsaim to revolutionize this foundation by introducing a conformalstructure uniquely characterizing geometry surfaces. Many laws ofphysics are governed by conformal structures. For example, heatdiffusion and electromagnetic field distribution on surfaces,tension in soap bubbles and parts of string theory in theoreticalphysics are determined by conformal surface structures. Encouragedby the existing success, the scientists strive to explore and unveilthe potentials of conformal structures for computer graphics,geometric modeling and much of scientific computing. To illustratethis potential, consider one aspect of conformal structures, namelythe canonical flattening of a surface into a plane, resulting in animage like representation of seemingly complicated three dimensionalgeometry. Overall, the tools developed in this project can helpboost the development of effective techniques to deal with theemerging problems related to scientific simulation, dataexploration, and identity matching or shape analysis forsurveillance and biological discovery.