CAREER: Conformal Geometry Applied to Shape Analysis and Geometric Modeling

保角结构是与黎曼曲面(一般曲面是黎曼曲面)联系在一起的一种自然结构，在几何和物理中起着重要的作用。共形结构已被证明在图形和视觉中很有用，例如，共形参数化提供了高质量的无局部失真的纹理映射，它被用于曲面匹配和变形，并应用于脑部映射等应用中。研究者探索了共形几何在计算机图形学和视觉中的潜力，最终使计算共形几何这一跨学科领域变得可接触和有用。关于它的直接影响，研究者将保角几何应用于几何建模和形状分析。具体地说，1.基于TeichMuller空间理论构造形状空间，其中所有曲面的空间被建模为有限维流形，每个点表示一个曲面的共形等价类(Riemann曲面)，形状空间的度量度量两个形状的共形结构的偏差。另一个相关的目标是建立一个几何搜索引擎。2.找到一种系统的方法将平面区域上定义的几何结构推广到流形上，如流形三角B样条和流形Powell-Sabin曲面。通过在流形B样条线曲面中插入节点来设计新的曲面细分方案。在教育方面，研究人员找到了一种有效的方法来教授非数学专业的共形几何(黎曼曲面理论)，通过实施实用的算法来可视化抽象的概念，建立对深刻理论的理解。研究人员建立并传播了一套具体的软件工具来计算和可视化任意实曲面的共形结构，这使得学生可以理解该理论，并使其实际应用对更广泛的社区有用。基于保角几何的流形B-Spline工具弥补了传统图形多边形网格与CAGD中的样条曲面之间的鸿沟。通用几何搜索引擎应用于几何数据库和互联网搜索引擎。研究人员通过系统地开发经典材料来补充软件开发，允许将其整合到非数学专业的课程中。计算共形几何的研究和教育直接惠及计算机图形学、视觉、科学计算、医学成像、数学和物理等领域。计算共形几何已经对图形行业产生了影响，并将在未来得到更广泛的应用。

Conformal structure is a natural structure associated with Riemann surfaces (General surfaces are Riemann surfaces) and plays fundamental roles in geometry and physics. Conformal structure has proven useful in graphics and vision, for example, conformal parameterization provides high-quality texture mapping without local distortion, and it is used in surface matching and morphing with applications such as brain mapping.The investigator explores the potential of conformal geometry in computer graphics and vision and ultimately makes the interdisciplinary field of computational conformal geometry accessible and useful to the society. With respect to its direct impacts, the investigator applies conformal geometry to geometric modeling and shape analysis. Specifically, 1.Construct shape spaces based on Teichmuller space theory, where the space of all surfaces is modeled as a finite dimensional manifold, each point represents a conformal equivalence class of surfaces (a Riemann surface), and the metric of the shape space measures the deviation of conformal structures of the two shapes. An additional related goal is to build a geometric search engine. 2. Find a systematic way to generalize geometric constructions defined on planar domains to manifolds, such as manifold triangular BSplines and manifold Powell- Sabin surfaces. Design new surface subdivision schemes by inserting knots into manifold BSpline surfaces. With aspect to education, the investigator finds an effective way to teach conformal geometry (Riemann surface theory) to non-math majors by implementing practical algorithms to visualize the abstract concepts and establish the understanding of the profound theories.Conformal geometry theory is fully developed but very abstract. The investigator builds and disseminates a concrete set of software tools to compute and visualize the conformal structure of arbitrary real surfaces, which makes the theory accessible to students and its practical applications useful to the broader community. Manifold BSpline tools based on conformal geometry bridge the gap between traditional polygonal meshes in graphics and spline surfaces in CAGD. The generic geometric search engine is applied to a geometric database and an Internet search engine. The investigator complements the software development with a systematic development of the classic material in a context that permits integration into the curriculum of non-math majors. The fields of computer graphics, vision, scientific computing, medical imaging, mathematics and physics all benefit from the research and education of computational conformal geometry directly. Computational conformal geometry has already made impacts on the graphics industry and will be more broadly applied in the future.