CAREER: Signal Processing for Surfaces

复杂几何在不同领域的应用中变得越来越普遍。整形外科医生使用计算机模型来计划和执行手术。机械工程师几乎组装和测试汽车和飞机。动画师创造出与现实难以区分的复杂角色和效果。几何建模已经找到了艺术保护的方法：现在可以创建捕捉最细微细节的雕塑的计算机模型。几何数据的日益复杂要求新的高效表示和算法。我们长期研究计划的目标是研究复杂几何数据的表示，特别是曲面。我们计划推出一个广泛而基本的程序来探索三种类型的表示：分段光滑曲面、多尺度曲面和组合体-曲面表示。我们设计复杂曲面演示的方法是基于细分的概念，这是一种建模光滑曲面的有效技术。细分算法，虽然更灵活和通用，但保留了许多有用的性质的样条线。最重要的是，这些算法生成的曲面具有自然的层次结构，并使用高效的局部规则进行计算；细分算法可以很容易地扩展以表示多尺度数据。我们研究的总体目标是为曲面数据的构建、绘制、操作、处理、传输和存储设计支持高效算法的表示法。我们将密切合作地探索这些表示的理论性质(正则性、收敛性、公平性、稳定性)及其实际应用。拟议项目的具体目标是将设计并广泛用于图像的信号处理工具推广到表面。研究计划与教育计划相结合，教育计划有三个主要目标：(1)为计算机图形学及相关领域的研究培养研究生；(2)将研究的最新进展纳入各级课程，包括本科生，并鼓励本科生从事研究；(3)将最新研究进展的知识传授给该行业的专业人员。

Complex geometry becomes increasingly common in applicationsarising in diverse areas. Plastic surgeons use computermodels to plan and perform operations. Mechanical engineersvirtually assemble and test cars and planes. Animatorscreate complex characters and effects indistinguishable fromreality. Geometric modeling has found its way into artconservation: it is now feasible to create computer modelsof sculptures capturing the most subtle of details.Increasing complexity of geometric data calls for newefficient representations and algorithms. The goal of ourlong-term research plan is to study representations ofcomplex geometric data, surfaces in particular. We plan tolaunch a broad and fundamental program to explore threetypes of representations: piecewise-smooth surfaces,multiscale surfaces and combined volume-surfacerepresentations. Our approach to designing complex surfacerepresentations is based on the concept of subdivision, apowerful technique for modeling smooth surfaces. Subdivisionalgorithms, while being more flexible and general, retain anumber of useful properties of splines. Most importantly,the surfaces generated by these algorithms have a naturalhierarchical structure and are computed using efficientlocal rules; subdivision algorithms can be easily extendedto represent multiscale data. The general goal of ourresearch is to design representations supporting efficientalgorithms for construction, rendering, manipulation,processing, transmission and storage of surface data. Wewill explore in close coordination both theoreticalproperties of these representations (regularity, convergencerates, fairness, stability) and their practicalapplications. The specific goal of the proposed project isto generalize to surfaces the signal processing tools thatwere designed and are extensively used for images. Theresearch program is integrated with the education plan,which has three main goals: (1) prepare graduate studentsfor research in computer graphics and related areas, (2)incorporate the latest advances in research into curriculumat all levels, including undergraduate, and engage motivatedundergraduate students in research (3) transfer knowledgeabout latest research advances to professionals in theindustry.