CAREER: Subdivision Schemes and Wavelets: New Tools, New Settings

多分辨率方法的研究主要集中在三个部分相关的方向：(I)几何环境下的细分模式和小波，(Ii)冗余系统的优势和(Iii)在高维环境下的理想数据表示。几何环境中的细分和多分辨率方法是最近开始的研究领域。在地形建模和三维扫描技术等应用领域的极高兴趣的推动下，这一研究方向引起了计算机科学家和应用数学家的注意。现有的曲面生成细分方法并不能保证得到的曲面上到处都有定义良好的曲率。在各种应用中，需要处理曲率光滑的曲面.在正则背景下，作者最近构造了一族具有小插补模板的FC^2细分格式.在这种构造下，应用了最优化、小波理论和多值插值理论的各种工具.本项目将进一步改进这些工具，以解决最初的公开问题，使这些工具也可以应用到其他环境中。对于图像重建/处理中的各种应用，研究发现，高度冗余的类小波图像表示法的性能显著优于图像编码界普遍使用的标准非冗余正交小波变换。一方面，目前还没有一个完整的理论来解释冗余的基本优势。另一方面，研究人员刚刚开始认识到小波变换在高维上的基本极限，并提出了各种新的数据表示方案来解决这些基本极限。在这个项目下的研究活动寻求在表示和分析图像和其他多维数据的自适应方法的一般领域中发展数学理论和计算工具。该教育计划提出了一种教授计算科学的新方法。该项目将为我们的数值计算课程开发更适合计算机科学专业学生的需求、兴趣和数学能力的教材。这个项目的目标是为我们未来的高中数值计算课程开发一套电子讲稿。这些讲稿将为任何现有教科书增加两个新的维度：(I)笔记中将集成一个交互式计算环境(基于QPE，如MatLab或Octave)，以便学生可以很容易地再现所给出的任何计算结果，并推导出任何新的计算实验。(Ii)计算机图形学、计算机视觉、数据挖掘和计算流体力学等应用将在本说明的不同地方使用，以说明数值方法在工业中的作用。

The proposed research will focus on three partly relateddirections in multiresolution methods: (I) subdivision schemesand wavelets in the geometric setting, (II) the advantage of redundantsystems and (III) ideal data representation in higher dimensions.Subdivision and multiresolution methods in the geometric settingis a recently launched research area.Prompted by the extremely high interests in applications such as terrainmodeling and 3-D scanning technology, this research direction has attractedthe attention of both computer scientists and applied mathematicians.Existing subdivision schemes for surface generation do not promisewell-defined curvature everywhere in a resulted surface. In variousapplications it is desirable to work with curvature smooth surfaces.In the regular setting the proposer recently constructed a family ofC^2 subdivision schemes with a small interpolating stencil.Underlying this construction is a variety of tools from optimization,wavelet theory and multivairate interpolation theory. This project willfurther refine these tools in order to resolve the original open problemand such that these tools can be applied to other settings as well.For various applications in image reconstruction/processing, itwas found that highly redundant wavelet-like image representationssignificantly outperform the standard non-redundant orthogonalwavelet transforms used ubiquitously in the image codingcommunity. On the one hand, there is currently no complete theoryfor explaining the fundamental advantages of redundancy. On theother hand, researchers have just started to realize thefundamental limits of wavelet transforms in higher dimensions andhave proposed a wide variety of novel data representation schemesthat address these fundamental limits. Research activityunder this project seeks to develop mathematical theory andcomputational tools in the general area of adaptive methods ofrepresenting and analyzing images and other multidimensional data.The educational plan proposes a new way of teachingcomputational science. The projectwill develop teaching materials for our numerical computing coursesthat better suit the needs, interests and mathematical abilityof computer science students. The goal ofthis project is to develop a set ofelectronic lecture notes for our future senior numerical computingcourse. These lecture notes will add two new dimensions to any ofthe existing textbooks: (i) An interactive computationalenvironment (based on QPE's such as Matlab or Octave)will be integrated into the notes so that a student can easily reproduceany computational result presented and derive any newcomputational experiments. (ii) Applicationsareas such as computer graphics, computer vision, data-mining andcomputational fluid dynamics will beused in various places ofthis notes to illustrate the role of numerical methods in industry.