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

职业：细分方案和小波：新工具，新设置

• 批准号: 9984501
• 负责人: Thomas Yu
• 金额: $ 24.5万
• 依托单位: Rensselaer Polytechnic Institute
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-04-15 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9984501&HistoricalAwards=false
• 关键词: CAREER; Subdivision; Schemes; Wavelets; New

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.

提出的研究将集中在多分辨率方法的三个部分相关方向：(I)几何设置中的细分方案和小波，（II）冗余系统的优势和（III）高维理想数据表示。几何背景下的细分和多分辨率方法是一个新兴的研究领域。由于对地形建模和三维扫描技术等应用的极大兴趣，这一研究方向引起了计算机科学家和应用数学家的关注。现有的曲面生成细分方案并不能保证在生成的曲面上处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表面。在常规情况下，利用一个小插值模板构造了一组c ^2细分方案。这种结构的基础是各种工具从优化，小波理论和多变量插值理论。本项目将进一步完善这些工具，以解决最初的开放问题，使这些工具也可以应用于其他设置。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明显优于图像编码社区中普遍使用的标准非冗余正交小波变换。一方面，目前还没有完整的理论来解释冗余的基本优势。另一方面，研究人员刚刚开始意识到高维小波变换的基本限制，并提出了各种新颖的数据表示方案来解决这些基本限制。本项目的研究活动旨在发展数学理论和计算工具，用于表示和分析图像和其他多维数据的自适应方法。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学生需求、兴趣和数学能力的教材。本计画的目标是为我们未来的高级数值计算课程开发一套电子课堂讲稿。这些课堂笔记将为现有的教科书增加两个新的维度：(i)交互式计算环境（基于QPE，如Matlab或Octave）将集成到笔记中，以便学生可以轻松地再现任何计算结果并推导任何新的计算实验。（ii）应用领域，如计算机图形学、计算机视觉、数据挖掘和计算流体动力学将在本说明的各个地方被用来说明数值方法在工业中的作用。