RUI: Development of Frame Extensions and their Application

RUI：框架扩展的开发及其应用

• 批准号: 0103762
• 负责人: Shidong Li
• 金额: $ 7.86万
• 依托单位: San Francisco State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103762&HistoricalAwards=false
• 关键词: RUI; Development; Frame; Extensions; their

0103762LiA program of research related to the application of pseudoframes for subspaces (PFFS) and relevant interpolation techniques is proposed. Theme 1 of the program studies the contribution of PFFSs to artifact reduction in data/image compression. Key observations lie in the fact that there are symmetric and compactly supported biorthogonal wavelets, but not within an MRA (of compactly supported wavelets). It shows the "lack of room" within the MRA subspace structure. PFFS is a tool to explore the insight story of such a phenomenon. Theme 2 focuses on a new construction approach of interpolatory multiresolution analysis (I-MRA). The key observation is that an interpolatory sequence can be defined by sampled reproducing kernel function of a subspace V of an MRA. It is expected that such an interpolatory sequence can easily form a basis or frame of V, giving rise to an interpolatory "interface" to the MRA, and a construction methodology of I-MRA. Theme 3 studies applications of PFFSs in the Sinc-type methods for interpolations and quadratures. The motivation is to explore the advantage of fast decaying properties of PFFSs in the Sinc-type applications. In theme 4, the P.I. expects to apply similar interpolation techniques as described in theme 2 to study multi-band samplings using frames and/or PFFSs as fundamental tools. The technique of PFFSs with their flexibility is expected to play a role.Each research topic is highly motivated by its applications in data/image compression, sampling and interpolation for efficient numerical implementations. PFFS is a tool to study a subspace V while "standing" outside of it. In doing so, one is given the freedom and additional "room" to perform tasks in V by using advanced tools not available in V. In signal/image processing, such advanced properties of PFFS are extremely useful because it is known that biorthogonal wavelets of finite length with symmetry are desirable for reducing ``blocking'' and ``ringing'' effects, and yet such biorthogonal wavelets do not reside in its dual wavelet subspace. We believe that PFFS shall not only provide a class of biorthogonal wavelet examples, the additional information (known from the construction) contributed by PFFS will provide practical insight about "blocking" and "ringing" artifacts, and thereby delivers new solutions and enhanced signal/image compression rate and communication speed. Other applications of PFFSs in irregular sampling (for multi-band functions) as well as in the Sinc-type methods (for interpolation and quadratures) are all based on applying flexible properties of PFFSs. The expectations include enhanced practical efficiency and improved rate of convergence. The project on interpolatory multiresolution analysis has practical applications in data acquisition/sampling while simultaneously performing the wavelet transform. One of the key ingredients of the study is to build an interpolatory "interface" to a given MRA and still keep the underlying desirable wavelet properties.

0103762Li 提出了与子空间伪帧（PFFS）应用和相关插值技术相关的研究计划。该计划的主题 1 研究 PFFS 对减少数据/图像压缩中的伪影的贡献。 关键的观察结果在于存在对称且紧支持的双正交小波，但不在 MRA（紧支持的小波）内。它显示了 MRA 子空间结构内“缺乏空间”。 PFFS 是一个探索这种现象的洞察故事的工具。 主题 2 重点关注插值多分辨率分析 (I-MRA) 的新构建方法。 关键的观察是插值序列可以通过 MRA 子空间 V 的采样再现核函数来定义。 预计这样的插值序列可以很容易地形成V的基础或框架，从而产生MRA的插值“接口”以及I-MRA的构建方法。 主题 3 研究 PFFS 在 Sinc 型插值和求积方法中的应用。 其动机是探索 PFFS 在 Sinc 型应用中快速衰减特性的优势。 在主题 4 中，P.I.期望应用主题 2 中描述的类似插值技术来研究使用帧和/或 PFFS 作为基本工具的多频带采样。 PFFS 技术以其灵活性有望发挥作用。每个研究主题都因其在数据/图像压缩、采样和插值中的应用而受到高度的推动，以实现高效的数值实现。 PFFS 是一种在“站在”子空间 V 之外时研究子空间 V 的工具。 这样做时，人们通过使用 V 中不可用的高级工具获得了自由和额外的“空间”来执行 V 中的任务。在信号/图像处理中，PFFS 的这种高级属性非常有用，因为众所周知，具有对称性的有限长度的双正交小波对于减少“阻塞”和“振铃”效应是理想的，但这种双正交小波并不驻留在其对偶中 小波子空间。我们相信，PFFS 不仅会提供一类双正交小波示例，PFFS 贡献的附加信息（从构造中已知）还将提供有关“阻塞”和“振铃”伪像的实用见解，从而提供新的解决方案和增强的信号/图像压缩率和通信速度。 PFFS 在不规则采样（对于多频带函数）以及 Sinc 型方法（对于插值和求积）中的其他应用都是基于应用 PFFS 的灵活特性。 期望包括提高实际效率和提高收敛速度。 插值多分辨率分析项目在数据采集/采样方面具有实际应用，同时执行小波变换。 这项研究的关键要素之一是为给定的 MRA 构建一个插值“接口”，同时仍然保持潜在的理想小波特性。