RUI: Advanced Theories of Frames, Pseudoframes, and Optimality Issues with Applications

RUI：框架、伪框架和应用程序的最优性问题的高级理论

• 批准号: 9803679
• 负责人: Shidong Li
• 金额: $ 7.5万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-06-15 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803679&HistoricalAwards=false
• 关键词: RUI; Advanced; Theories; Frames; Pseudoframes

This is a project to study several problems related to the theory and applications of frames and pseudoframes arising in the signal processing area. Frames and pseudoframes are basis-like systems. The use of frames and pseudoframes in signal processing comes naturally in problems such as irregular sampling, signal restoration/recovery, Gabor time-frequency representations, etc. The topics in this research program are largely linked to their practical values in applications. The first topic relates to the construction of affine frames with a generalized multiresolution structure (GMS) and applications, in which the investigator seeks to construct a GMS built upon affine pseudoframes for subspaces. The results of this study are expected to have practical values in terms of numerical efficiency and smooth filter design, and will be used in the study of applications in industrial problems such as VLSI design; The second topic is on pseudoframes for subspaces (PFFS) and their applications to irregular sampling and subspace signal processing problems. The key feature of PFFS is that (pseudoframe) elements are allowed to be outside of a concerned subspace, a freedom that permits the construction of simpler and implementable solutions such as sampling functions, which is otherwise difficult or impossible when restricted to the concerned subspace only; The third topic is on the role of various dual frames and their applications. The investigator will study optimality issues of frame applications using the dual frame formula derived by the PI. The results of the study serve as a foundation of applied optimal frame representations associated with (infinitely many) dual frames; The fourth topic focuses on refined time-frequency analysis via multi-Gabor expansions (MGE), which generalize the metaplectic representations that are useful in time-frequency (TF). The investigation will focus on theoretical and computational studies of refined and efficient TF representations of a signal using MGEs.

本课题主要研究信号处理领域中出现的框架和伪框架的理论与应用问题。 框架和伪框架是类基系统。 在信号处理中使用帧和伪帧来自然的问题，如不规则采样，信号恢复/恢复，Gabor时频表示等，在这个研究计划的主题在很大程度上是链接到他们的实际应用价值。 第一个主题涉及到建设的仿射框架与广义多分辨率结构（GMS）和应用程序，其中调查人员试图构建一个GMS建立在仿射伪框架的子空间。 本研究的结果有望在数值效率和平滑滤波器设计方面具有实用价值，并将用于超大规模集成电路设计等工业问题中的应用研究;第二个主题是子空间伪框架（PFFS）及其在不规则采样和子空间信号处理问题中的应用。PFFS的主要特点是允许（伪框架）元素在相关子空间之外，这种自由允许构造更简单和可实现的解决方案，例如采样函数，这在仅限于相关子空间时是困难或不可能的;第三个主题是各种对偶框架及其应用的作用。 研究者将使用PI推导的双框架公式研究框架应用的最优性问题。该研究的结果作为基础的应用最佳的框架表示与（无限多）双框架;第四个主题侧重于精细的时频分析，通过多伽柏展开（MGE），推广的metaplectic表示，是有用的时频（TF）。 调查将集中在理论和计算研究的精细和有效的TF表示的信号使用MGE。