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

本课题主要研究信号处理领域中出现的帧和伪帧的理论和应用方面的几个问题。帧和伪帧是类基系统。在信号处理中使用帧和伪帧是很自然的，比如不规则采样、信号恢复/恢复、Gabor时频表示等问题。本研究计划的课题在很大程度上与其在应用中的实际价值有关。第一个主题涉及具有广义多分辨率结构（GMS）的仿射框架的构建及其应用，其中研究者试图构建基于子空间仿射伪帧的GMS。本研究结果预计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的应用研究；第二个主题是子空间伪帧及其在不规则采样和子空间信号处理问题中的应用。PFFS的关键特征是允许（伪框架）元素在相关子空间之外，这种自由允许构建更简单和可实现的解决方案，如采样函数，这在仅限于相关子空间时是困难或不可能的；第三个主题是关于各种双框架的作用及其应用。研究者将使用PI导出的双框架公式研究框架应用的最优性问题。研究结果为应用与（无限多）对偶帧相关的最优帧表示奠定了基础；第四个主题侧重于通过多gabor展开式（MGE）进行精细时频分析，该展开式推广了在时频（TF）中有用的元表示。该研究将侧重于使用MGEs对信号进行精炼和有效的TF表示的理论和计算研究。

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.