RUI: Development of Frame Extensions and their Applications, II

提出了一个关于子空间伪帧（PFFS）应用的后续项目。伪框架是用于子空间x的线性分析和合成的类框架系统。它们与框架的不同之处在于，分析和合成序列不一定包含在x中。该项目的主题1研究双正交小波的构造（通过PFFS），该构造将正则性要求和完美的重构原则与实际的优化设计准则结合起来。主题2研究了PFFS几何特性在同步去噪和信号重建中的应用。PFFS沿着X的任何互补子空间生成X上的非正交投影，这使得通过使投影的零空间包含噪声子空间来实现噪声去除技术。主题3将研究PFFS在噪声环境下采样和线性逆问题中的应用。给定卷积或（非理想）采样操作，可以应用PFFS来指定最佳（重构）结果的可用重构子空间X。我们将研究如何选择可访问的重建子空间X以及如何通过PFFS进行重建或反卷积的示例。这些技术的性能也将被研究。所有的信号分析系统都将信号分解成多个分量。我们引入信号分析的主要新颖之处是PFFS系统的思想。PFFS是一种站在X之外分析子空间X的工具。这样做，一个人被赋予了自由和额外的“空间”，通过使用X中不可用的先进手段来执行X中的信号分解和分析。因此，PFFS系统比大多数其他系统更灵活，因为它们可以被设计成产生最有用的组件，用于去噪和信号重建目的。本研究将开发多个PFFS系统，旨在解决宽带通信、目标识别、信号/图像恢复/去模糊等领域的许多不同信号分析问题。特别是，我们希望推进图像压缩技术，以实现高压缩率和最少的人为失真，如检查板效应（在解压后的图像上显示棋盘）和环形效应（在边缘上显示模糊）。同样，我们期望通过减少从一个频带到另一个频带的泄漏来开发改进语音信号压缩的工具。更一般地说，我们将开发的方法可以在许多不同的信号处理环境中针对噪声去除和数字信号的精确重建进行定制。

A continuing project relating to applications of pseudoframes for subspaces (PFFS) is proposed. Pseudoframes are frame-like systems for the linear analysis and synthesis of a subspace X. They differ from frames in that both analysis and synthesis sequences are not necessarily contained in X. Theme 1 of the project studies constructions of biorthogonal wavelets (via PFFS) that incorporate the regularity requirement and the perfect reconstruction principle with practical optimal design criteria. Theme 2 studies the application of the geometrical properties of PFFS in simultaneous noise removal and signal reconstruction. PFFS generate non-orthogonal projections onto X along any complementary subspace of X. This enables a technique of noise removal by having the null space of the projection to contain the noise subspace. Studies in theme 3 will be on the applications of PFFS to sampling and linear inverse problems in noisy environments. Given a convolution or (non-ideal) sampling operation, one can apply PFFS to specify an available reconstruction subspace X for the best (reconstruction) result. We will study examples of how accessible reconstruction subspaces X should be selected and how reconstruction or deconvolution is carried out via PFFS. Performance of these techniques will also be studied.All signal analysis systems break a signal into components. The principal novelty that we introduce to signal analysis is the idea of PFFS systems. PFFS is a tool to analyze a subspace X while standing outside of X. In doing so, one is given the freedom and additional "room" to perform signal decomposition and analysis in X by using advanced means not available in X. Consequently, PFFS systems are more flexible than most others in that they can be designed to produce the most useful components for denoising and signal reconstruction purposes. This study will develop multiple PFFS systems designed to solve a number of different signal analysis problems in broadband communications, target recognition, and signal/image restoration/de-blurring, etc. In particular, we hope to advance the art of image compression to achieve high compression rate with the least artificial distortions such as check-boarding effect (showing checkerboards on the decompressed image) and ringing effect (showing blurring on the edges). Similarly, we expect to develop tools for improving speech signal compression by reducing leakage from one frequency band to another. More generally, the methods we will develop can be tailored for noise removal and accurate reconstruction of digital signals in a number of different signal processing environments.