RUI: Development of Frame Extensions and their Applications, II

提出了对子空间伪框(PFF)应用的一个继续项目。伪帧是用于子空间X的线性分析和综合的类框架系统。它们与框架的不同之处在于，分析和综合序列不一定包含在X中。本项目的主题1研究双正交小波的构造(通过PFF)，它结合了正则性要求和具有实用的优化设计准则的完美重构原理。主题2研究了滤波器的几何性质在同时去噪和信号重建中的应用。PFF沿着X的任何补子空间在X上生成非正交投影。这使得能够通过使投影的零空间包含噪声子空间来去除噪声的技术。主题3的研究将是在噪声环境中的抽样和线性逆问题中的应用。在给定卷积或(非理想)采样操作的情况下，可以应用PFF来指定可用的重建子空间X以获得最佳(重建)结果。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PFF执行重建或去卷积的例子。这些技术的性能也将被研究。所有的信号分析系统将信号分解成分量。我们引入信号分析的主要新奇之处在于PFFS系统的概念。PFFS是一种在X之外分析子空间X的工具。通过这样做，人们获得了自由和额外的“空间”，通过使用X中没有的先进手段在X中执行信号分解和分析。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为产生用于去噪和信号重建目的的最有用的组件。这项研究将开发多个PFFS系统，旨在解决宽带通信、目标识别和信号/图像恢复/去模糊等领域中的一些不同的信号分析问题。特别是，我们希望推动图像压缩技术的发展，以实现高压缩比和最小的人为失真，例如检查板效应(在解压缩的图像上显示棋盘)和振铃效应(在边缘显示模糊)。同样，我们希望开发工具，通过减少从一个频段到另一个频段的泄漏来改善语音信号压缩。更广泛地说，我们将开发的方法可以在许多不同的信号处理环境中为数字信号的噪声去除和准确重建量身定做。

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.