RUI: Development of Sparsity-inducing Dual Frames and Algorithms with Applications, II

RUI：稀疏性双框架和算法的开发及其应用，II

• 批准号: 1615288
• 负责人: Shidong Li
• 金额: $ 17.49万
• 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: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1615288&HistoricalAwards=false
• 关键词: RUI; Development; Sparsity; inducing; Dual

This award supports the ongoing research program of the Principal Investigator in the general area of signal processing. Signal processing is concerned with making signals easier to handle or better in quality. It is widely used in modern society, such as in cell-phone communications. In this project, the principal novelties introduced to signal processing are developments of the notion of "sparsity-inducing dual frames" ("sparse duals") and advanced ideas of recovering underlying signals from much smaller "samples" of the signal. Sparsity-inducing dual frames are a set of signal components used in obtaining the sparsest (the most concise) decomposition of a signal. The substance of sparse duals lies in the fact that they have the capacity to greatly advance the effectiveness of signal recovery through a small number of samples of a signal. The objective is to advance the art of new signal-detection techniques. This study has the potential to impact information technology, ranging from simplified sampling devices to radar systems, commercial imaging techniques, geographic survey, mapping, wireless communication, surveillance systems, medical image requisitions and combinations, and a number of signal requisition applications.Topics proposed for study include the development of tail-minimization techniques in the sparse-dual-based l-one analysis approach and applications, together with the development of a soft-thresholding-infused null-space tuning algorithm with feedbacks for more effective denoising and signal smoothness. Several themes will be developed. One theme aims at directly reducing the tail coefficients of sparse frame expansions. This is motivated by the observation that the signal recovery error bound is directly proportional to the "tails" of the signal coefficients. Another theme is to construct sparsity-inducing dual frames that minimize the tail coefficients for a class of s-sparse signals and nearly s-sparse signals. Such sparse duals are clearly ultimate analysis operators in the analysis approach. A third theme is designed to understand an equivalent l-one synthesis problem derived from a sparse-dual-based l-one analysis formulation. The new synthesis problem decouples the product of the sensing matrix and the frame matrix, and the kernel of the new sensing matrix is smaller. A final theme is aimed at infusing soft thresholding into the iterative "Null Space Tuning Algorithm with Feedbacks" (NST+FB). The NST+FB algorithm is known to converge in finitely many steps. The Principal Investigator will enrich the NST+FB algorithm with a soft-thresholding mechanism for enhanced denoising and smoothness, for applications such as image processing and radar imaging.

该奖项支持首席调查员在信号处理一般领域的持续研究计划。信号处理涉及使信号更易于处理或质量更好。它在现代社会中被广泛使用，例如在手机通信中。在这个项目中，信号处理的主要创新是“稀疏诱导双帧”(“稀疏对帧”)概念的发展，以及从小得多的信号“样本”中恢复潜在信号的先进思想。稀疏诱导双帧是用于获得信号的最稀疏(最简明)分解的一组信号分量。稀疏对偶的实质在于它们能够通过少量的信号样本来极大地提高信号恢复的有效性。其目的是促进新信号检测技术的发展。这项研究可能会影响信息技术，从简化的采样设备到雷达系统，商业成像技术，地理测量，测绘，无线通信，监控系统，医学图像请求和组合，以及一些信号请求应用。建议研究的主题包括稀疏-基于双重L-One分析方法和应用中的尾部最小化技术的发展，以及为更有效地去噪和信号平滑而开发的具有反馈的软阈值注入零空间调整算法。将制定几个主题。其中一个主题旨在直接降低稀疏帧扩展的尾部系数。这是因为观察到信号恢复误差界与信号系数的“尾部”成正比。另一个主题是构造稀疏诱导对偶框架，使一类S-稀疏信号和近S-稀疏信号的尾部系数最小化。这种稀疏对偶显然是分析方法中的终极分析运算符。第三个主题旨在理解一个等价的L一综合问题，该问题源于基于稀疏对偶的L一分析公式。新的综合问题将感知矩阵和框架矩阵的乘积解耦，新的感知矩阵的核更小。最后一个主题是将软阈值注入迭代的“带反馈的零空间调整算法”(NST+FB)。众所周知，NST+FB算法在有限多个步骤中收敛。首席调查员将使用软阈值机制来丰富NST+FB算法，以增强去噪和平滑，用于图像处理和雷达成像等应用。