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

该奖项支持首席研究员在信号处理一般领域正在进行的研究项目。信号处理涉及使信号更容易处理或质量更好。它在现代社会中被广泛使用，比如在手机通信中。在这个项目中，引入信号处理的主要新奇之处是“稀疏诱导双帧”（“稀疏对偶”）概念的发展，以及从信号的更小“样本”中恢复潜在信号的先进思想。稀疏性诱导双帧是一组用于获得信号最稀疏（最简洁）分解的信号组件。稀疏对偶的实质在于，它们能够通过少量的信号样本，极大地提高信号恢复的有效性。目标是推进新的信号检测技术的艺术。这项研究有可能影响信息技术，从简化采样设备到雷达系统、商业成像技术、地理调查、制图、无线通信、监控系统、医学图像请求和组合，以及许多信号请求应用。提出的研究主题包括基于稀疏双元的l- 1分析方法和应用中的尾部最小化技术的发展，以及用于更有效去噪和信号平滑的带有反馈的注入软阈值的零空间调谐算法的发展。将拟定若干主题。其中一个主题旨在直接降低稀疏框架展开的尾部系数。这是由于观察到信号恢复误差界限与信号系数的“尾部”成正比。另一个主题是构造稀疏诱导对偶帧，使一类s-稀疏信号和近s-稀疏信号的尾系数最小化。这种稀疏对偶显然是分析方法中的终极分析算子。第三个主题旨在理解由稀疏双基l- 1分析公式导出的等效l- 1合成问题。新的综合问题解耦了感知矩阵和帧矩阵的乘积，并且新感知矩阵的核更小。最后一个主题是将软阈值注入到迭代的“带反馈的零空间调谐算法”（NST+FB）中。已知NST+FB算法收敛于有限多步。首席研究员将用软阈值机制丰富NST+FB算法，以增强去噪和平滑性，用于图像处理和雷达成像等应用。

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.