RUI: Development of Sparsity-Inducing Dual Frames and Applications

研究者研究了稀疏诱导双框架的新概念及其应用。主题1定义了非精确框架的稀疏诱导双框架。初步研究表明，如果信号具有稀疏帧表示，则存在一个稀疏诱导对偶帧（稀疏对偶），该对偶帧在应用于信号时产生精确的稀疏系数。研究了稀疏对偶和普通稀疏对偶框架对一类信号的最优性问题。主题2开发了稀疏信号恢复中基于稀疏双元分析方法的迭代算法。改进和收敛分析替代迭代算法是这里的目标之一。主题3旨在了解基于稀疏双元的分析方法大大提高了稀疏信号恢复能力。这似乎与稀疏超平面的一些几何性质有关——稀疏表示的仿射子空间融合了稀疏对偶信息。一个目标是推导出适应稀疏超平面几何特性的算法，以获得更高的效率。主题4为实际场景发展了稀疏对偶的概念，其中信号仅仅是近似稀疏的帧。主题5开发了“任意”滤波器组，用于多通道信号/图像请求和多通道通信应用。这些系统表现出任意的信道间关系，因为它们通常不满足基于小波理论的众所周知的多速率滤波器组约束。挑战在于为这些系统实际构建现实或FIR合成滤波器。信号处理一般是指使信号易于处理或具有更好的质量。在传输或存储信息的现代系统中，这是一项常见的任务。在这个项目中，引入信号处理的主要新奇之处是稀疏性诱导对偶帧（稀疏对偶）的新概念和从更小的信号样本中恢复潜在信号的想法。稀疏诱导对偶帧是一组最优和基本的信号分量，用于获得最稀疏（非零数量最少）的信号分解系数。稀疏对偶的一个优点是它们可以从少量的信号样本中提高信号的恢复。研究者开发了有效的稀疏信号恢复方法。该项目的成果可能导致更简单和更有效的采样设备、雷达系统、商业成像技术、地理调查和测绘系统、无线通信系统、多传感器/摄像头监控系统和多通道医学图像请求方法。该项目为本科生提供了培训机会。

Li1313490 The investigator studies the new notion of sparsity-inducing dual frames and their applications. Theme 1 defines sparsity-inducing dual frames of a non-exact frame. Preliminary studies show that if a signal has a sparse frame representation, then there is a sparsity-inducing dual frame (sparse dual) that produces the exact sparse coefficient when applied to the signal. Optimality issues of sparse duals and common sparse dual frames to classes of signals are examined. Theme 2 develops iterative algorithms for a sparse-dual-based analysis approach in sparse signal recovery. Improvement and convergence analysis of an alternative iterative algorithm are among the goals here. Theme 3 aims to understand the greatly improved sparse signal recovery capacity of the sparse-dual-based analysis method. This appears to be related to some geometric properties of the sparse hyperplane -- the affine subspace of the sparse representation fused with sparse dual information. One goal is to derive algorithms that adaptable to the geometric properties of the sparse hyperplane for greater effectiveness. Theme 4 develops the notion of sparse duals for practical scenarios where signals are merely approximately sparse with frames. Theme 5 develops "arbitrary" filter banks for multi-channel signal/image requisition and multi-channel communication applications. These systems exhibit arbitrary inter-channel relationships because they generally do not satisfy the commonly known multi-rate filter bank constraints rooted in wavelet theories. The challenge lies in actual constructions of realistic or FIR synthesis filters for these systems. Signal processing generally refers to making a signal easy to handle or of better quality. It is a common task in modern systems that transmit or store information. In this project, the principal novelties introduced to signal processing are the new notion of sparsity-inducing dual frames (sparse duals) and ideas of recovering underlying signals from much smaller samples of the signal. Sparsity-inducing dual frames are sets of optimal and basic signal components used in obtaining the sparsest (the smallest number of nonzero) signal decomposition coefficients. An advantage of sparse duals is that they can improve the recovery of a signal from a small number of samples of the signal. The investigator develops effective methods of sparse signal recovery. Results of the project could lead to simpler and more effective sampling devices, radar systems, commercial imaging techniques, geographic survey and mapping systems, wireless communication systems, multi-sensor/camera surveillance systems, and multi-channel medical image requisition methods. The project provides training opportunities for undergraduate students.