平移不变空间中压缩采样理论及算法研究

项目名称： 平移不变空间中压缩采样理论及算法研究

项目编号： No.11301052

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 徐敏

作者单位： 大连理工大学

项目金额： 22万元

中文摘要： 信号采样是模拟的物理世界通向数字的信息世界的桥梁。多年来，指导信号采样的理论基础一直是著名的Shannon采样定理，但其产生的大量数据造成了存储空间的浪费。近年来出现了一种新颖的采样理论-压缩采样，它能够以远低于Nyquist速率采样信号。然而，标准的压缩采样是一种有限维理论，其仅适用于在某一个基下为稀疏的或可压缩的有限长离散信号。压缩采样理论框架与连续时间信号的采样之间有一个明显的空白。本项目试图针对有限时间范围上平移不变空间中的信号，结合平移不变空间生成函数的性质，在标准压缩采样理论框架下讨论由不完全采样数据重建连续时间信号的理论及算法。平移不变空间的优点是在保留带限信号类的结构与简单性的基础上，它还可以建模更易于数值实现的非带限信号类，对真实数据的逼近具有很强的灵活性。该项目的成功实施，将为进一步从理论和实际的结合上研究连续时间信号采样问题开拓新的思路。

中文关键词： 压缩采样；平移不变核；L1 范数正则化；稀疏性；

英文摘要： Signal sampling is a bridge from the analog world to digital world. It is well known that the Shannon sampling theorem has been the dominance of signal sampling over the years, with the shortcoming of large volume data transmission and storage. Recently, compressive sampling is emerging as a novel sampling theory based on which one can realize the signal sampling at a speed large below the Nyquist. However, the standard compressive sampling is currently only a finite-dimensional theory devoting to the recovery of finite-length signals that are sparse or compressible in a particular basis. There remains a prominent gap between the standard compressive sampling framework and the problem of sampling a continuous-time signal. By dealing with the signals in shift-invariant spaces over a finite interval and combining with the properties of generating functions, we try to develop the theories and algorithms of acquiring continuous-time signals from incomplete measurements within the standard compressive sampling framework. In addtion to retaining the structure and simplicity of bandlimited signals, shift-invariant spaces has the advantage of modeling non-bandlimited signals that are more amenable to numerical implementation, and are more flexible for approximating real data. The successful implement of the project woul

英文关键词： Compressive sampling；Shift-invariant kernel；L1 norm regularization；Sparsity；