基于子空间分析的低复杂度联合稀疏恢复方法研究

Joint sparse recovery is a newly emerged problem in the signal processing area with lots of applications including communication, radar, and image processing. Compared to conventional methods (e.g., Capon and MUSIC), joint sparse recovery can offer better resolution and deal with the scenarios of small number of samples and coherence sources. However, the computational complexity of existing algorithms of joint sparse recovery is very high, since they require high-dimensional optimization or large matrix inversion. This restricts the applications of joint sparse recovery in practical systems. By using the sparse structure information to reduce the dimension of the unknown variable or the constraints and approximate the ideal zero norm minimization, this project aims to give consideration to both the performance of the algorithm and the computational complexity. Specifically, this project will study the credibility of the sparse structure information obtained from the subspace analysis and its influence on joint sparse recovery. And then, this project will analyze the performance of the proposed methods. Consequently, the project will accomplish a unity of the method for joint sparse recovery based on the subspace analysis, which contains advanced processing methods, fast implementations of algorithms and theoretical analyses. The research results will be useful for designing fast algorithms of joint sparse recovery.

由于在通信、雷达、图像处理等诸多方面有着广阔的应用前景，联合稀疏恢复正受到信号处理界的极大关注。和传统的信号处理方法如Capon、MUSIC等相比，联合稀疏恢复在分辨性能、处理小样本或相干源的能力等方面有明显的优势。然而，当前主流的联合稀疏恢复算法往往涉及高维度的优化或大矩阵求逆问题，运算开销极其庞大，这已成为制约其工程应用的主要瓶颈。本项目旨在通过稀疏结构信息的有效运用，在压缩未知数或优化约束的维数的同时逼近理想的零范数优化问题，实现低运算复杂度和高性能的兼顾。具体而言，本项目拟通过子空间分析获取信号的稀疏结构信息，分析稀疏结构信息的可信度及其对稀疏恢复的影响，探索运用稀疏结构信息设计快速联合稀疏恢复算法的途径与方式，研究基于子空间分析的联合稀疏恢复算法的性能，从而建立起包括先进处理方法、快速实现算法和性能分析在内的完备的算法体系，为快速联合稀疏恢复算法的构建奠定理论与技术基础。

和传统的超分辨信号处理方法相比，稀疏恢复在分辨性能、处理小样本或相干源的能力等方面有明显的优势，因而在通信、雷达、声呐、图像处理等诸多方面有着广阔的应用前景。然而，基于凸优化的稀疏恢复算法往往为大规模的优化问题，运算开销极其庞大，这制约着其的实际应用。本项目通过稀疏结构信息的有效运用，在逼近理想的零范数优化问题的同时，或压缩了未知数的维数，或降低了约束方程的维数，或采用新的求解框架，实现低运算复杂度和高性能的兼顾。具体而言，对于常用的二阶锥规划求解框架，项目通过线性变换和Khatri-Rao积，实现了未知数维数和约束方程维数的压缩，有效降低了算法的求解复杂度；考虑到空间谱的非负性，通过目标函数的改造，将子空间加权稀疏恢复问题转化为非负最小二乘的求解问题，提出了快速模基础上的求解算法，使得单次迭代的运算复杂度仅为O(N)，这有力地促进了算法的工程实现。总而言之，本项目分析了子空间加权稀疏恢复算法的性能，设计了快速实用的子空间加权稀疏恢复算法，并将其应用到雷达、声呐等实际数据之中，验证了算法的优越性能。项目建立起了包括快速实现算法设计、性能分析和雷达、声呐实际应用在内的完备的算法体系，为快速子空间加权稀疏恢复算法的应用奠定了理论与技术基础。