基于LDPC码的压缩感知测量矩阵构造及性能分析

压缩感知作为一种新颖的信号采样理论，已成为近年来信息论和信号处理领域研究的一大热点。测量矩阵的构造是压缩感知基本理论的主要问题之一，目前常用的随机测量矩阵存在存储开销大、不易于进行硬件实现等弱点，同时其性能也不能得到确定性的保证，故有必要构造确定性测量矩阵。在测量矩阵的确定性构造中，基于纠错码、特别是LDPC码构造的二元稀疏矩阵引起了人们的极大兴趣，好的LDPC码校验矩阵常常是好的压缩感知测量矩阵。本课题致力于将LDPC码与压缩感知理论有机结合，着重研究二元测量矩阵的确定性构造并分析其重建性能，主要包括：揭示LDPC码与压缩感知之间的联系；构造性能优良、结构性强、参数灵活的二元稀疏准循环测量矩阵；分析二元测量矩阵在基追踪、OMP等重建算法下的性能；结合二元测量矩阵的结构化特性，降低现有测量或重建算法的复杂度，等等。这些关于测量矩阵的工作对于拓展压缩感知基本理论并推进其实用化具有重要意义。

Compressed sensing (CS) is a novel signal sampling theory and becomes a research focus in the areas of information theory and signal processing recently. The construction of measurement matrices is one of the main concerns in CS. At present, the frequently-used random measurement matrix has many drawbacks, e.g., large storage costs, not easy to be realized by hardware, and there is no guarantee that a specific realization of random matrices works. As a result, the deterministic construction of measurement matrices is necessary. Among them, binary measurement matrices from error-correcting codes, such as LDPC codes, attract lots of interests. Good parity-check matrices of LDPC codes are usually good measurement matrices in CS. In this project, we study the CS theory together with LDPC codes, focus on the the deterministic construction and reconstruction performance analysis of binary measurement matrices. The main contents of the project include revealing the relations between LDPC codes and CS; constructing binary sparse quasi-cyclic measurement matrices with good performance, strong structures and flexible parameters; evaluating the performance of binary measurement matrices under the basis pursuit and OMP algorithms; by employing the strong structures of binary measurement matrices, lowering the complexity of measuring and reconstructing algorithms, etc. These works on measurement matrices are very important for the CS basic theory and its practical applications.