基于Berlekamp-Justesen码的压缩感知确定性测量矩阵的构造

Deterministic Constructions of Compressive Sensing Matrices Based on Berlekamp-Justesen Codes

确定性测量矩阵构造是近期压缩感知领域的一个重要研究问题。该文基于Berlekamp-Justesen(B-J)码,构造了两类确定性测量矩阵。首先,给出一类相关性渐近最优的稀疏测量矩阵,从而保证其具有较好的限定等距性(RIP)。接着,构造一类确定性复测量矩阵,这类矩阵可以通过删除部分行列使其大小灵活变化。第1类矩阵具有很高的稀疏性,第2类则是基于循环矩阵,因此它们的存储开销较小,编码和重构复杂度也相对较低。仿真结果表明,这两类矩阵常常有优于或相当于现有的随机和确定性测量矩阵的重建性能。

Nowadays the deterministic construction of sensing matrices is a hot topic in compressed sensing. Two classes of deterministic sensing matrices based on the Berlekamp-Justesen（B-J） codes are proposed. Firstly, a class of sparse sensing matrices with near-optimal coherence is constructed. It satisfies the Restricted Isometry Property（RIP） well. Afterwards, a class of deterministic complex-valued matrices is proposed. The row and column numbers of these matrices are tunable through the row and column puncturing. Moreover, the first proposed matrices are high sparsity and the second matrices are able to obtain from the cyclic matrices, thus the storage costs of them are relatively low and both the sampling and recovery processes can be simpler. The simulation results demonstrate that the proposed matrices often perform comparably to, or even better than some random matrices and deterministic measurement matrices.