Deterministic Construction of Toeplitzed Structurally Chaotic Matrix for Compressed Sensing

The construction of sensing matrix is a fundamental issue in compressed sensing (CS). This paper introduces a new deterministic construction, referred to as Toeplitzed structurally chaotic matrix (TSCM), which possesses the advantages of both random and structural sensing matrices. We derive the matrix by first multiplying an orthonormal matrix with a chaotic-based Toeplitz matrix, and then subsampling the resultant matrix to obtain the structural one. Theoretically, we show that the entries of the TSCM are asymptotically normally distributed with that of arbitrary sparsifying matrices, yielding low mutual coherence that guarantees faithful recovery. Moreover, the proposed scheme is implementation friendly and hardware efficient, since its entries have almost no randomness and are easy to generate. Extensive numerical results via Matlab suggest that the TSCM outperforms the state-of-the-art matrix schemes and demonstrate its promising potentials.