An innovative orthogonal matrix based on nonlinear chaotic system for compressive sensing

Compressed sensing exploits the signal's sparsity by non-uniform sampling to achieve high-quality signal reconstruction at low sampling rates. This work aims to show the efficient performance of chaotic binary orthogonal matrices (CBOM) in compressed sensing. The nonlinear, high dimensional, irregular, and high complexity properties of chaotic systems can provide more diverse and efficient ways of sampling and reconstructing signals. The CBOM construction method proposed in this paper is divided into two steps, in the first step, the real-valued sequence of a one-dimensional chaotic map is binarised using the proposed Threshold-Matching Symbol Algorithm (TMSA) to obtain a chaotic binary sequence (CBS). The i.i.d properties of the CBS were proved using the Perron-Frobenius operator and the properties of the joint probability. The binarized CBS conditionally preserves the pseudo-random of chaotic sequences, as evidenced by the derivation of the well-distribution measure and the k-order correlation measure. In the second step, the binarised sequence CBS was split and orthogonalized to construct CBOM, which satisfies low storage and correlation. We prove that CBOM obeys the Restricted Isometric Condition (RIP). The orthogonalization of the matrix will further reduce matrix column correlation and improve the quality of the reconstruction. Numerical simulation results show that the proposed matrix has considerable sampling efficiency, comparable to Gaussian and partial Hadamard matrices, close to the theoretical limit. Meanwhile, the generation and reconstruction time of the proposed matrix is smaller than other matrices. Our framework covers partial one-dimensional chaotic maps, including Chebyshev, Tent, Logistic, and so on. We can easily apply this paradigm to various fields.