Construction of a Deterministic Binary Chaotic Compressed Sensing Measurement Matrix Utilizing Householder Orthogonalization and Pseudorandom Number Transformation

The development of measurement matrices remains a pivotal focus within the domain of compressed sensing theory. This paper introduces an innovative methodology for the construction of a deterministic binary measurement matrix, harnessing the properties of chaotic sequences. This approach streamlines the process and only requires a fixed number of initial variables for matrix construction. The construction method of the measurement matrix is divided into two distinct stages. Initially, the matrix elements are extracted through gapless sampling of chaotic sequences. Recognizing the inherent correlation among chaotic sequence elements due to gapless sampling, this paper introduces a novel nonlinear binary transformation. This innovative approach effectively mitigates the correlation and concurrently addresses storage limitations. Thereafter, the entries are integrated into a Toeplitz matrix, leveraging its structured row and columnar properties to augment storage efficiency. In subsequent steps, the approach integrates the Householder transformation to generate an orthogonal basis, thereby enhancing the matrix’s performance. Additionally, an XOR operation is employed to optimize the distribution of entries across each column vector. The research substantiates that the resultant matrix exhibits low correlation, is facile in generation, and amenable to reconstruction. Comprehensive experimental data corroborate the efficacy of the constructed measurement matrix in processing both one-dimensional and two-dimensional signals. The matrix demonstrates a strong potential to meet the Restricted Isometry Property (RIP), thereby underscoring its substantial advantages across a myriad of considerations.