Compressed sensing measurement matrix construction method based on uniform chaotic sequence and matrix factorization

Abstract

Based on the limitations of traditional measurement matrix reconstruction performance in non-ideal environments, as well as the high complexity and slow speed of matrix construction, large initial variables, and memory space requirements, this study proposes a measurement matrix construction method based on chaotic sequences. This method utilizes chaotic sequences as the foundation for constructing the measurement matrix, leveraging their pseudo-randomness and determinism to achieve the construction of a deterministic matrix. The conventional approach to creating a measurement matrix from chaotic sequences has limitations in sampling intervals, resulting in data redundancy. To address this issue, this paper presents a deterministic compressed sensing measurement matrix construction method based on a uniform chaotic sequence. Initially, an irreversible function is employed to conduct binary conversion on the chaotic sequence without interval sampling. This nonlinear transformation reduces the complexity and storage challenges of the sequence while decreasing correlation. Subsequently, the binary sequence is stored in a Toeplitz matrix with structured characteristics, further diminishing storage costs. Moreover, the nonlinear transformation guarantees the linear independence of each column of the Toeplitz matrix, serving as a foundation for Singular Value Decomposition (SVD) processing. SVD is utilized to produce a high-dimensional orthogonal matrix from a low-dimensional Toeplitz matrix. In comparison to alternative orthogonalization methods like Gram-Schmidt orthogonalization and eigenvalue decomposition, SVD does not necessitate the matrix to be square. It operates at a faster speed and requires lower matrix dimensions under identical conditions. This creates an orthogonal measurement matrix that performs well. A significant number of simulation experiments demonstrate that the measurement matrix formulated in this paper exhibits excellent performance.