Compressed Sensing Algorithm with Hadamard Measurement Matrix and OMP Reconstruction

In compressed sensing algorithms, we employ the Hadamard matrix as the measurement matrix and adopt the Orthogonal Matching Pursuit (OMP) algorithm for signal reconstruction. The implementation involves generating Hadamard matrices of varying dimensions through recursive construction or built-in functions like hadamard() in MATLAB, while OMP iteratively selects the most correlated atoms from the measurement matrix to approximate sparse signals. Experimental tests were conducted with different dimensions of the observation matrix, and the relative error of measurements was calculated using norm-based comparisons between original and reconstructed signals. Results indicate a correlational relationship between the dimension of the observation matrix and the measurement relative error. Specifically, as the dimension of the observation matrix increases, the measurement relative error correspondingly rises. This relationship provides valuable insights for further research on optimizing compressed sensing algorithms, particularly in selecting appropriate matrix dimensions to balance accuracy and computational efficiency. Key implementation considerations include controlling sparsity levels, adjusting iteration thresholds in OMP, and validating results with metrics like relative error formula: ‖x_original - x_reconstructed‖₂ / ‖x_original‖₂.