Orthogonal Matching Pursuit (OMP) Algorithm

This paper investigates compressed sensing technology, which aims to recover signals from a small number of measurements to reduce measurement costs and complexity. In practical applications, this often involves solving linear systems represented by equations of the form y=Ax, where y is the measurement vector and A is the sensing matrix. When both y and A are known, we can recover the sparse signal x. In compressed sensing research, solving this type of equation is crucial as it provides essential information for reconstructing the original signal, thereby improving both efficiency and accuracy of the recovery process.

The OMP algorithm implements this through an iterative greedy approach that selects the most correlated atom from the dictionary matrix A in each iteration. The key steps involve: computing correlations between the residual and dictionary atoms, updating the support set with the index of the maximum correlation, solving a least-squares problem on the selected support, and updating the residual by subtracting the reconstructed signal contribution. This process continues until meeting stopping criteria like reaching a predefined sparsity level or achieving a sufficiently small residual error.

Critical implementation considerations include efficient matrix operations for correlation calculations, proper handling of the least-squares solution using pseudo-inverse or QR decomposition, and careful residual updating to ensure orthogonal projections. The algorithm's performance depends on factors such as the restricted isometry property of matrix A and the actual sparsity of signal x.