One-Dimensional Discrete Signal Reconstruction Using Orthogonal Matching Pursuit in Compressed Sensing

In the field of signal processing, compressed sensing technology is widely applied for signal reconstruction and compression. Orthogonal Matching Pursuit (OMP) serves as a fundamental compressed sensing signal reconstruction algorithm, operating through iterative selection of the measurement matrix column most correlated with the current residual. The algorithm procedurally updates the signal estimate by solving a least-squares problem at each iteration, ensuring orthogonality between the residual and selected atoms. This approach provides high reconstruction accuracy and robustness through greedy atom selection combined with orthogonal projection. This article details OMP's application in 1D discrete signal reconstruction, explaining its algorithmic principles and implementation steps with emphasis on key components: residual calculation, atom selection criteria using inner products, and orthogonal projection via pseudoinverse operations. The implementation typically involves initializing a residual vector equal to the measurement vector, iteratively building a support set through maximum-correlation selection, and projecting measurements onto the subspace spanned by selected atoms. Code implementation would require functions for matrix column indexing, least-squares solution (e.g., numpy.linalg.lstsq in Python), and stopping criteria based on sparsity level or residual threshold. This comprehensive explanation aims to provide reference value for researchers in related fields.