Compressive Sensing Reconstruction Algorithms Including OMP

The core of Compressive Sensing (CS) lies in reconstructing sparse signals with high probability from a limited number of linear measurements, where reconstruction algorithms play a critical role. OMP (Orthogonal Matching Pursuit), as a classic greedy algorithm, iteratively selects atoms most correlated with the residual to gradually approximate the signal, making it suitable for moderately sparse scenarios. In code implementation, OMP typically involves calculating correlations between the measurement matrix and residual, followed by least-squares estimation and residual update. Its variant SOMP (Simultaneous OMP) extends to multiple measurement vector problems, while ROMP (Regularized OMP) enhances stability by limiting the candidate set in each iteration through regularization constraints.

SAMP (Sparsity Adaptive Matching Pursuit) eliminates the need for prior knowledge of sparsity by adaptively adjusting the support set size, addressing the practical challenge of unknown sparsity levels. Implementation-wise, SAMP uses stage-wise thresholding to progressively refine the sparsity estimate. CoSaMP (Compressive Sampling Matching Pursuit) maintains multiple atoms per iteration combined with backtracking pruning, achieving a balance between accuracy and computational load through its multi-candidate selection and refinement process. GPSR (Gradient Projection for Sparse Reconstruction) operates within a convex optimization framework, solving L1-norm minimization problems using gradient projection techniques with iterative shrinkage-thresholding operations.

Transform domain techniques such as wavelet transforms and DCT (Discrete Cosine Transform) commonly serve as sparsifying bases. Wavelet transforms excel at capturing local transient features (like image edges) through multi-resolution analysis, while DCT demonstrates superior performance for global periodic signals (such as JPEG compression) via energy compaction properties. The synergy between these algorithms and transforms forms a complete CS pipeline from theory to application, addressing key requirements including computational efficiency, robustness, and scenario adaptation through proper algorithm-transform combinations.