All Reconstruction Algorithms for Compressed Sensing

Compressed sensing reconstruction algorithms represent core techniques in signal processing for recovering sparse signals from limited linear measurements. When signals exhibit sparsity in certain transform domains, these algorithms can break through the limitations of the Nyquist sampling theorem. Mainstream algorithms can be classified into three major categories:

Greedy Pursuit Algorithms These algorithms iteratively select atoms that best match the measurement vectors to gradually approximate the original signal. OMP (Orthogonal Matching Pursuit) serves as a typical example, selecting the most correlated atom in each iteration and orthogonalizing the residual. Implementation typically involves: - Correlation computation between residual and measurement matrix columns - Orthogonal projection using least squares Improved algorithms include: ROMP (Regularized OMP) enhances stability by selecting atoms in batches CoSaMP maintains multiple potential atoms per iteration using a pruning approach SP (Subspace Pursuit) employs backtracking mechanisms to optimize the support set

Thresholding Algorithms StoMP (Stagewise Matching Pursuit) screens atoms by setting fixed thresholds, making it suitable for noisy environments. The algorithm implementation involves threshold-based atom selection and residual updates. IHT (Iterative Hard Thresholding) directly retains the largest K coefficients in each iteration through hard thresholding operations, which can be efficiently implemented using sorting algorithms.

Adaptive Algorithms SAMP (Sparsity Adaptive Matching Pursuit) eliminates the need for prior knowledge of sparsity level K by setting stage thresholds to automatically adjust support set size. The implementation typically involves dynamic step size adjustment and stage transition conditions. Variants include improved versions that dynamically adjust step sizes in each phase using adaptive learning rates.

The selection of these algorithms requires balancing computational complexity, reconstruction accuracy, and noise robustness. Current research trends focus on integrating deep learning to optimize traditional algorithms, such as using neural networks to predict iteration parameters or atom selection strategies, where convolutional networks can learn optimal threshold values or recurrent networks can model iterative processes.