Iterative Recovery Algorithms in Compressed Sensing

Iterative recovery algorithms in compressed sensing represent a crucial class of methods for reconstructing sparse signals from a small number of linear measurements. These algorithms progressively approximate the original signal through iterative processes, overcoming the limitations of traditional sampling theorems. From an implementation perspective, these methods typically involve iterative loops that update signal estimates while maintaining sparsity constraints.

Matching Pursuit serves as the fundamental prototype for this category, reconstructing signals by sequentially selecting basis vectors that exhibit the highest correlation with the current residual. In code implementations, this involves calculating inner products between the residual and dictionary atoms, then updating the support set appropriately. CoSaMP (Compressive Sampling Matching Pursuit) represents an enhanced version of Matching Pursuit with superior recovery performance, featuring a more sophisticated selection mechanism during each iteration.

The distinctive characteristic of CoSaMP algorithm lies in selecting multiple candidate components during each iteration while maintaining a fixed number of non-zero coefficients. Its workflow comprises four critical steps implemented sequentially in practical code: candidate selection (identifying promising indices), support set union (merging current and previous supports), least-squares estimation (computing optimal coefficients via pseudo-inverse operations), and pruning (retaining only the most significant components). This structured approach ensures the algorithm fully utilizes measurement information while preserving sparsity constraints through systematic thresholding operations.

The advantage of these iterative recovery algorithms lies in their ability to handle underdetermined linear systems while maintaining robustness against measurement noise. They have found successful applications in MRI imaging, single-pixel cameras, radar systems, and other domains, serving as a vital bridge connecting compressed sensing theory with practical implementations. From a programming perspective, these algorithms typically require efficient linear algebra operations and careful parameter tuning for optimal performance.