Compressed Sensing Reconstruction Algorithms Collection

Compressed sensing is a signal processing technique that breaks through traditional sampling theorem constraints, enabling efficient acquisition and reconstruction by leveraging signal sparsity. Its core lies in reconstruction algorithms that can recover original sparse signals with high probability from limited observational data. Here are seven representative reconstruction algorithms with their characteristics:

Matching Pursuit (MP) Uses a greedy iterative strategy, selecting the atom most correlated with the residual at each step. Simple to compute but may converge to suboptimal solutions. Ideal for real-time applications. Implementation tip: The core function typically involves computing inner products between residuals and dictionary atoms, with complexity O(mn) per iteration.

Orthogonal Matching Pursuit (OMP) An improved version of MP that orthogonalizes selected atoms to enhance convergence. Performs reliably with moderate sparsity levels and serves as a benchmark algorithm in compressed sensing. Algorithm note: Requires QR decomposition or least-squares solution at each iteration, increasing computational cost but improving accuracy.

Subspace Pursuit (SP) Balances accuracy and efficiency by dynamically adjusting candidate set size. Reduces iteration count compared to OMP, suitable for high-dimensional signal processing. Code consideration: Maintains a fixed-size support set throughout iterations, with backtracking mechanism for optimal atom selection.

Iterative Hard Thresholding (IHT) A non-greedy algorithm based on gradient descent, applying hard thresholding at each iteration to maintain sparsity. Computationally efficient but requires strict Restricted Isometry Property (RIP) conditions for measurement matrices. Key operation: Involves gradient descent step followed by thresholding operator, often implemented as x_{k+1} = H_k(x_k + Φ^T(y-Φx_k)).

Compressive Sampling Matching Pursuit (CoSaMP) Incorporates backtracking mechanism by retaining multiple potential atoms and reassessing them each iteration. Significantly outperforms OMP in reconstruction accuracy, especially for highly sparse signals. Implementation feature: Uses pruning strategy to maintain sparsity constraint while considering larger candidate sets than OMP.

Generalized Belief Propagation (GBP) A message-passing algorithm derived from probabilistic graphical models, particularly effective for structured sparse signals. Achieves global optimization through information exchange between nodes, though computationally intensive. Algorithm insight: Operates on factor graphs with message updates between variable and factor nodes, suitable for Bayesian compressive sensing implementations.

Iteratively Reweighted Least Squares (IRLS) Transforms L0 norm approximation into weighted L2 problems through dynamic weight adjustment. Demonstrates strong robustness in noisy environments, commonly used in error-sensitive applications like medical imaging. Computational approach: Solves sequence of weighted least-squares problems with weights updated based on current solution, effectively approximating lp-norms with p→0.

These algorithms complement each other in computational complexity, convergence speed, and application scenarios. Practical selection should consider signal characteristics (sparsity level, noise conditions) and hardware constraints (real-time requirements). Current research trends focus on adaptive reconstruction methods integrating deep learning techniques.