MATLAB Implementation of Compressed Sensing Classical Algorithms

The Matching Pursuit (MP) algorithm is a classic sparse signal reconstruction method in the field of compressed sensing, suitable for recovering high-dimensional sparse signals from a small number of observations. This algorithm iteratively selects atoms that best match the residual to gradually approximate the original signal, offering advantages of computational simplicity and ease of implementation. Algorithm Core Concept MP algorithm belongs to the greedy iterative algorithm family. At each step, it selects the atom from a predefined dictionary that has the maximum correlation with the current residual, calculates its contribution coefficient, and updates the residual. This process repeats until stopping conditions are met (such as the residual being sufficiently small or reaching the maximum iteration count). Mathematically, it achieves sparse signal representation through projection decomposition. MATLAB Implementation Key Points Dictionary Construction: Typically uses overcomplete dictionaries (such as DCT, wavelet bases, or random Gaussian matrices), which need to be predefined or dynamically generated in the code. In MATLAB, this can be implemented using functions like dctmtx() for DCT or randn() for Gaussian matrices. Iteration Process: Initialize the residual as the observed signal and the reconstructed signal as a zero vector. At each step, compute the inner products between the residual and all atoms, selecting the atom corresponding to the maximum inner product. Update the reconstructed signal and residual by subtracting the projection component of the selected atom from the residual. Stopping Criteria: Can set relative error thresholds or fixed iteration counts to prevent infinite loops. MATLAB implementation typically uses while loops with conditional break statements. Application Extensions Although simple, the MP algorithm suffers from over-matching issues (where subsequent iterations may correct earlier errors). Improved versions like Orthogonal Matching Pursuit (OMP) enhance stability through global orthogonalization. These algorithms find wide applications in image reconstruction, wireless communications, and are particularly suitable for scenarios requiring high real-time performance. Implementation Considerations Practical implementation requires attention to dictionary redundancy and atom normalization, as these factors can affect convergence speed. For large-scale data, matrix operations can be combined to optimize inner product calculation efficiency. In MATLAB, vectorized operations and precomputation of dictionary correlations can significantly improve performance. Key MATLAB functions involved: norm() for normalization, dot products for correlation calculations, and efficient matrix operations for large-scale implementations. The algorithm structure typically involves initializing variables, main iteration loop with atom selection and residual update, and convergence checking.