Simulation Algorithms for Compressed Sensing

Compressed sensing is a signal processing technique that breaks through the Nyquist sampling theorem, enabling signal reconstruction from a small number of non-adaptive linear measurements by leveraging signal sparsity. The SOMP (Simultaneous Orthogonal Matching Pursuit) algorithm is a classical reconstruction method particularly suitable for Multiple Measurement Vector (MMV) problems. In code implementation, this typically involves matrix operations for correlation computation and least-squares optimization.

Core Concept SOMP employs a greedy iterative strategy to reconstruct sparse signals by progressively selecting atoms (basis functions in the dictionary) that exhibit the highest correlation with the residual. Unlike OMP for single measurement vectors, SOMP simultaneously processes multiple measurement vectors, enhancing reconstruction accuracy through joint sparsity across signals. Algorithmically, this requires maintaining a shared support set across all measurement vectors during iteration.

Implementation Workflow Initialization: Input measurement matrix, observed data, and sparsity level; initialize residual as observed data with an empty support set. Code implementation typically involves matrix initialization and parameter validation. Atom Selection: Compute correlations between current residual and dictionary atoms, select the atom with maximum correlation to add to the support set. This step can be optimized using vectorized operations for efficiency. Signal Estimation: Estimate sparse coefficients on the support set via least-squares method and update the reconstructed signal. Implementation requires solving a linear system using pseudo-inverse or QR decomposition. Residual Update: Subtract the reconstructed component from observed data to obtain new residual. This involves matrix subtraction and norm calculation for convergence checking. Iteration Termination: Stop when preset sparsity level is reached or residual becomes sufficiently small. Typically implemented with a while-loop structure containing convergence criteria.

Advantages and Limitations SOMP's advantages include relatively low computational complexity and ease of implementation, especially for signal groups with common support sets. However, its performance depends on signal sparsity and measurement matrix coherence, potentially degrading in high-noise or non-strictly sparse scenarios. Code optimization may involve thresholding techniques for robustness.

Application Extensions The algorithm finds widespread applications in image compression, medical imaging (e.g., MRI acceleration), and wireless communications. Future improvements may involve integrating deep learning for optimized atom selection strategies or adaptive sparsity adjustment, which could be implemented through neural network modules combined with traditional optimization loops.