MATLAB Implementation of Weighted L1 Algorithm for Compressed Sensing

The Weighted L1 algorithm is a widely used optimization method in compressed sensing signal processing. Its core concept involves introducing weights to emphasize or de-emphasize the importance of different signal components, thereby achieving more effective sparse representation and reconstruction of signals. Implementing the Weighted L1 algorithm in MATLAB typically involves these key steps: Problem Formulation: Transform the signal reconstruction problem into an optimization framework where the objective function consists of a data fidelity term and a weighted L1 regularization term. The weighted regularization applies varying penalty strengths to different signal coefficients based on their positions, better adapting to the signal's structural characteristics. In MATLAB, this can be implemented using function handles to define the objective function with weighted norms. Weight Selection: Weight design forms the algorithm's core, commonly employing methods based on prior signal information or dynamically adjusting weights according to coefficient magnitudes estimated during iteration. For instance, larger weights suppress insignificant coefficients while smaller weights preserve critical information. MATLAB implementation often uses adaptive weight updates through iterative reweighting schemes, where weights are inversely proportional to current coefficient estimates. Optimization Solution: MATLAB provides built-in optimization tools like `fmincon` or third-party toolboxes such as CVX and SPGL1 to solve weighted L1 optimization problems. These tools efficiently handle convex optimization and output reconstructed sparse signals. When using CVX, the weighted L1 norm can be directly expressed as `norm(W.*x,1)` where W is the weight vector and x represents the signal coefficients. Performance Verification: Validate reconstruction accuracy and computational efficiency through simulation experiments or real-data testing. Common evaluation metrics include Signal-to-Noise Ratio (SNR), reconstruction error, and convergence speed. MATLAB scripts can automate these tests using functions like `snr()` for quality assessment and `tic/toc` for timing comparisons. The Weighted L1 algorithm finds broad applications in compressed sensing domains including medical imaging, radar signal processing, and image denoising. Proper weight design significantly enhances reconstruction quality and algorithm robustness, with MATLAB providing an ideal platform for prototyping and testing these implementations through its comprehensive mathematical and visualization capabilities.