Gradient Projection Method for Compressed Sensing Signal Reconstruction - Implementation Code

This documentation presents a comprehensive implementation of gradient projection-based signal reconstruction for compressed sensing systems. The core algorithm addresses the fundamental challenge of reconstructing signals from significantly fewer measurements than required by traditional Nyquist sampling theory. The implementation utilizes a mathematical framework that exploits signal sparsity in transform domains, where the gradient projection method iteratively refines the signal estimate by projecting gradient updates onto constraint sets. Key implementation aspects include: - Measurement matrix design using random Gaussian or Bernoulli matrices - Sparsity domain transformation through wavelet or Fourier basis - Gradient descent optimization with projection steps enforcing sparsity constraints - Convergence criteria based on relative error thresholds or maximum iterations Notably, this compressed sensing approach demonstrates particular advantages in scenarios where high-rate sampling is impractical, such as medical imaging, seismic data processing, and wireless communications. The code structure typically involves: 1. Initialization of sensing matrix and sparse transformation 2. Measurement simulation through linear projection 3. Iterative reconstruction using gradient updates with sparsity projection 4. Performance evaluation via reconstruction error metrics While alternative methods like Fourier and wavelet transforms remain valuable tools, gradient projection for compressed sensing offers distinct benefits in measurement efficiency. The implementation includes configurable parameters for sparsity levels, measurement ratios, and optimization thresholds to accommodate various application requirements. This implementation provides researchers and engineers with a practical foundation for exploring compressed sensing applications, with modular code design allowing easy integration of different sparsity bases and optimization enhancements.