Basis Pursuit Algorithm with DCT Basis in Compressed Sensing

This document presents a Basis Pursuit algorithm implementation utilizing the Discrete Cosine Transform (DCT) basis within the compressed sensing framework. The algorithm reconstructs original signals by solving an optimization problem that minimizes the L1-norm of the coefficient vector while maintaining measurement consistency. The implementation typically involves formulating the reconstruction as a linear programming problem, where the DCT basis serves as the sparsifying transform to represent signals efficiently. In compressed sensing applications, this approach enables recovery of incomplete signals by acquiring a small number of random measurements and then reconstructing the original signal using optimization techniques. The algorithm's core functionality includes measurement matrix generation, DCT basis computation, and solving the basis pursuit optimization problem using methods like linear programming or gradient descent. This methodology finds extensive applications in signal processing and image processing domains, particularly in scenarios requiring efficient data compression and reconstruction. Understanding the algorithm's mathematical foundation and implementation details is crucial for developing advanced signal processing systems.