Basis Pursuit Sparse Reconstruction Algorithm

This paper presents a MATLAB implementation of the Basis Pursuit sparse reconstruction algorithm. The algorithm consists of two main processes: signal sparsification and signal reconstruction. In the sparsification phase, the algorithm performs signal sampling and compression to transform the signal into a sparse representation format using techniques like compressive sensing. During the reconstruction phase, the original signal is progressively recovered through iterative algorithms such as L1-norm minimization, utilizing the sampled sparse representation and sparse basis matrices. The implementation details cover key MATLAB functions including signal preprocessing, dictionary creation using orthogonal matching pursuit principles, and optimization solvers for linear programming problems. The algorithm employs convex optimization techniques to solve the basis pursuit problem, typically formulated as min||x||₁ subject to Ax=b, where A represents the measurement matrix and b contains the observed samples. This paper provides comprehensive explanations of the algorithm's theoretical foundation and step-by-step implementation procedures, accompanied by practical MATLAB code examples. Key code components include sparse representation generation, measurement matrix design, and reconstruction error evaluation metrics. Through studying this material, readers can understand the fundamental principles of Basis Pursuit sparse reconstruction algorithms, master their implementation in MATLAB, and apply optimization techniques for practical applications requiring signal compression and recovery.