Basis Pursuit (BP) Algorithm for Compressed Sensing

The Basis Pursuit (BP) algorithm represents a classical signal reconstruction method in compressed sensing, designed to recover sparse signals from a small number of linear measurements. Its core principle transforms the signal reconstruction problem into a convex optimization formulation, seeking the sparsest solution by minimizing the L1-norm of the signal vector. In code implementation, this translates to solving min||x||₁ subject to Ax=b, where A is the measurement matrix and b represents the observed data.

Algorithmically, BP solves a constrained optimization problem to find the sparsest solution consistent with measurement data. The mathematical rigor of this approach ensures exact reconstruction of original signals when the measurement matrix satisfies specific conditions like the Restricted Isometry Property (RIP). Compared to matching pursuit algorithms, BP typically achieves higher reconstruction accuracy at the cost of increased computational complexity. Implementation often requires linear programming solvers such as interior-point methods or specialized optimization toolboxes like CVX in MATLAB.

Practical applications of BP involve solving linear programming problems through various optimization tools. Modern optimization advances enable BP to handle medium-scale problems, while higher-dimensional signals may necessitate more efficient variants like Basis Pursuit Denoising (BPDN) or hardware acceleration through GPUs. Key implementation considerations include memory management for large matrices and convergence criteria for iterative solvers.

A critical parameter in BP algorithm is the regularization coefficient (often denoted as λ in BPDN formulations: min||x||₁ + λ||Ax-b||₂²), which balances data fidelity and sparsity promotion. Parameter selection significantly impacts reconstruction quality, typically determined through cross-validation or empirical methods like the L-curve analysis. Code implementations often include automated parameter tuning routines.

The algorithm finds extensive applications in image processing, medical imaging, and wireless communications, particularly excelling in measurement-constrained scenarios requiring high-precision reconstruction. With compressed sensing theory evolution, BP has spawned numerous variants including Iterative Reweighted L1 minimization and ADMM-based implementations, enhancing both performance and applicability across diverse domains.