Bayesian Compressed Sensing: Theory and Implementation

This article explores Bayesian Compressed Sensing (BCS), a sophisticated technique that enables efficient signal recovery during the sampling process with significant practical applications across various domains. By integrating Bayesian statistical methods with compressed sensing algorithms, BCS provides deeper insights into signal structures and facilitates more straightforward reconstruction approaches. The implementation typically involves probabilistic modeling of sparse signals using prior distributions, where key functions like Bayesian inference algorithms (e.g., Relevance Vector Machines) estimate sparse coefficients through evidence maximization.

BCS finds extensive applications in image processing (e.g., MRI reconstruction), speech recognition systems, and general signal processing workflows. The core algorithm operates by solving optimization problems through iterative Bayesian updates, often implemented using Gaussian priors to enforce sparsity constraints. Practical code implementations may include MATLAB functions for sparse signal recovery with automatic relevance determination (ARD) techniques, where hyperparameters are automatically tuned during the reconstruction process.

Understanding Bayesian Compressed Sensing is essential for developing efficient sampling systems, as it provides theoretical foundations for handling underdetermined linear systems while quantifying reconstruction uncertainty through posterior distributions. The technique's ability to incorporate prior knowledge makes it particularly valuable for real-world applications requiring robust signal recovery from limited measurements.