MATLAB Code Implementation of Compressed Sensing

Compressed sensing is a signal acquisition technique that breaks through traditional sampling theorems by leveraging signal sparsity properties to efficiently reconstruct original signals through a small number of non-adaptive linear measurements. Implementing compressed sensing in MATLAB primarily involves three core components: sparse representation, measurement matrix design, and signal reconstruction.

First, sparse representation forms the foundation of compressed sensing. Many natural signals exhibit sparsity in specific transform domains (such as Fourier transform, wavelet transform, or discrete cosine transform). MATLAB's built-in transform functions (like dct or wavedec) can conveniently project signals into sparse domains using simple function calls like dct(x) for discrete cosine transforms or wavedec(x, level, 'wname') for wavelet decomposition.

Second, the design of measurement matrices is crucial. Commonly used measurement matrices include Gaussian random matrices, Bernoulli matrices, etc. In MATLAB, the randn function can generate random matrices following Gaussian distribution through syntax like M = randn(m,n) where m represents measurement dimensions and n denotes signal length. These matrices must satisfy the Restricted Isometry Property (RIP) condition to ensure effective preservation of signal information.

Finally, signal reconstruction algorithms constitute the core component. Common reconstruction algorithms include Basis Pursuit, Orthogonal Matching Pursuit (OMP), and iterative thresholding algorithms. MATLAB's Optimization Toolbox provides linear programming solvers (such as linprog) that can implement Basis Pursuit algorithms, while OMP algorithms can be implemented through custom iterative programs that select atoms most correlated with residuals in each iteration using operations like residual correlation computation and support set updating.

During implementation, attention must be paid to the incoherence between measurement matrices and sparse bases, as well as selecting appropriate reconstruction algorithms based on signal characteristics. The MATLAB implementation of compressed sensing demonstrates its application potential in fields like image compression and medical imaging through practical code examples involving sparse transforms, random matrix generation, and optimization-based reconstruction.