MATLAB Code Implementation of Compressed Sensing

Compressed sensing is a signal processing technique that breaks through the traditional Nyquist sampling theorem. By leveraging signal sparsity, it enables efficient data acquisition and reconstruction at sampling rates significantly lower than the Nyquist rate. MATLAB provides powerful numerical computation tools and extensive algorithm support for implementing compressed sensing. ### Core Implementation Approach Sparse Representation: Select appropriate sparse bases (such as DCT or wavelet transforms) to convert the original signal into a transform domain where it exhibits sparse characteristics. In MATLAB, this can be implemented using functions like `dct()` for discrete cosine transform or `wavedec()` for wavelet decomposition. Observation Matrix Design: Construct random measurement matrices (e.g., Gaussian random matrices, Bernoulli matrices) that satisfy the Restricted Isometry Property (RIP) to perform linear projection on sparse signals. MATLAB code typically uses `randn()` to generate Gaussian random matrices or `rand()` with thresholding for Bernoulli matrices. Optimization Reconstruction Algorithms: Recover the original signal from limited observations through L1-norm minimization (such as basis pursuit) or greedy algorithms (like OMP, CoSaMP). MATLAB implementations often utilize optimization toolboxes or custom functions like `l1eq_pd()` from the l1-magic package for basis pursuit, or implement OMP using iterative least-squares solutions with `lsqr()`. ### MATLAB Application Scenarios Image Compression and Recovery (e.g., single-pixel camera simulation) Medical Imaging (accelerated MRI sampling) Wireless Sensor Networks (reducing node energy consumption) The advantage of this technology lies in significantly reducing hardware sampling requirements, though it requires balancing reconstruction accuracy with computational complexity. MATLAB's matrix operations and optimization toolboxes (such as `l1-magic`) can rapidly validate algorithm performance through efficient matrix computations and convex optimization solvers.