MATLAB Code for Compressed Sensing Image Processing

Compressed Sensing (CS) is a signal processing technique that leverages signal sparsity to enable efficient sampling, widely applied in image processing, medical imaging, and communications. This article introduces several common CS reconstruction algorithms and their core concepts, typically implemented in MATLAB.

### 1. Orthogonal Matching Pursuit (OMP) OMP is a classic greedy algorithm that reconstructs sparse signals by iteratively selecting atoms (basis functions) most correlated with the residual. In image processing, OMP is often used with DCT or wavelet-based sparse representations. Key steps include: initializing the residual as observed data, iteratively selecting optimal bases, and updating coefficients via least squares. MATLAB implementation typically involves orthogonal projection using QR decomposition for efficient residue updates.

### 2. Compressive Sampling Matching Pursuit (CoSaMP) CoSaMP improves upon OMP's stability by selecting multiple atoms per iteration and retaining the most significant sparse components. It excels in high-dimensional signal reconstruction, particularly for image recovery. The algorithm progressively approximates the true signal through preselection, pruning, and least-squares optimization. Code implementation requires maintaining a candidate set with dynamic size adjustment using thresholding operations.

### 3. Iterative Hard Thresholding (IHT) IHT enforces sparsity through hard thresholding operations, suitable for L0-norm constrained problems. The core approach combines gradient descent with thresholding, preserving the largest K coefficients each iteration. IHT offers high computational efficiency for large-scale image reconstruction. MATLAB code typically uses matrix-vector multiplication with proximal operator implementation for thresholding.

### 4. Basis Pursuit (BP) BP reformulates reconstruction as an L1-norm optimization problem, achieving sparse solutions through convex relaxation. Commonly solved via linear programming (e.g., interior-point methods), it demonstrates strong noise robustness despite computational complexity. MATLAB implementations often utilize the `l1-magic` toolbox with CVX optimization package integration.

### 5. Subspace Pursuit (SP) SP is a CoSaMP variant that enhances convergence speed through dynamic candidate set adjustment. The algorithm maintains a fixed-size support set during iterations, making it suitable for moderately sparse image signals. Code implementation involves efficient set operations and orthogonal projections similar to OMP.

### 6. Generalized Belief Propagation (GBP) GBP employs probabilistic graphical models, using message passing to approximate sparse Bayesian solutions. It performs well in distributed compressed sensing but has higher implementation complexity. MATLAB code typically requires factor graph construction and iterative message updates between nodes.

### Application Scenarios and Selection Guidelines Fast reconstruction: Prioritize OMP or IHT algorithms. High-precision requirements: BP or CoSaMP are more suitable. Non-convex optimization: Consider SP or GBP for improved stability.

These algorithms' MATLAB implementations generally rely on linear algebra tools (e.g., SVD, QR decomposition) and optimization toolboxes. Practical applications require customization with measurement matrices (e.g., random Gaussian matrices) and sparse transformations (e.g., wavelets). Key functions include sparse matrix operations, optimization solvers, and transform-specific toolboxes like wavelet packages.