Image Compressed Sensing with Orthogonal Matching Pursuit (OMP) Implementation

Image compressed sensing is a technique that leverages signal sparsity for efficient sampling, breaking through the limitations of traditional Nyquist sampling theorem. The core principle involves reconstructing original signals from a small number of measurements, making it particularly suitable for image processing applications.

During implementation, the selection of sparse basis is crucial. Commonly used sparse bases include Fast Fourier Transform (FFT) and Discrete Wavelet Transform (DWT): FFT achieves compression by constructing orthogonal transformation matrices to convert images from spatial domain to frequency domain, utilizing sparsity in frequency domain coefficients. In code implementation, this typically involves applying fft2() for 2D images and thresholding small coefficients. DWT performs multi-scale decomposition of images, preserving low-frequency components while measuring high-frequency coefficients to extract sparse representations. Practical implementation uses wavelet families like 'db4' through functions such as wavedec2() for decomposition thresholding.

The reconstruction phase employs the Orthogonal Matching Pursuit (OMP) algorithm. OMP is a greedy algorithm that iteratively selects atoms most correlated with the residual to gradually approximate the original signal. Key implementation steps include: initializing residual, computing correlations with dictionary atoms, updating support set, and solving least squares problem. Advantages include high computational efficiency and ease of implementation, making it suitable for large-scale image data processing.

By combining FFT/DWT sparse representations with OMP reconstruction, image compressed sensing significantly reduces data volume while maintaining quality, providing efficient solutions for image transmission and storage systems.