Image Processing via Compressed Sensing Using L1 Minimum Norm Optimization

Compressed sensing is a revolutionary signal acquisition and processing technique that overcomes the limitations of traditional Nyquist sampling theorem, enabling high-quality signal reconstruction at significantly lower sampling rates than conventional methods. In image processing, compressed sensing technology combined with L1 minimum norm optimization can reconstruct high-quality images from a small number of measurement data. The core principle of L1 minimum norm optimization leverages signal sparsity. Natural images typically exhibit sparse representations in certain transform domains (such as wavelet transform, discrete cosine transform, etc.), meaning most transform coefficients are zero or near-zero. By solving an L1 norm minimization problem using algorithms like basis pursuit or iterative thresholding, these sparse coefficients can be efficiently recovered to reconstruct the original image. In the compressed sensing framework, image reconstruction can be modeled as a convex optimization problem. Unlike traditional L2 norm minimization, L1 norm optimization tends to produce sparse solutions that align well with the sparse characteristics of natural images. Implementation typically involves solving the optimization problem: minimize ||x||_1 subject to y = Ax, where A is the measurement matrix, y represents measured data, and x contains sparse coefficients. The integration of compressed sensing and L1 optimization offers numerous potential applications in image processing, including medical imaging, remote sensing image acquisition, and video compression - fields often challenged by high data acquisition costs or limited transmission bandwidth. This approach not only reduces hardware implementation complexity but also improves image acquisition efficiency through algorithms like orthogonal matching pursuit (OMP) or LASSO regularization that balance reconstruction quality and computational requirements.