FISTA Algorithm

In this section, we can provide a more detailed introduction to the FISTA algorithm and its applications in compressed sensing. FISTA, or Fast Iterative Shrinkage-Thresholding Algorithm, is an optimization algorithm widely used in image restoration and signal processing. By combining least squares minimization with L1-norm regularization, it achieves high-quality recovery while preserving sparsity in images or signals. The algorithm implements an accelerated version of the iterative soft-thresholding approach, where each iteration involves two key computational steps: a gradient descent update followed by a soft-thresholding operation that promotes sparsity through proximal mapping. Compressed sensing is an emerging signal acquisition and reconstruction technique that enables signal recovery using significantly fewer samples than traditional Nyquist-rate sampling. When applied to compressed sensing problems, FISTA effectively enhances both the accuracy and efficiency of signal reconstruction through its fast convergence properties. Consequently, the FISTA algorithm holds significant practical value in the field of compressed sensing, particularly in implementations requiring sparse signal recovery with convex optimization constraints.