Kernel FISTA and Application to Image Deblurring

Abstract Image deblurring is a challenging inverse problem due to its ill-posed nature, leading to various strategies for enhancing stability. The Rudin-Osher-Fatemi (ROF) model is a seminal framework recognized for its effectiveness in image denoising and deblurring, inspiring numerous variational techniques. Despite existing approaches focusing on solving analysis-based, synthesis-based, and balanced models, achieving a more accurate solution to the ROF model remains an open question. The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) is a widely applied numerical algorithm known for its efficiency in large-scale optimization. However, FISTA’s effectiveness in solving the ROF model is constrained because it is primarily designed for synthesis models when the sparsifying transform is orthonormal. To overcome this limitation, we propose a novel Kernel FISTA (K-FISTA), integrated with a structured Pseudoinverse Image Formulation (PIF) model. In practice, our approach outperforms classical methods like ADMM and sPADMM by achieving lower final energy values, offering a more accurate numerical approximation to the common minimizer while maintaining fast convergence. Extensive experiments validate the effectiveness of the K-FISTA method in image deblurring, and we provide theoretical guarantees for its convergence.