The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) is widely used to solve inverse problems arising from various image processing applications. Although FISTA may achieve \(O(1/k^{2})\) convergence rate in theory, it can be slow in actual implementation due to the fact that the algorithm employs only first-order information of the objective function. This motivates us to propose a Diagonal-Newton Fast Iterative Shrinkage-Thresholding Algorithm (DFISTA), by incorporating partial second-order information in the form of diagonal matrix into the standard FISTA. The incorporation of second-order information through diagonal matrix avoids the implementation difficulty of storing a dense matrix and hence rendering the algorithm suitable for large-scale problems. The diagonal components are obtained by approximating the spectrum of eigenvalues of the Hessian matrix via the least change updating technique under the log-determinant norm subject to the weak secant relation. Convergence properties of DFISTA are also established under standard assumptions. Four images are used to test the efficiency of DFISTA. The full-reference quality metrics including SSIM, PSNR, and RMSE and no-reference quality metrics including NIQE and BRISQUE of the recovered images are also calculated. Both quantitative and visual results indicate that DFISTA performs better than FISTA. In addition, DFISTA is tested on an industrial defect image and a medical image. Moreover, the computational time of DFISTA is shown to be comparable to that of FISTA, indicating that the computational cost of both algorithms is similar. The numerical results suggest that DFISTA can be used as an alternative algorithm for image restoration problems.
