A two-step iterative framework for signal and image deblurring using G-I-Nonexpansive Mappings

This study introduces G-I- nonexpansive mapping by combining I- nonexpansive mapping with a directed graph. It also establishes convergence results for a two-step Ishikawa-type iteration. Numerical experiments were conducted on benchmark image deblurring problems, in which images were degraded by motion blur and additive Gaussian noise. The proposed method achieves competitive restoration performance, with peak signal-to-noise ratio values of up to 24.51 dB. It outperforms classical approaches such as Wiener filtering, Lucy-Richardson and the Fast Iterative Shrinkage-Thresholding Algorithm while remaining comparable to Total variation (TV)-based methods. The method reliably enhances signals in 1D, achieving a peak signal-to-noise ratio of 29.73 dB and high structural similarity index measure values. These results suggest that the framework is an effective tool for restoring signals and images degraded by blur and noise.