Brezinski Inverse and Geometric Product-Based Steffensen's Methods for Image Reverse Filtering

Abstract

This work develops extensions of Steffensen's method to provide new tools for solving the semi-blind image reverse filtering problem. Two extensions are presented: a parametric Steffensen's method for accelerating the Mann iteration, and a family of 12 Steffensen's methods for vector variables. The development is based on Brezinski inverse and geometric product vector inverse. Variants of these methods are presented with adaptive parameter setting and first-order method acceleration. Implementation details, complexity, and convergence are discussed, and the proposed methods are shown to generalize existing algorithms. A comprehensive study of 108 variants of the vector Steffensen's methods is presented in the Supplementary Material. Representative results and comparison with current state-of-the-art methods demonstrate that the vector Steffensen's methods are efficient and effective tools in reversing the effects of commonly used filters in image processing