Matrix Forms of Gradient Descent Algorithms Applied to Restoration of Blurred Images

Abstract

Iterative gradient descent algorithms to solve algebraic linear equations are well known in the literature. These algorithms can be interpreted as dynamical closed loop control systems, where the step sizes are the control variables that can be optimally calculated by the Liapunov Control Functions (CLF) and Liapunov Optimizing Control (LOC) methods. In this paper matrix versions of gradient descent algorithms are deduced, including an unpublished matrix form of the Barzilai-Borwein algorithm. The step sizes (control variables) are also calculated by CLF/LOC. The main utility of these matrix algorithms is in image deblurring in the cases where the matrix that represents the blurring process is too large to be stored in the memory of the majority of conventional computational systems.