Fractional steepest decent optimization method: application to image restoration problem

The steepest decent optimization (SDO) is one of the widely used simple and efficientoptimization techniques which works based on the fact of moving in the oppositedirection of the steepest ascent to reach a minimum. Although the original SDO findssome acceptable solutions for the problem, it suffers from low speed and convergenceto local minima. After the introduction of fractional calculus many useful applicationsin science and engineering fields have been realized via non-integer-order derivativesand integrals. In this paper, a novel optimization method is proposed based on thefractional calculus and SDO algorithm. We use the memorization property of thefractional derivatives to provide a history of the past events of SDO procedure to helpSDO to escape from local minima traps. An adaptive accelerator is also introduced tospeed up the convergence rate of the algorithm. The proposed fractional steepestdecent optimization (FSDO) is applied to the image restoration problem. In this case,the inverse of the corruption function is estimated using a linear kernel whose elementsare being optimized using the introduced FSDO. Some comparative examples areprovided to show the superiority of the proposed FSDO in finding the optimumrestoration filters of noisy images.