QMGI algorithm for solving quaternion equation and its application in color image encryption

Abstract

In this present work, in order to solve the numerical solution of quaternion matrix equation \(EY=F\), the quaternion modified gradient-based algorithm (QMGI) is proposed by applying the real presentation of quaternion matrix. The proposed method can be applied to solve the quaternion solution, pure imaginary solution, and real solution of the studied equation \(EY=F\). If the studied equation is consistent, it is proved that the proposed algorithm converges to the exact solution for given any initial quaternion matrix under appropriate conditions. If the studied equation is not consistent, it is found that the QMGI algorithm converges to the least squares solution. And some numerical examples are examined to confirm the feasibility and efficiency of the proposed algorithms, which all indicate that the proposed QMGI algorithm is much more effective than QGI algorithm and QRGI algorithm in computational time and accuracy. Moreover, QMGl algorithm is applied to color image encryption and evaluated the encryption effectiveness from four aspects. All metrics are close to the ideal values. lt is demonstrated that the effectiveness of the encryption scheme and the accuracy of the obtained theory results in this paper.