The Hermitian Solutions of the Reduced Biquaternion Matrix Equation and Its Applications in Color Image Restoration

ABSTRACT In view of the advantages of the solution of the reduced biquaternion matrix equation, which was originally employed to solve the color image deblurred problems, we study the minimal norm Hermitian solution, pure imaginary Hermitian solution and pure real Hermitian solution of the reduced biquaternion matrix equation by presenting a new real representation method and establish the necessary and sufficient conditions for the existence of the above solutions. It is demonstrated that our real representation method can also obtain the minimal norm least squares Hermitian solution, the minimal norm least squares pure imaginary Hermitian solution and the minimal norm least squares pure real Hermitian solution of the studied reduced biquaternion matrix equation when it is inconsistent. Finally, we provide some numerical examples to illustrate the feasibility and validity of our method in comparison with the complex representation method, especially in terms of computing error and CPU time. Experimental results demonstrate that the proposed method is feasible and much more effective for color image restoration than the complex representation method.