Least-squares solutions of the elliptic quaternion matrix equation AX=B and their applications in colour image restoration

Keywords: Elliptic numbers, eliptik quaternions, elliptic quaternion matrices, least squares solution method, color image restoration. This study consists of five parts. The first part is an introduction devoted to the literature knowledge. In the second part of this study the fundamental definitions and theorems related to the elliptic numbers and their matrices are given. In the third part, the Frobenius norm of elliptic matrices is defined and its related properties are given. Later, the least-squares solution of the elliptic matix equation AX=B is investigated in the set of elliptic matrices, pure imaginary elliptic matrices and pure real elliptic matrices. Moreover, some illustrative examples have been resolved to prove the accuracy of our results and to distinguish them from the current results. In the fourth part, after some algebraic properties of elliptic quaternions and their matrices, which are the generalized form of elliptic numbers, are given, the Frobenius norm of elliptic quaternion matrices is defined and related properties are given. The least-squares solution of the elliptic quaternion matrix equation AX=B is studied in the set of elliptic quaternion matrices, pure imaginary elliptic quaternion matrices and pure real elliptic quaternion matrices with the help of elliptic matrix representations corresponding to elliptic quaternion matrices. Moreover, some illustrative examples have been resolved to prove the accuracy of our results and to distinguish them from the current results. In the last part, a new image restoration model using the algebra of elliptic quaternions, which we call ELSI distortion model, is obtained. In this context, a c/d/c model based EQLSM quaternionic least-squares restoration filter is applied to a color image sampled using elliptic quaternion algebra and passed through an elliptic valued PSF filter. Then, the improved image was transferred from digital to analog environment by selecting the most suitable space that minimizes the end-to-end MSD between the input image and the improved output image.