Euler-De Moivre formulas for fundamental matrices corresponding to commutative quaternions and their applications in color image restoration

Keywords: Commutative quaternions, fundamental matrices corresponding to commutative quaternions, Euler and De Moivre formula, RGB color space, image enhancement. This study consists of four 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 commutative quaternions and commutative matrices are given. In the third part, Euler and De Moivre formulas of real matrices of the 4x4 type corresponding to commutative quaternions are obtained. With the help of these formulas obtained, n. order roots and power of fundamental matrices are calculated. Finally, pseudo codes and classes in the MATLAB environment that calculate the roots and power of matrices corresponding to commutative quaternions have been developed. In the fourth part, the color image is sampled with the help of basic matrices corresponding to commutative quaternions and the distortion- enhancement processes of the blocks of this image with filters, which are basic matrices. In addition, the relative errors in the image reconstruction process are examined and the efficiency of the method is revealed.