Quaternion Filtering Based on Quaternion Involutions and its Application in Signal Processing

The quaternion gradient plays an important role in quaternion signal processing, and has undergone several modifications. Recently, three methods for obtaining the quaternion gradient have been proposed based on generalized HR calculus, the quaternion product, and quaternion involutions, respectively. The first method introduces the quaternion rotation, which is difficult to calculate and often relies on lookup tables; the second is cumbersome because it depends on all the real and imaginary parts of the variables and parameters; the third transforms the gradient from the quaternion domain to the real domain, and uses real derivatives and the real chain rule. In this paper, we generalize the quaternion involutions method to calculate the quaternion gradient of the quaternion matrix function. Several examples are presented in which the proposed gradient is applied to the adaptive filter, Kalman filter, and kernel filters to solve the associated optimization problems.