Abstract
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.
Highlights
Quaternions provide a compact model of mutual information between data channels, and have been used in fields such as traditional navigation [1], [2], Kalman filtering [3]–[5], neural networks [6], [7], spectral estimation [8], [9], communication [10], system control [11], motion tracking [5], [12], biomedical engineering [13], and adaptive filtering [7], [14]– [29]
In the derivation of quaternion signal processing algorithms, the mean square error (MSE) criterion is commonly used as the cost function
The real-valued cost function based on MSE for the quaternion linear filter in the discrete time domain i is given by JMSE (w) = |e(i)|2 = e(i)e∗(i) = e∗(i)e(i)
Summary
Quaternions provide a compact model of mutual information between data channels, and have been used in fields such as traditional navigation [1], [2], Kalman filtering [3]–[5], neural networks [6], [7], spectral estimation [8], [9], communication [10], system control [11], motion tracking [5], [12], biomedical engineering [13], and adaptive filtering [7], [14]– [29]. Cheong Took and Mandic [14] introduced the GHR calculus, and derived a systematic framework for calculating the derivatives of a quaternion matrix function with respect to quaternion variables [33]–[35] It has been pointed out [36], [37] that the full derivation of the gradient in [14] is incorrect, and the correct quaternion gradient is obtained using the product method in [36] and the quaternion involutions method in [37]. The product method [36] is simple but cumbersome, because it depends on all the real and imaginary parts of the variables and parameters, and it is difficult to calculate the derivatives of a quaternion matrix function.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
