A Class of Quaternion Kalman Filters

The existing Kalman filters for quaternion-valued signals do not operate fully in the quaternion domain, and are combined with the real Kalman filter to enable the tracking in 3-D spaces. Using the recently introduced HR-calculus, we develop the fully quaternion-valued Kalman filter (QKF) and quaternion-extended Kalman filter (QEKF), allowing for the tracking of 3-D and 4-D signals directly in the quaternion domain. To consider the second-order noncircularity of signals, we employ the recently developed augmented quaternion statistics to derive the widely linear QKF (WL-QKF) and widely linear QEKF (WL-QEKF). To reduce computational requirements of the widely linear algorithms, their efficient implementation are proposed and it is shown that the quaternion widely linear model can be simplified when processing 3-D data, further reducing the computational requirements. Simulations using both synthetic and real-world circular and noncircular signals illustrate the advantages offered by widely linear over strictly linear quaternion Kalman filters.