The Iterated Extended Kalman Filter (IEKF) could be applied either globally or locally.
The global IEKF is applied to the whole observed data. Given a set of n observations . The initial state estimate is with covariance matrix . After applying the EKF to the set , we get an estimate with covariance matrix (the superscript, 1 here, denotes the number of iteration). Before performing the next iteration, we must back propagate to time , denoted by . At iteration 2, is used as the initial state estimate, but the original initial covariance matrix is again used as the initial covariance matrix at this iteration. This is because if we use the new covariance matrix, it would mean we have two identical sets of measurements. Due to the requirement of the back propagation of the state estimate, the application of the global IEKF is very limited. Maybe it is interesting only when the state does not evolve over time . In that case, no back propagation is required. In the problem of estimating 3D motion between two frames, the EKF is applied spatially, i.e., it is applied to a number of matches. The 3D motion (the state) does not change from one match to another, thus the global IEKF can be applied.
The local IEKF [8, 15] is applied to a single sample data by redefining the nominal trajectory and relinearizing the measurement equation. It is capable of providing better performance than the basic EKF, especially in the case of significant nonlinearity in the measurement function . This is because when is generated after measurement incorporation, this value can serve as a better state estimate than for evaluating and in the measurement update relations. Then the state estimate after measurement incorporation could be recomputed, iteratively if desired. Thus, in IEKF, the measurement update relations are replaced by setting (here, the superscript denotes again the number of iteration) and doing iteration on
for iteration number and then setting . The iteration could be stopped when consecutive values and differ by less than a preselected threshold. The covariance matrix is then updated based on .