Closed-Form Distance Estimators under Kalman Filtering Framework with Application to Object Tracking

In this paper, the minimum mean square error (MMSE) estimation problem for calculation of distances between two signals via the Kalman filtering framework is considered. The developed algorithm includes two stages: the Kalman estimate of a state vector computed at the first stage is nonlinearly transformed at the second stage based on a distance function and the MMSE criterion. In general, the most challenging aspect of application of the distance estimator is calculation of the multivariate Gaussian integral. However, it can be successfully overcome for the specific metrics between two points in line, between point and line, between point and plane, and others. In these cases, the MMSE estimator is defined by an analytical closed-form expression. We derive the exact closed-form bilinear and quadratic MMSE estimators that can be effectively applied for calculation of an inner product, squared norm, and Euclidean distance. A novel low-complexity suboptimal estimator for special composite functions of linear, bilinear, and quadratic forms is proposed. Radar range-angle responses are described by the functions. The proposed estimators are validated through a series of experiments using real models and metrics. Experimental results show that the MMSE estimators outperform existing estimators that calculate distance and angle in nonoptimal manner.