Each time an outfielder tracks and catches a fly ball, he intuitively solves a problem of target tracking. That problem has stymied engineers, mathematicians, and computer scientists for years. Two great mathematicians, American, Norbert Wiener and Russian, Andrey Kolmogorov, first approached the problem during World War II. Rather than catching fly balls, Kolmogorov and Wiener were trying, in the days before computers, to develop mathematical algorithms that would help to track enemy aircraft by radar and automatically guide an anti-aircraft gun to shoot it down. The research started by Kolmogorov and Wiener has developed into a thriving area of applied mathematics known as Filtering Theory. Filtering, estimation of a signal or an image from noisy data, is the basic component of the data assimilation in target tracking.
Research in nonlinear filtering performed in CAMS placed USC among the world leaders in this important field. Recently, a group of CAMS scientists, S. Lototsky, R. Mikulevicius and B. Rozovskii, have found a complete solution of "The Last Wiener Problem", development of a Wiener type optimal nonlinear filter. This theoretical breakthrough led the CAMS group to invention of a new algorithm (S3) for tracking objects with possibly nonlinear dynamics. Optimal nonlinear tracking filters are much more efficient than the standard extended Kalman filter in many practically important situations (e.g. infrared search and track, radar warning receiver, noise jammed radar, etc.)
The commercial applications include air traffic control, human-computer interfaces based on motion-capture, advanced optical and magnetic registration systems for virtual reality simulation etc. All the above studies were funded by the Office of Naval Research and the Army Research Office.
Principal collaborators and partners: Signal Processing and Applied Mathematics Group, Space and Naval Warfare Systems Center (SPAWAR Systems Center), San Diego, Lockheed Martin Tactical Defense Systems, TRW.
Movies
The following movies demonstrate the performances of EKF and ONF in angle-only tracking:
Movie 1. Angle-Only Tracking: Target Trajectories and Angular Observations
In this example we consider the 2D angle-only tracking of a maneuvering target. The state equation is highly nonlinear and the target performs an evasive maneuver. The only available observation is a noisy time series of the target azimuth angle. The upper picture shows the true trajectory of the target ("pure" target’s dynamics – red line) and the perturbed trajectory (with noise – yellow line). The lower picture shows the angular observations (non-perturbed – pink; perturbed – light blue).
Movie 2: Angle-Only Tracking: EKF Versus ONF
This movie compares the two nonlinear filtering algorithms in the problem outlined above – the conventional extended Kalman filter and the spectral optimal nonlinear filter (ONF) developed in USC. Performance of EKF is shown in the left part and performance of ONF in the right part. The trajectories are similar to that in the Movie 1. The upper pictures show the real location of the target (solid lines) and the plots of the posterior densities. The lower ones show the confidence regions. It is seen that the optimal nonlinear filter "catches" the target and tracks it quite accurately while EKF completely "diverges" (misses the target and does not follow its track).