Mathematical analysis and computational methods for Probabilistic Multi-Hypothesis Tracking (PMHT)

Abstract

This thesis is concerned with the mathematical analysis of Probabilistic Multi-Hypothesis Tracking (PMHT), which represents an approach how multi-target tracking (MTT) can be modeled mathematically and state estimates for tracked objects can be calculated. First of all, we examine the tracking model itself and its placement into the different standard MTT model approaches. We continue the work with the more practical aspects, namely the development of a totally automatic track management system that is based on PMHT, and investigate how the different tasks involved can effectively be realized especially for situations of ‘closely-spaced’ targets. The practical considerations are supplemented by a theoretical analysis of the Expectation-Maximization (EM) algorithm, the standard numerical optimization method employed to solve the state estimation problem of PMHT. We derive statements concerning the connection of EM to other well-known iterative schemes as well as its convergence rate and apply this to the PMHT model. It is shown that for ‘well-separated’ targets EM already reaches Newton-convergence. For other scenarios, alternatives and acceleration techniques are considered to improve convergence. A specially designed hybrid Newton-EM combination turns out to be robust and most efficient in cases of ‘closely-spaced’ targets. Furthermore, a second approach to accelerate the PMHT computations is investigated. This approach consists in the application of model reduction methods. We derive a reduced order model for PMHT based on Proper Orthogonal Decomposition (POD) together with corresponding stability, approximation error and convergence results. Numerical examples show that the established reduced order PMHT algorithm is even able to speed up the calculations for ‘well-separated’ targets without taking a noticeable tracking performance loss.