Performance bounds for track-before-detect target detection

The use of Track-Before-Detect (TBD) has been shown to be effective for enhanced target detection in certain cases but performance has been elusive in others. The TBD concept enhances detection by integrating multiple frames of data collected over a period of several seconds. Since the integration period is so long, the target's potential motion must be compensated for. TBD hypothesizes different target trajectories to account for target motion. The problem in analyzing or quantifying TBD behavior is the statistical correlation or dependence among the hypotheses. In this paper we present a theoretical framework for analyzing the TBD concept by utilizing the theory of Simultaneous Statistical Inference. We address the problem by adapting the method of simultaneous confidence intervals developed by Henry Scheffe. This method uses the fact that the confidence ellipsoid of a Gaussian random vector may be constructed as an infinite intersection of linear hyperplanes. In this paper we show that a single TBD hypothesis can be represented as a hyperplane in the detection space. We declare a detection whenever the observation falls on one side of the hyperplane (an ay from the origin). Then,using the Scheffe construction we can approximate the distribution for the TBD detection statistic. Finally, we establish performance bounds and quantify the relationship between the number of hypotheses and detection performance.