A Unified Analysis for Dynamic Programming Track-Before-Detect Algorithms: Error Convergence and Spatial Uncertainty

Abstract

The Dynamic Programming Track-Before-Detect (DP-TBD) class of algorithms is a core approach to the small low signal-to-noise ratio (SNR) target detection problem. These methods detect targets by recursively accumulating data through a sequence of iterative maximizations, a process that has traditionally limited their theoretical analysis. We propose a novel spatial analysis for the general DP-TBD class of algorithms where we derive a fundamental inverse relationship between detection uncertainty and location uncertainty using specific threshold constructions. Our analysis explicitly incorporates spatial distance from the target state into the probability bounds and allow this distance to vary as a function of iteration count, i.e. the number of processed frames. Integrating additional observations increases confidence in target existence while reducing certainty about the target's location. Our framework precisely details how each parameter affects performance and establishes the necessary conditions under which this analysis holds. Within this framework, we propose Normalized Path Integration (NPI), a DP-TBD algorithm that achieves broad applicability by tracking targets based on the similarity between observations as opposed to directly integrating the observations themselves. We experimentally validate this theory and compare different DP-TBD constructions on the Sequential Infrared Small Target Detection (SIRSTD) dataset: a real dataset consisting of small aerial infrared targets.