A Unified Theory of Dynamic Programming Algorithms in Small Target Detection

Small target detection is inherently challenging due to the minimal size, lack of distinctive features, and the presence of complex backgrounds. Heavy noise further complicates the task by both obscuring and imitating the target appearance. Weak target signals require integrating target trajectories over multiple frames, an approach that can be computationally intensive. Dynamic programming offers an efficient solution by decomposing the problem into iterative maximization. This, however, has limited the analytical tools available for their study. In this paper, we present a robust framework for this class of algorithms and establish rigorous convergence results for error rates under mild assumptions. We depart from standard analysis by modeling error probabilities as a function of distance from the target, allowing us to construct a relationship between uncertainty in location and uncertainty in existence. From this framework, we introduce a novel algorithm, Normalized Path Integration (NPI), that utilizes the similarity between sequential observations, enabling target detection with unknown or time varying features.