# Dynamic Programming Ring for Point Target Detection
## 1 Introduction  
As a very important technology in the field of radar or infrared systems, point target detection has been widely studied, but its development has still been limited by certain challenges. First, the point target features, such as texture and color, cannot be captured because of a far observation distance. Second, the amplitude of a point target may be less than that of its surrounding background. The point target detection algorithms can be roughly divided into detect-before-track (DBT) [1; 2; 3] and track-before-detect (TBD) [4; 5; 6; 7] algorithms, or single-frame and multi-frame target detection algorithms. When the signal-to-noise ratio (SNR) of a point target is low, the DBT algorithms can easily lose the target. In contrast, the TBD algorithms process a number of frames before making a decision on target existence. Therefore, the TBD algorithms are particularly useful when the SNR is low. Since [PERSON] et al. [8] first proposed the dynamic programming (DP) TBD (DP-TBD) algorithm, the DP-TBD has been the mainstream method in the field of dim point target detection because DP lowers the requirements on data storage and search range. The DP-TBD algorithms can be roughly divided into probability density accumulation-based algorithms and energy accumulation-based algorithms according to the merit function type [9; 10]. The merit functions of the probability density accumulation-based algorithms [11] require information on statistical characteristics of a signal and background noise. However, for non-cooperative targets, this prior knowledge is difficult to obtain. In contrast, the energy accumulation-based algorithms [12; 13] use the gray accumulation value of each stage of a target as a merit function, which simplifies the iteration steps. For this reason, the energy density accumulation-based algorithms have a wider application scope than the probability accumulation-based algorithms. However, both types of DP-TBD algorithms have the same deficiencies, which can be summarized as follows. First, a pseudo trajectory is formed when the merit function accumulates to the nearby, strong noise. At each stage of the DP-TBD algorithms, only one optimal trajectory is retained, while other trajectories are discarded; thus, the incorrect state transition may occur in each stage due to the presence of strong noise. Second, the target merit function diffuses in each stage of the probability density or energy accumulation, which could obscure the target position. However, diffusions of the merit function cannot be completely avoided because of inherited characteristics of the DP-TBD. To address the aforementioned problems, researchers have designed many improved algorithms for merit function. [PERSON] et al. [14] proposed a merit function with system memory coefficient to reduce the influence of noise. However, the system memory coefficient is dependent on the SNR, and the parameter choice is challenging to detect non-cooperative targets. In recent years, the DP-TBD algorithms based on state transition constraints, including trajectory constraints [15; 16; 17], amplitude constraints [18], penalty function constraints [19], and multi-level thresholds [20; 21], have been extensively studied. However, all of the above-mentioned improved algorithms represent first-order Markov chains. In a state-transition stage, these algorithms use only the merit function of the previous frame, which often results in a large error of pixel state estimation. Recently, the second-order DP-TBD has emerged. Unlike the first-order DP-TBD, the second-order DP-TBD uses the observation data of the previous two frames [22; 23] or the subsequent frame [24; 25]. However, for dim point targets, high detection accuracy is challenging to achieve when unreliable observation data are used directly for state transition decisions. This study focuses on improving the detection efficiency of dim point targets using the DP-TBD. First, a second-order DP, named the merit function filtering DP (MFF-DP), is proposed.