# Dynamic Programming Ring for Point Target Detection
## 3 MFF-DPR-Based Point Target Detection
### Mff-Dpr  
The MFF-DP shares the same drawback with the traditional DPs. As the merit function diffuses (Figure 2), the target will be drowned in the diffused bright spots, which complicates target detection. Therefore, suppressing the merit function diffusion of DP is of great significance to a point target detection algorithm. The study on the diffusion of the merit function of the DP-TBD has shown that the distribution of the merit function at the target position is similar to the shape of a comet after the sequential or reverse operation. The nuclei of two comets coincide, and the comet  
Figure 1: Sketchy diagram of the correlation between the observation data \(X_{\mathbf{p}}^{(t)}\) of the current frame (painted green) and the data of the other frames; the regions participating in the state transition decision process are represented in red. (**a**) First-order DP; (**b**) second-order DP with the correlation with the observation data of the previous two frames; (**c**) second-order DP with backtracking; (**d**) MFF-DP. A point target moves at a speed not greater than \(v_{\max}=1\ \text{pixel/frame}\). tails are symmetrical about the position of a point target, as shown in Figure 2a,b,d,e. In this paper, the above property is named as the time reversal symmetry of DP. If the merit functions accumulated in sequential and reverse orders are superimposed together (Figure 2c,f), the merit function of a point target is enhanced after superimposition due to the coincidence of comet nuclei, and other areas are well suppressed. Consequently, the target position becomes more pronounced and is consistent with the extreme value position of the merit function on the local image plane. Based on the time reversal symmetry of the DPs, this paper connects the sequential and reverse observation data head to tail to form a ring structure and employs the DP to update the pixel state on the ring structure, as shown in Figure 3, thus obtaining a DP ring (DPR) algorithm. Taking the MFF-DP as an example, this paper provides the steps of the corresponding DPR algorithm, i.e., the MFF-DPR algorithm, as shown in Algorithm 1. The same approach can be applied to other DPs. To distinguish the sequential MFF-DP, reverse MFF-DP and MFF-DPR, \"+\", \"-\", and \"+\" are used in time expressions, respectively. Figure 2: The distributions of different merit functions. (**a,d**) Distributions of merit functions with sequential accumulation at time \(t=50\); (**b,e**) distributions of merit functions with reverse accumulation at time \(t=50\); (**c,f**) distributions of averaged merit functions with sequential accumulation and reverse accumulation at time \(t=50\); (**a–c**) the calculation results of the algorithm proposed in Ref. [15]; (**d–f**) the calculation results of the MFF-DP algorithm; the size of simulation image is 128 \(\times\) 128 pixels, the sequence length is \(N=100\), the background noise is \(n_{p}^{(t)}\sim N(0,1)\), the point target is \(SNR=1.8\), the initial position is (30, 20), and motion velocity is (0.6, 0.8) pixel/frame. **Input:** Assume the upper limit of the point target motion speed is \(v_{\text{max}}\), the sequence length is \(N\), the iteration counter is \(k=1\), and the present time \(t=+3\). **Output:** The merit function of the MFF-DPR \(I_{\mathbf{\Omega}}^{(st)}\). **Step 1:** Pixel state updating. ```
Pixel state \(S_{\mathbf{\Omega}}^{(t)}=\left\{\left(l_{\mathbf{p}}^{(t)},v_{\mathbf{p}}^{(t)}\right)| \mathbf{p}\in\mathbf{\Omega}\right\}\) is updated weekly along the data ring, as shown in Figure 3.

If \(t=+3\) or \(+4\), due to the lack of prior information on the velocity, according to Equations (2), (3) and (5), the state estimation is performed in the following way:

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