# Dynamic Programming Ring for Point Target Detection
## 3 MFF-DPR-Based Point Target Detection
### Mff-Dpr
\begin{array}{l}v_{\mathbf{p}}^{(t)}=\text{argmax}_{\|\mathbf{p}\|_{\leq}\leq \eta_{\text{max}}}\left\{X_{\mathbf{p}_{1}}^{(t_{1})}+X_{\mathbf{p}_{2}}^{(t_{2})} \right\}\\ l_{\mathbf{p}}^{(t)}=X_{\mathbf{p}}^{(t)}+\frac{1}{2}\text{max}_{\|\mathbf{p}\|_{\geq}\leq \eta_{\text{max}}}\left\{X_{\mathbf{p}_{1}}^{(t_{1})}+X_{\mathbf{p}_{2}}^{(t_{2})} \right\}\end{array} \right., \tag{6}\]

where \(\mathbf{p}_{1}\) and \(\mathbf{p}_{2}\) denote the pixel coordinates at the previous two frames, namely, at times \(t_{1}=t-1\) and \(t_{2}=t-2\), respectively; \(\mathbf{p},\mathbf{p}_{1}\), and \(\mathbf{p}_{2}\) satisfy the straight-line constraint. If \(t=+(N+1)\), let \(t=-(N-1)\), and the first two states of the reverse MFF-DP are initialized in the following way:

\[\left\{ \begin{array}{l}v_{\mathbf{\Omega}}^{(-N)}=-v_{\mathbf{\Omega}}^{(+N)},v_{ \mathbf{\Omega}}^{(-N-1)}=-v_{\mathbf{\Omega}}^{(+(N-1))}\\ I_{\mathbf{\Omega}}^{(-N)}=I_{\mathbf{\Omega}}^{(+N)},I_{\mathbf{\Omega}}^{(-(N-1))}=I_{ \mathbf{\Omega}}^{(+(N-1))}\end{array} \right.. \tag{7}\]

If \(t=0\), let \(t=+2\), the first two states of the sequential MFF-DP are defined in the following way:

\[\left\{ \begin{array}{l}v_{\mathbf{\Omega}}^{(+1)}=-v_{\mathbf{\Omega}}^{(-1)},v_{ \mathbf{\Omega}}^{(+2)}=-v_{\mathbf{\Omega}}^{(-2)}\\ I_{\mathbf{\Omega}}^{(+1)}=I_{\mathbf{\Omega}}^{(-1)},I_{\mathbf{\Omega}}^{(+2)}=I_{\mathbf{ \Omega}}^{(-2)}\end{array} \right.. \tag{8}\]

Otherwise, the pixel state \(S_{\mathbf{\Omega}}^{(t)}\) is updated in the way of iteration using Equations (2)-(5). The iteration counter increased by one, i.e., \(k=k+1\); if \(k\leq 2N,t=t+1\), the algorithm returns to the Step 1 of pixel state updating; otherwise, it proceeds to the Step 2 of MFF-DPR merit function derivation. **Step 2:** MFF-DPR merit function derivation. The merit function of the MFF-DPR \(I_{\mathbf{\Omega}}^{(st)}\) can be obtained by averaging merit functions of the sequential and reverse MFF-DPs at the same time, which can be expressed as follows:

\[I_{\mathbf{\Omega}}^{(st)}=\frac{1}{2}\left(I_{\mathbf{\Omega}}^{(+t)}+I_{\mathbf{\Omega}} ^{(-t)}\right),\;1\leq t\leq N. \tag{9}\]

Figure 3: Sketchy diagram of the DPR; \(S_{\mathbf{\Omega}}^{(+t)}\) and \(S_{\mathbf{\Omega}}^{(-t)}\) denote the pixel state estimation results of the sequential and reverse DPs at time \(t\), respectively. ### Multi-Target Detection

After the energy accumulation of point targets based on the MFF-DPR, the point targets can be detected by correlating the extreme points of the merit function of the MFF-DPR \(I_{\mathbf{\Omega}}^{(st)}\). However, when there are multiple targets in the field of view, trajectories of different point targets can intersect, causing the merit functions of multiple point targets to interfere with each other, so a wrong correlation of target identifications can easily occur. To address this problem, this paper uses the target coordinates extracted in multiple iterations to detect the target trajectories one by one, and reduces the target identification error through trajectory regularization, as shown in Figure 4.

The steps of the multi-target detection algorithm based on the MFF-DPR are given in Algorithm 2. The same approach can be applied to other multi-target detection algorithms based on the DP/DPR. ```
Input: Set the number of iterations to be equal to the target number, i.e., \(L=N_{\text{tar}}\); the sequence length is \(N\), and the iteration counter is set to \(l=1\). Output: The multi-target trajectories \(\left\{\mathbf{p}_{i}^{(l)}\middle|1\leq i\leq N_{\text{tar}},1\leq t\leq N\right\}\). Step 1: Energy accumulation. Run Algorithm 1 for the merit functions of the MFF-DPR of the \(l\)th iteration \(\left\{I_{\mathbf{\Omega}}^{(l,st)}\middle|1\leq t\leq N\right\}\). Step 2: Merit function maximum value coordinates extraction.