# Dynamic Programming Ring for Point Target Detection
## 3 MFF-DPR-Based Point Target Detection  
### Mrf-Dp  
According to the DP-based point target detection [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26], the merit function of the DP of a pixel with coordinates of \(\mathbf{p}\) at time \(t\) can be obtained iteratively as follow:  
\[I_{\mathbf{p}}^{(t)}=X_{\mathbf{p}}^{(t)}+\max_{\left\|\mathbf{v}\right\|_{2}\leq v_{\max }}\Big{\{}E_{\mathbf{p},\mathbf{v}}^{(t)}\Big{\}}, \tag{2}\]  
where \(\left\|\ \right\|_{2}\) represents the Euclidean norm, \(\mathbf{v}\) denotes the pixel transition velocity on the image plane, \(v_{\max}\) denotes the maximum speed of the point target, and \(E_{\mathbf{p},\mathbf{v}}^{(t)}\) represents the optimization function. The main difference between the traditional DPs lies in \(E_{\mathbf{p},\mathbf{v}}^{(t)}\). The optimization function of the first-order DPs [14; 15; 16; 17; 18; 19; 20; 21] can be rewritten as \(E_{\mathbf{p},\mathbf{v}}^{(t)}=I_{\mathbf{p}-\mathbf{v}}^{(t-1)}\), which means the state transition of the first-order DPs depends only on the merit function of the previous frame (Figure 1a), and the estimation error of pixel state will be large. For traditional second-order DPs [22; 23; 24; 25], the correlation with the observation data (Figure 1b,c) is introduced to \(E_{\mathbf{p},\mathbf{v}}^{(t)}\), which represents an improvement compared to the first-order DPs. However, for dim point targets, direct involvement of unreliable observation data in the state transition can affect the estimation of the pixel state. To address the aforementioned problems, this study sets the optimization function \(E_{\mathbf{p},\mathbf{v}}^{(t)}\) and depends on the merit functions of the previous two frames, as shown in Figure 1d. This avoids the direct involvement of observation data in the state transition decision as in the traditional second-order DPs, thus reducing the noise influence on the state estimation of pixels. The trajectory of a point target in the three-dimensional space formed by an image plane and a time axis is regarded as a continuous curve, and the geometric characteristics of the trajectory are equivalent to the dynamic characteristics of the point target. When the observation time is short enough, such as three consecutive frames, the trajectory of a point target can be approximated as a straight line in a three-dimensional space. In this study, the locally linearized target trajectory in a three-dimensional space is used as a dynamic model of a point target. Thus, under the local straight-line trajectory constraints, the optimization function \(E_{\mathbf{p},\mathbf{v}}^{(t)}\), depending on the merit functions of the previous two frames, can be defined as follows:  
\[E_{\mathbf{p},\mathbf{v}}^{(t)}=\frac{1}{2}w_{\mathbf{p},\mathbf{v}}^{(t)}\Big{(}I_{\mathbf{p}_{ 1}}^{(t_{1})}+I_{\mathbf{p}_{2}}^{(t_{2})}\Big{)}, \tag{3}\]where \(\frac{1}{2}\) represents the normalization coefficient of the merit function, and the velocity matching label can be expressed as:  
\[w_{\mathbf{p},\mathbf{p}}^{(t)}=\left\{ \begin{array}{ll}1,&\|\mathbf{p}-\mathbf{p}_{\mathbf{p}_{1}} ^{(t_{1})}\|_{\infty}\leq\frac{1}{2}\text{ and }\|\mathbf{p}-\mathbf{v}_{\mathbf{p}_{2}}^{(t_{2})}\|_{\infty}\leq \frac{1}{2}\\ 0,&\text{otherwise}\end{array} \right., \tag{4}\]  
where \(\|\ \|_{\infty}\) represents the Chebyshev norm; \(\frac{1}{2}\) indicates that the estimation error of each velocity component is not larger than the minimum speed resolution of the second-order DP, i.e., \(\frac{1}{2}\ \text{pixel/frame}\); \(\mathbf{p}_{1}\) 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 local straight-line constraint, namely, there exists a straight-line trajectory that passes through coordinates \(\mathbf{p},\mathbf{p}_{1}\), and \(\mathbf{p}_{2}\) at a velocity of \(\mathbf{v}\in\left\{\mathbf{v}\middle|\mathbf{2v}\in Z^{2}\right.\right\}\).