# Research on an Optimal Path Planning Method Based on A* Algorithm for Multi-View Recognition
## 3 Optimal Path Planning for Multi-View Recognition Based on the A\({}^{*}\) Algorithm
### Optimal Path Planning Based on \(A^{*}\) Algorithm for Multi-View Recognition
\begin{split} A_{ii}=\left[\begin{array}{cccccccc}0&R^{1}_{[1,2]}&R^{2}_{[1, 3]}&\dots&R^{\tilde{q}}_{[1,\tilde{\mu}(1+\tilde{\mu})]}&0&\dots&0\\ 0&0&R^{1}_{[2,2]}&R^{2}_{[2,4]}&\dots&R^{\tilde{q}}_{[2,\tilde{\mu}(2+\tilde{ \mu})]}&0&\dots\\ 0&0&0&R^{1}_{[3,4]}&R^{2}_{[3,5]}&\dots&0&\dots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&0&R^{1}_{[(M_{\tilde{\mu}}-\tilde{\mu}),(M_{\tilde{\mu}}-\tilde{\mu}+ 1)]}&\dots&R^{\tilde{q}-1}_{[\tilde{\mu}(M_{\tilde{\mu}}-\tilde{\mu}),\tilde{ \mu}M_{\tilde{\mu}}]}&\tilde{\mu}(M_{\tilde{\mu}}-\tilde{\mu}),\tilde{\mu}M_{ \tilde{\mu}}]\\ 0&0&0&0&R^{1}_{[(M_{\tilde{\mu}}-\tilde{\mu}+1),\tilde{\mu}(M_{\tilde{\mu} }-\tilde{\mu}+2)]}&\dots&R^{\tilde{q}-1}_{[\tilde{\mu}(M_{\tilde{\mu}}-\tilde{ \mu}+1),\tilde{\mu}M_{\tilde{\mu}}]}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&0&0&0&R^{1}_{[\tilde{\mu}(M_{\tilde{\mu}}-\tilde{\mu}+1),\tilde{\mu} M_{\tilde{\mu}}]}\\ 0&0&0&0&0&0&0\end{array}\right]\end{split}  \tag{21}\]  
\[ \begin{split} A_{ii=0}=\left[\begin{array}{cccccccc}R^{2}_{[1, 2]-\tilde{\mu}]}&0&\dots&0&R^{\tilde{q}}_{[1,\tilde{\mu}(1+\tilde{\mu})]}&0& \dots&0\\ 0&R^{2}_{[2,2]+\tilde{\mu}]}&0&\dots&0&R^{\tilde{q}}_{[2,\tilde{\mu}(2+\tilde{ \mu})]}&0&\dots&0\\ 0&0&R^{2}_{[2,2]+\tilde{\mu}]}&0&\dots&0&0&R^{\tilde{q}}_{[\tilde{\mu}(M_{ \tilde{\mu}}-\tilde{\mu}+1),\tilde{\mu}M_{\tilde{\mu}}]}&0&\dots\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots& \vdots&\vdots\\ 0&0&0&0&0&0&0&0&0&0\\ \end{array}\right]\end{split}  \tag{22}\]  
\[ \begin{split} A_{ii=0}=\left[\begin{array}{cccccccc}R^{2}_{[1, 2]-\tilde{\mu}]}&0&\dots&0&R^{\tilde{q}}_{[1,\tilde{\mu}(1+\tilde{\mu})]}&0& \dots&0\\ 0&R^{2}_{[2,2]+\tilde{\mu}]}&0&\dots&0&R^{\tilde{q}}_{[2,\tilde{\mu}(2+\tilde{ \mu})]}&0&\dots&0\\ 0&0&R^{2}_{[2,2]+\tilde{\mu}]}&0&\dots&0&R^{\tilde{q}}_{[2,\tilde{\mu}(2+ \tilde{\mu})]}&0&\dots&0\\ 0&0&0&R^{2}_{[2,2]+\tilde{\mu}]}&0&\dots&0&R^{\tilde{q}}_{[2,\tilde{\mu}(2+ \tilde{\mu})]}&0&\dots&0\\ 0&0&0&0&0&0&0&0&0&0\\ \end{array}\right]\end{split}  \tag{23}\]  
Given the starting imaging point, the multi-view target viewpoint optimization path planning in barrier-free environment is equivalent to the directed graph path search problem with a given starting point. The number of nodes to be searched is \(k-1\), and the maximum path corresponding to the sum of edge weights is the search path. In order to facilitate the graph search, the problem of maximizing the sum of edge weights is transformed into the problem of minimizing the sum of search path weights by finding the reciprocal and normalized to (0,1):  
\[R_{e}(\Phi_{ij},\Phi_{(i+d_{0})(j+d_{\lambda})})=\frac{\frac{1}{\vec{R}(\Phi_{ ij},\Phi_{(i+d_{0})(j+d_{\lambda})})}-\vec{R_{\text{emin}}}}{R_{\text{emax}}- \vec{R_{\text{emin}}}}+\varepsilon \tag{24}\]  
where \(\vec{R_{\text{emax}}}\vec{R_{\text{emin}}}\) are the maximum and minimum values of the reciprocal of edge weights, respectively. \(\varepsilon\) is the parameter to ensure the continuity of directed edges. The transformed matrix \(\mathbf{R_{e}}\) becomes a new recognition performance matrix. In addition to recognition performance, it also ensures the shortest end running distance to improve operation efficiency. When the end enters the set observation radius range, the fixed radius does not move, and the path planning is carried out on the semicircular sphere with this radius as the radius. Similarly, viewpoint intervals are set as the vertexes of the directed graph, and the distances between two viewpoints are defined as the edge weights of the directed graph. The adjacency matrix \(\mathbf{D}_{R}\) is constructed, which can be expressed as:\[\mathbf{D}_{R}=\left[