# 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
### Objective Functions and Constraints  
The path planning of viewpoints only needs to consider the shortest movement route and high efficiency of viewpoint recognition. The objective function of the mathematical model that transforms the viewpoint seeking path planning problem into a multi-objective optimization problem at this point can be expressed as:  
\[\left\{ \begin{array}{l}\max\widehat{R}(\mathbf{X})\\ \min\widehat{D}(\mathrm{P}_{c},\mathrm{P}_{1})\end{array} \right. \tag{12}\]  
where the monotonically increasing function \(\widehat{R}(\mathbf{X})\) represents the recognition performance of the multi-view target recognition system. The larger the function value is, the higher the recognition accuracy is. The information contained in the images obtained from different viewpoints at different locations is different, and the function \(\widehat{R}(\mathbf{X})\) is also different. \(\widehat{D}(\mathrm{P}_{c},\mathrm{P}_{1})\) denotes the distance from the starting point \(\mathrm{P}_{1}\) of the observation path to the target point \(\mathrm{P}_{c}\). It is very difficult to find a specific analytic function to represent to characterize the recognition performance of the system theoretically. Based on the previous study of multi-view recognition of targets, this section selects the Softmax classifier for classification and outputs the category posterior probabilities to characterize the recognition performance. For more accurate identification, while the design of the classifier is further improved, combined with integrated learning theory, the single classifier is effectively combined, as shown in Figure 3 for the basic structure of BM2 DKPCANet network-integrated multi-view classification. \(x_{i}\) is the original image acquired by the camera, and the imaging viewpoint of the pair of factors is P(\(x_{i}\)) = (\(R\)p, \(\theta_{p}\), \(\lambda_{p}\)). The multi-viewpoint image is decomposed into image pairs and input to the two-viewpoint base classifier for classification and recognition, and then all the reco_gnition results are fused using an integrated learning method. Figure 3: Multi-view classifier. The structure of the two-view classifier network is shown in Figure 4. To better describe the recognition performance optimization function, the Softmax classifier is chosen to calculate the posterior probability to classify the image attributes. In order to reduce the planning complexity and the number of divided intervals, this paper sets that the subsequent planning is carried out along the sphere with R as the radius when the viewpoint enters within the allowed boundary region, i.e., R is between R\({}_{l}\) and R\({}_{u}\). According to the region division of the target observation model, the sphere is homogenized into M\({}_{lo}\cdot\)M\({}_{la}\) regions, and the set of view intervals can be expressed as:  
\[\left\{\left(\theta_{i},\lambda_{j}\right)\left|\theta_{i}=\left[\frac{2\pi \cdot(i-1)}{\text{M}_{la}},\frac{2\pi\cdot i}{\text{M}_{la}}\right],\lambda_{j }=\left[\frac{\pi\cdot(i-1)}{2\text{M}_{la}},\frac{\pi\cdot i}{2\text{M}_{la}} \right]\right\} \tag{13}\] \[s.t.i=1,2,\ldots,\text{M}_{la},j=1,2,\ldots,\text{M}_{lo}\]  
\(\Phi_{ij}\) is used to express the above perspective range. The set of perspective range can be further expressed as:  
\[\left\{\Phi_{ij}|\Phi_{ij}=\left(\theta_{i},\lambda_{j}\right)\right\} \tag{14}\]  
Each original image \(\mathbf{x}_{i}\) belongs to only one of the view intervals. The classifier classifies only the image pairs within the viewpoint interval, and the original data set is reorganized into a series of image pairs as:  
\[\left\{\left.\left(\mathbf{x}_{u},\mathbf{x}_{v}\right)\right|\mathbf{x}_{u}\in\mathbf{X}_{i},\mathbf{x}_{v}\in\mathbf{X}_{i+d},\text{P}(\mathbf{X}_{i})\in\Phi_{ij},\text{P}(\mathbf{X}_{ i+d})\in\Phi_{(i+d_{\theta})(j+d_{\lambda})}\right\} \tag{15}\]  
where \(d_{\theta}\) is perspective interval difference in the dimensional direction and \(d_{\lambda}\) is perspective interval difference in longitude.