# Research on an Optimal Path Planning Method Based on A* Algorithm for Multi-View Recognition
## 2 Related Works
### Feature Extraction Based on BM2 DKPCANet Model  
The BM2 DKPCANet model for feature extraction of the input sequence images is divided into three stages. As shown in Figure 1, the output features are binary hashed and clustered after two-layer 2 DKPCA convolutional filtering, followed by histogram features output. The process of 2 DKPCA convolutional filtering for the first filtering layer is described as follows. Suppose that there are S multi-view image sequences \(\mathbf{I}_{i}\) (i = 1, 2,..., S) \(\in\mathds{R}^{m\times n}\) as samples for training. S local feature matrices \(\mathbf{T}_{i}\) sized \(k^{2}\times mn\) are obtained after \(k\times k\) windows for block sliding sampling. Then, the column direction kernel matrix dimension is \(Smn\times Smn\) when performing centralized learning training. To reduce the computation, the average column vector is computed instead of the original column vector to obtain a nonlinear mapping sample \(\mathbf{\overline{T}}_{i}\), which is mapped to the kernel feature space as follow:  
\[\mathbf{\Phi}_{1}=[\Phi(\mathbf{\overline{T}}_{1}),\Phi(\mathbf{\overline{T}}_{2}),\dots, \Phi(\mathbf{\overline{T}}_{S})] \tag{1}\]  
Figure 1: BM2 DKPCANet model. The kernel matrix can be expressed as:  
\[K_{1}=\boldsymbol{\Phi_{1}}^{T}\boldsymbol{\Phi_{1}}=[k(\overline{T}_{s}, \overline{T}_{s})]=[\left\langle\Phi(\overline{T}_{s}),\Phi(\overline{T}_{s}) \right\rangle] \tag{2}\]  
The dimensionality is reduced to \(S\times S\). The kernel matrix is centralized such that  
\[\overline{K}_{1}=K_{1}-\frac{1}{S}K_{1}I-\frac{1}{S}IK_{1}+\frac{1}{S^{2}}IK_{ 1}I \tag{3}\]  
where \(\boldsymbol{1}\) is the s-order matrix with all 1 components. The eigenvectors corresponding to the first \(L_{1}\) largest eigenvalues are taken as 2 DKPCA filter banks \(\boldsymbol{W_{1}}^{1}=[w_{1},w_{2},\ldots,w_{L1}]\), which are convolved with the training sample \(I_{i}\) after the zero-filled boundary and then output the first-layer filtering:  
\[\boldsymbol{I}_{i}^{l}=\boldsymbol{I}_{i}*\boldsymbol{W_{l}^{1}},i=1,2, \ldots,S;l=1,2,\ldots,L1 \tag{4}\]  
where \(*\) is two-dimensional convolution. It is also used as input for the second stage of convolutional filtering to further explore the effective classification features. The same steps as described previously are used to obtain the second stage filter set convolution kernel \(\boldsymbol{W_{\ell}^{2}}=[w_{1},w_{2},\ldots,w_{L2}]\), The second layer filtering output is obtained by convolving the training samples \(\boldsymbol{I}_{i}^{l}\) with the zero-padding boundary. \[ \begin{array}{l}O_{i}^{l}=I_{i}^{l}*W_{\ell}^{2}=I_{i}*W_{1}^{1}*W_{\ell}^{2 },\\ s.t.i=1,2,\ldots,S;l=1,2,\ldots,L_{1};\ell=1,2,\ldots,L_{2}\end{array}  \tag{5}\]  
In order to reduce the data redundancy caused in the process of multi-view image acquisition and reduce the computation time and storage space to solve problems such as large data storage and long-time operation of multiple views, the binary multi-view clustering algorithm [45] is used to optimize the binary encoding and cluster of multiple views jointly. The total optimization function is written as:  
\[ \begin{array}{l}\min F(U^{m^{\prime}},B^{\prime},C,G,\alpha)\\ =\sum_{m^{\prime}=1}^{M}\left(\alpha^{m^{\prime}}\right)^{\prime}(\left\|B^{ \prime}-U^{m^{\prime}}\Phi(O^{m^{\prime}})\right\|_{F}^{2}+\beta\left\|U^{m^{ \prime}}\right\|_{F}^{2}\\ -\gamma/\
u^{\prime}tr((U^{m^{\prime}}\Phi(O^{m^{\prime}}))(U^{m^{\prime}} \Phi(O^{m^{\prime}}))^{T})+\lambda\left\|B^{\prime}-CG\right\|_{F}^{2}\\ s.t.C^{T}1=0,\sum_{m^{\prime}}\alpha^{m^{\prime}}=1,\alpha^{m^{\prime}}>0,B^{ \prime}\in\{-1,1\}^{q^{\prime}\times r^{\prime}},\\ C\in\{-1,1\}^{q^{\prime}\times c},G\in\{0,1\}^{c\times r^{\prime}},\sum_{j}g _{ji}=1.\end{array}  \tag{6}\]  
where \(\alpha^{m^{\prime}}\) is the weight of the \(m^{\prime}\) th perspective of some fruit box, \(m^{\prime}=1,...,\) M. Different perspectives have different weights. r > 1 is the scalar controlling the weight.