# RBF-Based Monocular Vision Navigation for Small Vehicles in Narrow Space below Maize Canopy
## 2 Materials and Methods
### Navigation Method
#### 2.2.4 Detection of the Crop Row Line  
Many stalks of corn, given a number of man-made or natural factors, are not in the corn row. Trajectory planning by the system is needed to navigate the vehicle. The optimal trajectory based on the global map has been widely studied in the field of robot navigation [30; 31; 32]. The coordinate points of all stalks are mapped in accordance with the world coordinate. Among the methods of agricultural machinery navigation, Hough transform is one of the most commonly used. The results obtained by the Hough transform have larger errors. The RBF, which behaves like a local approximation of the neural network and has many advantages, is used for path planning in the corn fields. It is not susceptible to the problems associated with non-fixed input because of its hidden behavior unit. A regularization method may well reflect the \"geometric\" features when approaching [33]. The regularization network topology is shown in Figure 8. The RBF network with a nonlinear separable hidden layer first converts the input space into a linearly-separable feature space (usually a high-dimensional space) and then the output layer of the linear division, thereby completing the classification. \(X=(x_{1},\,x_{2}\,\dots\,x_{i})\) is the input data. \(y=[y_{1},\,y_{2}\,...\,y_{i}]^{T}\) is calculated as the final output. Figure 7: (**a**) The original image; (**b**) the results of Otsu threshold analysis; (**c**) the results of a block filter; and (**d**) the results of extracting the ridge line. RBF uses a Gaussian radial basis function. The RBF learning algorithm is used to solve the center vector \(C_{t}\), the width parameter \(\sigma_{i}\), and the connection weights between the values of the hidden and output layers \(\omega_{i}\). \[G(x,c_{i})=\exp(-\frac{m_{1}}{d_{\max}^{2}}\ |\ |\ x-c_{i}\ |\ |^{2})i=1,\ 2\ \dots\ m_{1} \tag{4}\]  
where \(m_{1}\) is the number of centers, and \(d_{\max}\) is the maximum distance between the selected centers. As can be seen, all of the standard deviations are fixed to ensure that each RBF are not too sharp, nor too shallow:  
\[\sigma=\frac{d_{\max}}{\sqrt{2m_{1}}} \tag{5}\]  
The inputs of the hidden layer are the combinations of the input vector \(x=[x_{1},\ x_{2}\ \dots\ x_{n}]^{T}\). The network of the hidden layer to the output layer is linearly mapped:  
\[y=\sum_{i=1}^{m_{1}}w_{i}G(x,c_{i}) \tag{6}\]  
Finally, the availability of the Gaussian kernel network structure is given by:  
\[y=\sum_{i=1}^{k}w_{i}\text{exp}(-\ |\ |\ x-u_{i}\ |\ |^{2}/2\sigma_{i}^{2}) \tag{7}\]  
In this article, a term which constrains the complexity of the approximation function is added based on the standard error term:  
\[e=d_{k}-y \tag{8}\]  
where \(d_{k}\) is the distance between the selected centers for the sample of \(k\). The Gaussian basis function is local to the center vector, in the sense that:  
\[\lim_{|\ |\ x\ |\ |\rightarrow\infty}\rho\ |\ |\ x-u_{i}\ |\ |\ =0 \tag{9}\]  
In this study, \(n\) is the number of centers, and \(d\) is the maximum distance between the chosen centers. Thus:  
\[\sigma=\frac{d}{\sqrt{2n}} \tag{10}\]  
The smaller the value of \(d\), the smaller the width of RBF and, therefore, the more selective the base function is. In a regularization network, the number of hidden behavior units is the same as the  
Figure 8: Regularization network structure. sample, and the data center of the basis function is the sample itself. The extension function and the connection weights are the only parameters that were taken into account. Stalk coordinates correspond with the world coordinate as the input for online learning. With a gradient of training methods, the learning rate was 0.001, and the target error was 0.05. While the error is less than 0.05, the network outputs the result \(G(x,c_{i})\); otherwise, it adds a set to the sample. It can adapt well to the complex diversity of the vehicle environmental planning exercise with a \(\sigma\) value of 0.3.