## 4 Results and Discussion
### Impact of Network Structure on Results  
To prove that the convolution neural network with multiple filters of different sizes is better, this paper compared the VGG-16, VGG-19, ResNet-18, ResNet-34, and AlexNet convolution networks commonly used in the field of image recognition with the convolution neural network in this paper using four indicators, namely, cross-entropy, validation set accuracy, time complexity, and space complexity.  
1. Time complexity  
Time complexity, that is, the number of operations of the model, can be measured by FLOPs [22], that is, floating point operations. Time complexity determines the training/prediction time of the model. If the complexity is too high, model training and prediction will consume a large amount of time. We can neither quickly verify ideas and improve models nor can we quickly predict. The time complexity formula for a single convolution layer is as follows:  
\[Time\sim O\Big{(}M^{2}\cdot K^{2}\cdot C_{in}\cdot C_{out}\Big{)} \tag{3}\]  
where \(M\) is the side length of the output characteristic graph of each convolution kernel; \(K\) is the side length of each convolution kernel; \(C_{in}\) is the number of channels of each convolution kernel, that is, the number of input channels, that is, the number of output channels of the upper layer; \(C_{out}\) is the number of convolution cores of the convolution layer, that is, the number of output channels;  
It can be seen that the time complexity of each convolution kernel is completely determined by the area of the output feature map \(M^{2}\), the area of the convolution kernel \(K^{2}\), the number of input channels \(C_{in}\), and the number of output channels \(\text{Cout}\). The output feature size itself is determined by the input matrix size \(X\), convolution kernel size \(K\), padding, and stride, which are shown as follows:  
\[M=\frac{(X-K+2\times Padding)}{Stride}+1 \tag{4}\]  
The overall time complexity of the convolutional neural network is the sum of the time complexity of all convolutional layers. The calculation formula is as follows:  
Figure 10: Recognition results of convolutional neural network.  
\[Time\sim O(\sum_{l=1}^{D}M_{l}^{2}\cdot K_{l}^{2}\cdot C_{l-1}\cdot C_{l}) \tag{5}\]  
where \(D\) is the number of convolution kernels possessed by the neural network, that is, the depth of the network; \(l\) is the \(l\)-th convolution layer of the neural network; \(C_{l}\) is the number of output channels of the \(l\)-th convolution layer of the neural network, \(C_{out}\), that is, the number of convolution cores of this layer; for the convolution layer, the number of input channels \(C_{in}\) is the number of output channels of the (\(l\)-1)-th convolution layer.  
2. Space complexity  
The number of parameters of the model determines the space complexity. Due to the limitation of the dimension curse, the more parameters of the model, the more data required for training the model. However, the data set in real life is usually not too large, which will lead to an easier overfitting of model training. The space complexity is calculated as follows:  
\[Space\sim O(\sum_{l=1}^{D}K_{l}^{2}\cdot C_{l-1}\cdot C_{l}+\sum_{l=1}^{D}M^{2 }\cdot C_{l}) \tag{6}\]  
The total parameters are only related to the size \(K\), channel number \(C\), and layer number \(D\) of the convolution kernel, and are independent of the size of the input data.  
Through the calculation of cross-entropy, model accuracy, network time complexity, and space complexity, the results of the six models are shown in Table 3. It can be seen from Figure 11 that the multi-scale perceptual convolutional neural network established in this paper has a high model accuracy. From Figure 12, it can be concluded that the convolutional neural network established in this paper is superior to other convolutional neural network models in terms of the time complexity and space complexity model.