## 3 The Proposed Method  
### Multi-Branch Training and Parameter Reconstruction  
Generally speaking (regardless of gradient vanishing or explosion), the deeper or wider a neural network is the more feature information it learns. In other words, under the same depth, wider networks learn more target features, such as GoogLeNet [22]; under the same width, deeper networks also improve the feature analysis, such as VGG16 [23] and ResNet [24]. Compared with VGG, multi-branch networks often have better regression and classification effects. However, the structure of the chain-convolutional neural network similar to VGG is simple and compact, and chain networks are easier to be distributed. Comprehensively considering the advantages of the multi-branch structure and the lightweight goal, the novel method of multi-branch training and parameter reconstruction inference was designed to improve the application efficiency of lightweight neural networks, similar to VGG. The novelty is summarized as follows: (1) Multi-branch training learns more feature information and reduces over-fitting due to enhanced feature maps. (2) Parameter reconstruction preserves the original structure after training. (3) It achieves the inference effects of a multi-branch network, compared with the original network. The total weight parameters do not increase while inferring, although the local structure becomes wider when training.  
Figure 1(a) shows the structural changes when training. One group of convolutions is appended with several different groups such as two extra groups of convolutions with 3 \(\times\) 3 and 1 \(\times\) 1 size, respectively. It makes the network learn more feature information than the original group. Figure 1(b) shows the process of parameter reconstruction, which remains as the structure of the original network after training. The new process of converting several trained groups into one group of convolution layers for inference is described as follows.  
Assuming \(\mathbf{W_{1}}\)\(\in\)\(\mathbb{R}^{\text{C1}\times\text{C2}\times 3\times 3}\) to denote the 3 \(\times\) 3 convolution layer with \(C_{1}\) inputs and \(C_{2}\) output channels, \(\mathbf{b_{i}}\)\(\in\)\(\mathbb{R}^{\text{C1}\times\text{C2}}\) denotes the offsets. Then, the output feature map \(\mathbf{Y_{i}}\) is:  
\[Y_{i}=W_{i}\otimes X+b_{i}, \tag{2}\]  
\[Y=\sum\
olimits_{i}Y_{i}=\sum\
olimits_{i}(W_{i}\otimes X+b_{i})=\sum \
olimits_{i}(W_{i})\otimes X+\sum\
olimits_{i}(b_{i})=\mathbf{W}\otimes\mathbf{X}+ \mathbf{b}, \tag{3}\]  
Figure 1: Remote sensing images with different landscapes used in the experiment.  
where \(\otimes\) is the convolution operator. \(\mathbf{Y}\) is the output of the structure of parallel convolutions. \(\mathbf{W}\) is the integrated weight parameters. \(\mathbf{b}\) is the integrated offset. The description of the gradient backward propagation is:  
\[\frac{dY}{dX}=\sum\frac{\partial Y_{i}}{\partial X}=\sum\frac{\partial(W_{i}X+b _{i})}{\partial X}=\sum\frac{\partial(W_{i}X)}{\partial X}+\sum\frac{\partial b _{i}}{\partial X}=\sum W_{i} \tag{4}\]  
Note that the number, size, and moving strip of convolution kernels must be the same. For lightweight networks, a kernel of a 3 \(\times\) 3 size is recommended. Small convolution kernels could achieve the effects of large size convolution kernels, and the amount of parameters is lower than a larger convolution kernel. For example, a 5 \(\times\) 5 size convolution operation is equivalent to two 3 \(\times\) 3 size convolution operations. In addition, small kernels expand perceiving areas [25].